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arXiv:1101.5747v2 [math.CO] 1 Feb 2011
Rainbow connections of graphs – A survey∗
Xueliang Li, Yuefang Sun
Center for Combinatorics and LPMC-TJKLC
Nankai University, Tianjin 300071, P.R. China
E-mails: lxl@nankai.edu.cn, syf@cfc.nankai.edu.cn
Abstract
The concept of rainbow connection was introduced by Chartrand et al. in 2008. It
is fairly interesting and recently quite a lot papers have been published about it. In this
survey we attempt to bring together most of the results and papers that dealt with it.
We begin with an introduction, and then try to organize the work into five categories,
including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow in-
dex, rainbow vertex-connection number, algorithms and computational complexity.
This survey also contains some conjectures, open problems or questions.
Keywords: rainbow path, (strong) rainbow connection number, rainbow k-connectivity,
k-rainbow index, rainbow vertex-connection number, algorithm, computational com-
plexity
AMS Subject Classification 2000: 05C15, 05C40
1 Introduction
1.1 Motivation and definitions
Connectivity is perhaps the most fundamental graph-theoretic subject, both in combi-
natorial sense and the algorithmic sense. There are many elegant and powerful results on
connectivity in graph theory. There are also many ways to strengthen the connectivity con-
cept, such as requiring hamiltonicity, k-connectivity, imposing bounds on the diameter, and
so on. An interesting way to strengthen the connectivity requirement, the rainbow connec-
tion, was introduced by Chartrand, Johns, McKeon and Zhang [12] in 2008, which is restated
as follows:
This new concept comes from the communication of information between agencies of
government. The Department of Homeland Security of USA was created in 2003 in response
∗Supported by NSFC.
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to the weaknesses discovered in the transfer of classified information after the September
11, 2001 terrorist attacks. Ericksen [25] made the following observation: An unanticipated
aftermath of those deadly attacks was the realization that law enforcement and intelligence
agencies couldn’t communicate with each other through their regular channels, from radio
systems to databases. The technologies utilized were separate entities and prohibited shared
access, meaning that there was no way for officers and agents to cross check information
between various organizations.
While the information needs to be protected since it relates to national security, there
must also be procedures that permit access between appropriate parties. This two-fold issue
can be addressed by assigning information transfer paths between agencies which may have
other agencies as intermediaries while requiring a large enough number of passwords and
firewalls that is prohibitive to intruders, yet small enough to manage (that is, enough so
that one or more paths between every pair of agencies have no password repeated). An
immediate question arises: What is the minimum number of passwords or firewalls needed
that allows one or more secure paths between every two agencies so that the passwords along
each path are distinct?
This situation can be modeled by graph-theoretic model. Let G be a nontrivial connected
graph on which an edge-coloring c : E(G) → {1,2,··· ,n}, n ∈ N, is defined, where adjacent
edges may be colored the same. A path is rainbow if no two edges of it are colored the same.
An edge-coloring graph G is rainbow connected if any two vertices are connected by a rainbow
path. An edge-coloring under which G is rainbow connected is called a rainbow coloring.
Clearly, if a graph is rainbow connected, it must be connected. Conversely, any connected
graph has a trivial edge-coloring that makes it rainbow connected; just color each edge with
a distinct color. Thus, we define the rainbow connection number of a connected graph
G, denoted by rc(G), as the smallest number of colors that are needed in order to make G
rainbow connected [12]. A rainbow coloring using rc(G) colors is called a minimum rainbow
coloring. So the question mentioned above can be modeled by means of computing the value
of rainbow connection number. By definition, if H is a connected spanning subgraph of G,
then rc(G) ≤ rc(H). For a basic introduction to the topic, we refer the readers to Chapter
11 in [16].
In addition to regarding as a natural combinatorial measure and its application for the
secure transfer of classified information between agencies, rainbow connection number can
also be motivated by its interesting interpretation in the area of networking [10]: Suppose
that G represents a network (e.g., a cellular network). We wish to route messages between
any two vertices in a pipeline, and require that each link on the route between the vertices
(namely, each edge on the path) is assigned a distinct channel (e.g. a distinct frequency).
Clearly, we want to minimize the number of distinct channels that we use in our network.
This number is precisely rc(G).
Let c be a rainbow coloring of a connected graph G. For any two vertices u and v of G,
a rainbow u − v geodesic in G is a rainbow u − v path of length d(u,v), where d(u,v) is
the distance between u and v in G. A graph G is strong rainbow connected if there exists a
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rainbow u−v geodesic for any two vertices u and v in G. In this case, the coloring c is called
a strong rainbow coloring of G. Similarly, we define the strong rainbow connection number
of a connected graph G, denoted src(G), as the smallest number of colors that are needed
in order to make G strong rainbow connected [12]. Note that this number is also called the
rainbow diameter number in [10]. A strong rainbow coloring of G using src(G) colors is
called a minimum strong rainbow coloring of G. Clearly, we have diam(G) ≤ rc(G) ≤
src(G) ≤ m, where diam(G) denotes the diameter of G and m is the size of G.
In a rainbow coloring, we only need to find one rainbow path connecting any two vertices.
So there is a natural generalizaiton: the number of rainbow paths between any two vertices
is at least an integer k with k ≥ 1 in some edge-coloring. A well-known theorem of Whitney
[55] shows that in every κ-connected graph G with κ ≥ 1, there are k internally disjoint u−v
paths connecting any two distinct vertices u and v for every integer k with 1 ≤ k ≤ κ. Similar
to rainbow coloring, we call an edge-coloring a rainbow k-coloring if there are at least k
internally disjoint u−v paths connecting any two distinct vertices u and v. Chartrand, Johns,
McKeon and Zhang [13] defined the rainbow k-connectivity rck(G) of G to be the minimum
integer j such that there exists a j-edge-coloring which is a rainbow k-coloring. A rainbow
k-coloring using rck(G) colors is called a minimum rainbow k-coloring. By definition,
rck(G) is the generalization of rc(G) and rc1(G) = rc(G) is the rainbow connection number
of G. By coloring the edges of G with distinct colors, we see that every two vertices of G
are connected by k internally disjoint rainbow paths and that rck(G) is defined for every
1 ≤ k ≤ κ. So rck(G) is well-defined. Furthermore, rck(G) ≤ rcj(G) for 1 ≤ k ≤ j ≤ κ.
Note that this new defined rainbow k-connectivity computes the number of colors, this
is distinct with connectivity (edge-connectivity) which computes the number of internally
(edge) disjoint paths. We can also call it rainbow k-connection number.
Now we introduce another generalization of rainbow connection number by Chartrand,
Okamoto and Zhang [15]. Let G be an edge-colored nontrivial connected graph of order n.
A tree T in G is a rainbow tree if no two edges of T are colored the same. Let k be a fixed
integer with 2 ≤ k ≤ n. An edge coloring of G is called a k-rainbow coloring if for every set
S of k vertices of G, there exists a rainbow tree in G containing the vertices of S. The k-
rainbow index rxk(G) of G is the minimum number of colors needed in a k-rainbow coloring
of G. A k-rainbow coloring using rxk(G) colors is called a minimum k-rainbow coloring.
Thus rx2(G) is the rainbow connection number rc(G) of G. It follows, for every nontrivial
connected graph G of order n, that rx2(G) ≤ rx3(G) ≤ ··· ≤ rxn(G).
The above four new graph-parameters are all defined in edge-colored graphs.
elevich and Yuster [36] naturally introduced a new parameter corresponding to rainbow
connection number which is defined on vertex-colored graphs. A vertex-colored graph G
is rainbow vertex-connected if any two vertices are connected by a path whose internal
vertices have distinct colors. A vertex-coloring under which G is rainbow vertex-connected
is called a rainbow vertex-coloring. The rainbow vertex-connection number of a connected
graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to
make G rainbow vertex-connected. The minimum rainbow vertex-coloring is defined sim-
Kriv-
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ilarly. Obviously, we always have rvc(G) ≤ n − 2 (except for the singleton graph), and
rvc(G) = 0 if and only if G is a clique. Also clearly, rvc(G) ≥ diam(G) − 1 with equality if
the diameter of G is 1 or 2.
Note that rvc(G) may be much smaller than rc(G) for some graph G. For example,
rvc(K1,n−1) = 1 while rc(K1,n−1) = n − 1. rvc(G) may also be much larger than rc(G) for
some graph G. For example, take n vertex-disjoint triangles and, by designating a vertex
from each of them, add a complete graph on the designated vertices. This graph has n
cut-vertices and hence rvc(G) ≥ n. In fact, rvc(G) = n by coloring only the cut-vertices
with distinct colors. On the other hand, it is not difficult to see that rc(G) ≤ 4. Just color
the edges of the Knwith, say, color 1, and color the edges of each triangle with the colors
2,3,4.
In Section 2, we will focus on the rainbow connection number and strong rainbow con-
nection number. We collect many upper bounds for these two parameters. From Section 3
to Section 5, we survey on the other three parameters: rainbow k-connectivity, k-rainbow
index, rainbow vertex-connection number, respectively. In the last section, we sum up the
results on algorithms and computational complexity.
1.2 Terminology and notations
All graphs considered in this survey are finite, simple and undirected. We follow the
notations and terminology of [7] for all those not defined here. We use V (G) and E(G) to
denote the set of vertices and the set of edges of G, respectively. For any subset X of V (G),
let G[X] denote the subgraph induced by X, and E[X] the edge set of G[X]; similarly, for
any subset F of E(G), let G[F] denote the subgraph induced by F. Let G be a set of graphs,
then V (G) =?
complete subgraph of G, and a maximal clique is a clique that is not contained in any larger
clique of G. For a set S, |S| denotes the cardinality of S. An edge in a connected graph
is called a bridge, if its removal disconnects the graph. A graph with no bridges is called
a bridgeless graph. A vertex is called pendant if its degree is 1. We call a path of G with
length k a pendant k-length path if one of its end vertex has degree 1 and all inner vertices
have degree 2 in G. By definition, a pendant k-length path contains a pendant ℓ-length path
(1 ≤ ℓ ≤ k). A pendant 1-length path is a pendant edge. We denote Cna cycle with n
vertices. For n ≥ 3, the wheel Wnis constructed by joining a new vertex to every vertex of
Cn. We use g(G) to denote the girth of G, that is, the length of a shortest cycle of G.
G∈GV (G), E(G) =?
G∈GE(G). We define a clique in a graph G to be a
Let G be a connected graph. Recall that the distance between two vertices u and v in G,
denoted by d(u,v), is the length of a shortest path between them in G. The eccentricity of a
vertex v is ecc(v) := maxx∈V (G)d(v,x). The diameter of G is diam(G) := maxx∈V (G)ecc(x).
The radius of G is rad(G) := minx∈V (G)ecc(x). Distance between a vertex v and a set
S ⊆ V (G) is d(v,S) := minx∈Sd(v,x). The k-step open neighbourhood of a set S ⊆ V (G)
is Nk(S) := {x ∈ V (G)|d(x,S) = k}, k ∈ {0,1,2,···}. A set D ⊆ V (G) is called a k-step
dominating set of G, if every vertex in G is at a distance at most k from D. Further, if D
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induces a connected subgraph of G, it is called a connected k-step dominating set of G. The
cardinality of a minimum connected k-step dominating set in G is called its connected k-step
domination number, denoted by γk
every vertex that is not dominated by it has at least k neighbors that are dominated by it.
In [11], Chandran, Das, Rajendraprasad and Varma made two new definitions which will be
useful in the sequel. A dominating set D in a graph G is called a two-way dominating set
if every pendant vertex of G is included in D. In addition, if G[D] is connected, we call D
a connected two-way dominating set. A (connected) two-step dominating set D of vertices
in a graph G is called a (connected) two-way two-step dominating set if (i) every pendant
vertex of G is included in D and (ii) every vertex in N2(D) has at least two neighbours in
N1(D). Note that if δ(G) ≥ 2, then every (connected) dominating set in G is a (connected)
two-way dominating set.
c(G). We call a two-step dominating set k-strong [36] if
A subgraph H of a graph G is called isometric if distance between any pair of vertices in
H is the same as their distance in G. The size of a largest isometric cycle in G is denoted by
iso(G). A graph is called chordal if it contains no induced cycles of length greater than 3.
The chordality of a graph G is the length of a largest induced cycle in G. Note that every
isometric cycle is induced and hence iso(G) is at most the chordality of G. For k ≤ α(G),
we use σk(G) to denote the minimum degree sum that is taken over all independent sets of
k vertices of G, where α(G) is the number of elements of an maximum independent set of
G.
2 (Strong) Rainbow connection number
2.1Basic results
In [12], Chartrand, Johns, McKeon and Zhang did some basic research on the (strong)
rainbow connection numbers of graphs. They determined the precise (strong) rainbow con-
nection numbers of several special graph classes including trees, complete graphs, cycles,
wheel graphs, complete bipartite graphs and complete multipartite graphs.
Proposition 2.1 [12] Let G be a nontrivial connected graph of size m. Then
(a) rc(G) = 1 if and only if G is complete, src(G) = 1 if and only if G is complete;
(b) rc(G) = 2 if and only if src(G) = 2;
(c) rc(G) = m if and only if G is a tree, src(G) = m if and only if G is a tree.
Proposition 2.2 [12] For each integer n ≥ 4, rc(Cn) = src(Cn) = ⌈n
2⌉.
Proposition 2.3 [12] For each integer n ≥ 3, we have
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rc(Wn) =
1 if n = 3,
2 if 4 ≤ n ≤ 6,
3 if n ≥ 7.
and src(Wn) = ⌈n
3⌉.
Proposition 2.4 [12] For integers s and t with 2 ≤ s ≤ t, rc(Ks,t) = min{⌈
for integers s and t with 1 ≤ s ≤ t, src(Ks,t) = ⌈
s√t⌉,4}, and
s√t⌉.
Proposition 2.5 [12] Let G = Kn1,n2,...,nkbe a complete k-partite graph, where k ≥ 3 and
n1≤ n2≤ ... ≤ nksuch that s =?k−1
i=1niand t = nk. Then
rc(G) =
1
2
min{⌈
if nk= 1,
if nk≥ 2 and s > t,
s√t⌉,3} if s ≤ t.
and
src(G) =
1
2
⌈
if nk= 1,
if nk≥ 2 and s > t,
s√t⌉ if s ≤ t.
By Proposition 2.1, it follows that for every positive integer a and for every tree T of size
a, rc(T) = src(T) = a. Furthermore, for a ∈ {1,2}, rc(G) = a if and only if src(G) = a. If
a = 3,b ≥ 4, then by Proposition 2.3, rc(W3b) = 3 and src(W3b) = b. For a ≥ 4, we have
the following.
Theorem 2.6 [12] Let a and b be positive integers with a ≥ 4 and b ≥
exists a connected graph G such that rc(G) = a and src(G) = b.
5a−6
3. Then there
Then, combining Propositions 2.1 and 2.3 with Theorem 2.6, they got the following result.
Corollary 2.7 [12] Let a and b be positive integers. If a = b or 3 ≤ a < b and b ≥5a−6
then there exists a connected graph G such that rc(G) = a and src(G) = b.
3
,
Finally, they thought the question that whether the condition b ≥5a−6
and raised the following conjecture:
3
can be deleted ?
Conjecture 2.8 [12] Let a and b be positive integers. Then there exists a connected graph
G such that rc(G) = a and src(G) = b if and only if a = b ∈ {1,2} or 3 ≤ a ≤ b.
In [19], Chen and Li gave a confirmative solution to this conjecture by showing a class of
graphs with given rainbow connection number a and strong rainbow connection number b.
From the above several propositions, we know rc(G) = src(G) hold for some special
graph classes. A difficult problem is following:
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Problem 2.9 Characterize graphs G for which rc(G) = src(G), or, give some sufficient
conditions to guarantee rc(G) = src(G).
Recall the fact that if H is a connected spanning subgraph of a nontrivial (connected)
graph G, then rc(G) ≤ rc(H). This fact is very useful to bounding the value of rc(G) by
giving bounds for its connected spanning subgraphs. We have noted that if, in addition,
diam(H) = 2, then src(G) ≤ src(H). The authors of [12] naturally raised the following
conjecture:
Conjecture 2.10 [12] If H is a connected spanning subgraph of a nontrivial (connected)
graph G, then src(G) ≤ src(H).
Recently, this conjecture was disproved by Chakraborty, Fischer, Matsliah and Yuster
[10]. They showed the following example: see Figure 2.1, here G is obtained from H by
adding the edge e = uv, then H is a connected spanning subgraph of G. It is easy to show
that there is a strong rainbow coloring of H which costs six colors, but the graph G costs at
least seven colors to ensure its strong rainbow connection.
H
u
v
Figure 2.1 A counterexample to Conjecture 2.10.
Suppose that G contains two bridges e = uv and f = xy. Then G−e−f contains three
components Gi(1 ≤ i ≤ 3), where two of these components contain one of u,v,x and y and
the third component contains two of these four vertices, say u ∈ V (G1), x ∈ V (G2) and
v,y ∈ V (G3). If S is a set of k vertices contains u and x, then every tree whose vertex set
contains S must also contain the edges e and f. This gives us a necessary condition for an
edge-colored graph to be k-rainbow colored.
Observation 2.11 [15] Let G be a connected graph of order n containing two bridges e and
f. For each integer k with 2 ≤ k ≤ n, every k-rainbow coloring of G must assign distinct
colors to e and f.
From Observation 2.11, we know that if G is rainbow connected under some edge-coloring,
then any two bridges obtain distinct colors.
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2.2 Upper bounds for rainbow connection number
We know that it is almost impossible to give the precise rainbow connection number of
a given arbitrary graph, so we aim to give some nice bounds for it, especially sharp upper
bounds.
In [9], Caro, Lev, Roditty, Tuza and Yuster investigated the extremal graph-theoretic
behavior of rainbow connection number. Motivated by the fact that there are graphs with
minimum degree 2 and with rc(G) = n−3 (just take two vertex-disjoint triangles and connect
them by a path of length n − 5), it is interesting to study the rainbow connection number
of graphs with minimum degree at least 3 and they thought of the following question: is it
true that minimum degree at least 3 guarantees rc(G) ≤ αn where α < 1 is independant of
n? This turns out to be true, and they proved:
Theorem 2.12 [9] If G is a connected graph with n vertices and δ(G) ≥ 3, then rc(G) <5
6n.
In the proof of Theorem 2.12, they first gave an upper bound for the rainbow connection
number of 2-connected graphs (see Theorem 2.23), then from it, they next derived an upper
bound for the rainbow connection number of connected bridgeless graphs (see Theorem 2.25).
The constant 5/6 appearing in Theorem 2.12 is not optimal, but it probably cannot be
replaced with a constant smaller than
rc(G) = diam(G) =
4
, and one of such graphs can be constructed as follows [53]:
Take two vertex disjoint copies of the graph K5− P3and label the two vertices of degree
2 with w1 and w2k+2, where k ≥ 1 is an integer. Next join w1 and w2k+2 by a path of
length 2k + 1 and label the vertices with w1,w2,··· ,w2k+2. Now for 1 ≤ i ≤ k every edge
w2iw2i+1 is replaced by a K4− e and we identify the two vertices of degree 2 in K4− e
with w2iand w2i+1. The resulting graph G4k+10is 3-regular, has order n = 4k + 10 and
rc(G4k+10) = diam(G4k+10) = 3k + 5 =
4
conjectured:
3
4, since there are 3-regular connected graphs with
3n−10
3n−10
. Then Caro, Lev, Roditty, Tuza and Yuster
Conjecture 2.13 [9] If G is a connected graph with n vertices and δ(G) ≥ 3, then rc(G) <
3
4n.
Schiermeyer proved the conjecture in [53] by showing the following result:
Theorem 2.14 [53] If G is a connected graph with n vertices and δ(G) ≥ 3, then rc(G) <
3n−1
4.
For 2-connected graphs Theorem 2.14 is true by Theorem 2.23. Hence it remains to
prove it for graphs with connectivity 1. Schiermeyer extended the concept of rainbow con-
nection number as follows: Additionally we require that any two edges of G have different
colors whenever they belong to different blocks of G. The corresponding rainbow connection
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number will be denoted by rc∗(G). Then they derived Theorem 2.14 by first proving the
following result: let G be a connected graph with n vertices, connectivity 1, and δ ≥ 3, then
rc∗(G) ≤3n−10
Not surprisingly, as the minimum degree increases, the graph would become more dense
and therefore the rainbow connection number would decrease.
Roditty, Tuza and Yuster also proved the following upper bounds in term of minimum
degree.
4
.
Specifically, Caro, Lev,
Theorem 2.15 [9] If G is a connected graph with n vertices and minimum degree δ, then
rc(G) ≤ min{nlnδ
δ
(1 + oδ(1)),n4lnδ + 3
δ
}.
In the proof, they used the concept of a connected two-dominating set (A set of vertices
S of G is called a connected two-dominating set if S induces a connected subgraph of G and,
furthermore, each vertex outside of S has at least two neighbours in S) and the probabilistic
method. They showed that in any case it cannot be improved below
constructed a connected n-vertex graph with minimum degree δ and this diameter: Take
m copies of Kδ+1, denoted by X1,··· ,Xmand label the vertices of Xiwith xi,1,··· ,xi,δ+1.
Take two copies of Kδ+2, denoted by X0,Xm+1 and similarly label their vertices. Now,
connect xi,2with xi+1,1for i = 0,··· ,m with an edge, and delete the edges (xi,1,xi,2) for
i = 0,··· ,m + 1. The obtained graph has n = (m + 2)(δ + 1) + 2 vertices, and minimum
degree δ (and maximum degree δ + 1). It is straightforward to verify that a shortest path
from x0,1to xm+1,2has length 3m + 5 =
3n
δ+1−δ+7
δ+1as they
3n
δ+1−δ+7
δ+1.
This, naturally, raised the open problem of determining the true behavior of rc(G) as a
function of δ.
In [10], Chakraborty, Fischer, Matsliah and Yuster proved that any connected n-vertex
graph with minimum degree Θ(n) has a bounded rainbow connection.
Theorem 2.16 [10] For every ǫ > 0 there is a constant C = C(ǫ) such that if G is a
connected graph with n vertices and minimum degree at least ǫn, then rc(G) ≤ C.
The proof of Theorem 2.16 is based upon a modified degree-form version of Szemer´ edi
Regularity Lemma (see [35] for a good survey on Regularity Lemma) that they proved.
The above lower bound construction suggests that the logarithmic factor in their upper
bound may not be necessary and that, in fact rc(G) ≤ Cn/δ where C is a universal constant.
If true, notice that for graphs with a linear minimum degree ǫn, this implies that rc(G) is at
most C/ǫ. However, Theorem 2.16 does not even guarantee the weaker claim that rc(G) is a
constant. The constant C = C(ǫ) they obtained is a tower function in 1/ǫ and in particular
extremely far from being reciprocal to 1/ǫ.
Finally, Krivelevich and Yuster in [36] determined the behavior of rc(G) as a function of
δ(G) and resolved the above-mentioned open problem.
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Theorem 2.17 [36] A connected graph G with n vertices has rc(G) <
20n
δ(G).
The proof of Theorem 2.17 uses the concept of connected two-step dominating set. Kriv-
elevich and Yuster first proved that for a connected graph H with minimum degree k and
n vertices, there exists a two-step dominating set S whose size is at most
is a connected two-step dominating set S′containing S with |S′| ≤ 5|S| − 4. They found
two edge-disjoint spanning subgraphs in a graph G with minimum degree at least ⌊δ−1
Then they derived a rainbow coloring for G by giving a rainbow coloring to each subgraphs
according to its connected two-step dominating set.
n
k+1and there
2⌋.
The authors noted that the constant 20 obtained by their proof is not optimal and can be
slightly improved with additional effort. However, from the example below Theorem 2.15,
one cannot expect to replace C by a constant smaller than 3.
Motivated by the results of Theorems 2.14, 2.15 and 2.17, Schiermeyer raised the following
open problem in [53].
Problem 2.18 [53] For every k ≥ 2 find a minimal constant ckwith 0 < ck≤ 1 such that
rc(G) ≤ ckn for all graphs G with minimum degree δ(G) ≥ k. Is it true that ck=
all k ≥ 2 ?
3
k+1for
This is true for k = 2,3 as shown before (c2= 1 and c3=3
4).
Recently, Chandran, Das, Rajendraprasad and Varma [11] nearly settled the above prob-
lem. They used the concept of a connected two-way two-step dominating set in the argument
and they first proved the following result.
Theorem 2.19 [11] If D is a connected two-way two-step dominating set in a graph G, then
rc(G) ≤ rc(G[D]) + 6.
Furthermore, they gave a nearly sharp bound for the size of D by showing that every
connected graph G of order n ≥ 4 and minimum degree δ has a connected two-way two-step
dominating set D of size at most
many connected graphs G such that γ2
3n
δ+1− 2; moreover, for every δ ≥ 2, there exist infinitely
c(G) ≥
3(n−2)
δ+1− 4. Then the following result is easy.
Theorem 2.20 [11] For every connected graph G of order n and minimum degree δ,
rc(G) ≤
3n
δ + 1+ 3.
Moreover, for every δ ≥ 2, there exist infinitely many connected graphs G such that rc(G) ≥
3(n−2)
δ+1− 1.
Theorem 2.20 answers Problem 2.18 in the affirmative but up to an additive constant
of 3. Moreover, this bound is seen to be tight up to additive factors by the construction
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mentioned in [9] (see the example below Theorem 2.15) and [23]. And therefore, for graphs
with linear minimum degree ǫn, the rainbow connection number is bounded by a constant.
Recently, Dong and Li [22] derived an upper bound on rainbow connection numbers of
graphs under given degree sum condition σ2. Recall that for a graph G, σ2(G) = min{d(u)+
d(v) | u,v are independent in G}. Clearly, the degree sum condition σ2is weaker than the
minimum degree condition.
Theorem 2.21 [22] For a connected graph G of order n, rc(G) ≤ 6n−2
σ2+2+ 7.
Similar to the method of Theorem 2.20, they derived that every connected graph G of
order n with at most one pendant vertex has a connected two-way two-step dominating set
D of size at most 6n−2
σ2+2+ 2. Then by using Theorem 2.19, they got the theorem.
From the example below Theorem 2.15, we know their bound are seen to be tight up to
additive factors. Note that by the definition of σ2, we know σ2≥ 2δ, so from Theorem 2.21,
we can derive rc(G) ≤ 6n−2
as an improvement of that in Theorem 2.20.
σ2+2+ 7 ≤
3(n−2)
δ+1+ 7. And the bound in Theorem 2.21 can be seen
With respect to the the relation between rc(G) and the connectivity κ(G), mentioned in
[53], Broersma asked a question at the IWOCA workshop:
Problem 2.22 [53] What happens with the value rc(G) for graphs with higher connectivity?
For κ(G) = 1, Theorem 2.14 means that if G is a graph of order n, connectivity κ(G) = 1
and δ ≥ 3. Then rc(G) ≤3n−1
above, Caro, Lev, Roditty, Tuza and Yuster derived:
4. For κ(G) = 2, in the proof of Theorem 2.12, as we mentioned
Theorem 2.23 [9] If G is a 2-connected graph with n vertices then rc(G) ≤2n
3.
That is, if G is a graph of order n, connectivity κ(G) = 2. Then rc(G) ≤2n
From Theorem 2.20, we can easily obtain an upper bound of the rainbow connection
number according to the connectivity:
3.
rc(G) ≤
3n
δ + 1+ 3 ≤
3n
κ + 1+ 3.
Therefore, for κ(G) = 3, rc(G) ≤3n
results in [9], and by using the Fan Lemma, Li and Shi [41] improved this bound by showing
the following result.
4+ 3; for κ(G) = 4, rc(G) ≤3n
5+ 3. Motivated by the
Theorem 2.24 ([41]) If G is a 3-connected graph with n vertices, then rc(G) ≤
3(n+1)
5
.
However, for general connectivity, there is no upper bound which is better than
3n
κ+1+3.
The following result is an important ingredient in the proof of Theorem 2.12 in [9].
Page 12
Theorem 2.25 [9] If G is a connected bridgeless graph with n vertices, then rc(G) ≤4n
5−1.
From Theorem 2.20, we can also easily obtain an upper bound of the rainbow connection
number according to the edge-connectivity λ:
rc(G) ≤
3n
δ + 1+ 3 ≤
3n
λ + 1+ 3.
Note that all the above upper bounds are determined by n and other parameters such as
(edge)-connectivity, minimum degree. Diameter of a graph, and hence its radius, are obvious
lower bounds for rainbow connection number. Hence it is interesting to see if there is an
upper bound which is a function of the radius r or diameter alone. Such upper bounds were
shown for some special graph classes in [11] which we will introduce in the sequel. But, for a
general graph, the rainbow connection number cannot be upper bounded by a function of r
alone. For instance, the star K1,nhas a radius 1 but rainbow connection number n. Still, the
question of whether such an upper bound exists for graphs with higher connectivity remains.
Basavaraju, Chandran, Rajendraprasad and Ramaswamy [4] answered this question in the
affirmative. The key of their argument is the following lemma, and in the proof of this
lemma, we can obtain a connected (k − 1)-step dominating set from a connected k-step
dominating set.
Lemma 2.26 [4] If G is a bridgeless graph, then for every connected k-step dominating set
Dkof G, k ≥ 1, there exists a connected (k − 1)-step dominating set Dk−1⊃ Dksuch that
rc(G[Dk−1]) ≤ rc(G[Dk]) + min{2k + 1,ζ},
where ζ = iso(G).
Given a graph G and a set D ⊂ V (G), A D-ear is a path P = (x0,x1,··· ,xp) in G
such that P ∩ D = {x0,xp}. P may be a closed path, in which case x0 = xp. Further,
P is called an acceptable D-ear if either P is a shortest D-ear containing (x0,x1) or P is
a shortest D-ear containing (xp−1,xp). Let A = {a1,a2,···} and B = {b1,b2,···} be two
pools of colors, none of which are used to color G[Dk]. A Dk-ear P = (x0,x1,··· ,xp) will
be called evenly colored if its edges are colored a1,a2,··· ,a⌈p
Basavaraju, Chandran, Rajendraprasad and Ramaswamy proved this lemma by constructing
a sequence of sets Dk= D0⊂ D1⊂ ··· ⊂ Dt= Dk−1and coloring the new edges in every
induced graph G[Di] such that the following property is maintained for all 0 ≤ i ≤ t: every
x ∈ Di\Dklies in an evenly colored acceptable Dk-ear in G[Di].
The following theorem can be derived from Lemma 2.26 easily.
2⌉,b⌊p
2⌋,··· ,b2,b1in that order.
Theorem 2.27 [4] For every connected bridgeless graph G,
rc(G) ≤
r
?
i=1
min{2i + 1,ζ} ≤ rζ,
where r is the radius of G.
Page 13
Theorem 2.27 has two corollaries.
Corollary 2.28 [4] For every connected bridgeless graph G with radius r,
rc(G) ≤ r(r + 2).
Moreover, for every integer r ≥ 1, there exists a bridgeless graph with radius r and rc(G) =
r(r + 2).
Corollary 2.29 [4] For every connected bridgeless graph G with radius r and chordality k,
rc(G) ≤
r
?
i=1
min{2i + 1,k} ≤ rk.
Moreover, for every two integers r ≥ 1 and 3 ≤ k ≤ 2r + 1, there exists a bridgeless graph
G with radius r and chordality k such that rc(G) =?r
i=1min{2i + 1,k}.
Corollary 2.28 answered the above question in the affirmative, the bound is sharp and is a
function of the radius r alone. Basavaraju, Chandran, Rajendraprasad and Ramaswamy also
demonstrated that the bound cannot be improved even if we assume stronger connectivity
by constructing a κ-vertex-connected graph of radius r whose rainbow connection number
is r(r + 2) for any two given integers κ,r ≥ 1: Let s(0) := 0,s(i) := 2?r−i+1
and t := s(r) = r(r + 1). Let V = V0⊎ V1⊎ ··· ⊎ Vtwhere Vi= {xi,0,xi,1,··· ,xi,κ−1} for
0 ≤ i ≤ t − 1 and Vt= {xt,0}. Construct a graph Xr,κon V by adding the following edges.
E(X) = {{xi,j,xi′,j′} : |i − i′| ≤ 1} ∪ {{xs(i),0,xs(i+1),0} : 0 ≤ i ≤ r − 1}.
Corollary 2.29 generalises a result from [11] that the rainbow connection number of any
bridgeless chordal graph is at most three times its radius as the chordality of a chordal graph
is three.
j=r
j for 1 ≤ i ≤ r
In [9], Caro, Lev, Roditty, Tuza and Yuster also derived a result which gives an upper
bound for rainbow connection number according to the order and the number of vertex-
disjoint cycles. Here χ′(G) is the chromatic index of G.
Theorem 2.30 [9] Suppose G is a connected graph with n vertices, and assume that there
is a set of vertex-disjoint cycles that cover all but s vertices of G. Then rc(G) < 3n/4 +
s/4 − 1/2. In particular:
(i). If G has a 2-factor then rc(G) < 3n/4.
(ii). If G is k-regular and k is even then rc(G) < 3n/4.
(iii). If G is k-regular and χ′(G) = k then rc(G) < 3n/4.
Another approach for achieving upper bounds is based on the size (number of edges) m
of the graph. Those type of sufficient conditions are known as Erd˝ os-Gallai type results.
Research on the following Erd˝ os-Gallai type problem has been started in [34].
Page 14
Problem 2.31 [34] For every k, 1 ≤ k ≤ n−1, compute and minimize the function f(n,k)
with the following property: If |E(G)| ≥ f(n,k), then rc(G) ≤ k.
In [34], Kemnitz and Schiermeyer gave a lower bound for f(n,k), i.e., f(n,k) ≥?n−k+1
(k−1). They also computed f(n,k) for k ∈ {1,n−2,n−1}, i.e., f(n,1) =?n
n − 1,f(n,n − 2) = n, and obtained f(n,2) =?n−1
In [48], Li and Sun provided a new approach to investigate the rainbow connection
number of a graph G according to some constraints to its complement graph G. They gave
two sufficient conditions to guarantee that rc(G) is bounded by a constant. By using the
fact that rc(G) ≤ rc(H) where H is a connected spanning subgraph of a connected graph G,
and the structure of its complement graph as well as Propositions 2.4 and 2.5, they derived
the following result.
2
?+
2
?,f(n,n−1) =
2
?+ 1 for k = 2.
Theorem 2.32 [48] For a connected graph G, if G does not belong to the following two
cases: (i) diam(G) = 2,3, (ii) G contains exactly two connected components and one of
them is trivial, then rc(G) ≤ 4. Furthermore, this bound is best possible.
For the remaining cases, rc(G) can be very large as discussed in [48]. So They add a
constraint: let G be triangle-free, then G is claw-free. And they derived the following result.
In their argument, Theorem 2.40 is useful.
Theorem 2.33 [48] For a connected graph G, if G is triangle-free, then rc(G) ≤ 6.
The readers may consider the rainbow connection number of a graph G according to
some other condition to its complement graph.
Chen, Li and Lian [17] investigated Nordhaus-Gaddum-type result. A Nordhaus-Gaddum-
type result is a (sharp) lower or upper bound on the sum or product of the values of a parame-
ter for a graph and its complement. The name “Nordhaus-Gaddum-type” is so given because
it is Nordhaus and Gaddum [49] who first established the following type of inequalities for
chromatic number of graphs in 1956.
Theorem 2.34 [17] Let G and G be connected with n ≥ 4, then
4 ≤ rc(G) + rc(G) ≤ n + 2.
Furthermore, the upper bound is sharp for n ≥ 4 and the low bound is sharp for n ≥ 8.
They also proved that rc(G)+rc(G) ≥ 6 for n = 4,5; and rc(G)+rc(G) ≥ 5 for n = 6,7
and these bounds are best possible.
Page 15
2.3 For some graph classes
Some graph classes, such as line graphs, have many special properties, and by these
properties we can get some interesting results on their rainbow connection numbers in terms
of some graph parameters. For example, in [9] Caro, Lev, Roditty, Tuza and Yuster derived
Theorem 2.23 according to the ear-decomposition of a 2-connected graph. In this subsection,
we will introduce some results on rainbow connection numbers of line graphs, etc.
In [42] and [43], Li and Sun studied the rainbow connection numbers of line graphs in
the light of particular properties of line graphs shown in [30] and [31]. They gave two sharp
upper bounds for rainbow connection number of a line graph and one sharp upper bound
for rainbow connection number of an iterated line graph.
Recall the line graph of a graph G is the graph L(G) (or L1(G)) whose vertex set
V (L(G)) = E(G), and two vertices e1, e2 of L(G) are adjacent if and only if they are
adjacent in G. The iterated line graph of a graph G, denoted by L2(G), is the line graph
of the graph L(G). More generally, the k-iterated line graph Lk(G) is the line graph of
Lk−1(G) (k ≥ 2). We also need the following new terminology.
For a connected graph G, we call G a clique-tree-structure, if it satisfies the following
condition: each block is a maximal clique. We call a graph H a clique-forest-structure,
if H is a disjoint union of some clique-tree-structures, that is, each component of a clique-
forest-structure is a clique-tree-structure. By the above condition, we know that any two
maximal cliques of G have at most one common vertex. Furthermore, G is formed by its
maximal cliques. The size of a clique-tree(forest)-structure is the number of its maximal
cliques. An example of clique-forest-structure is shown in Figure 2.2. If each block of a
Figure 2.2 A clique-forest-structure with size 6 and 2 components.
clique-tree-structure is a triangle, we call it a triangle-tree-structure. Let ℓ be the size
of a triangle-tree-structure. Then, by definition, it is easy to show that there are 2ℓ + 1
vertices in it. Similarly, we can give the definition of a triangle-forest-structure and there
are 2ℓ+c vertices in a triangle-forest-structure with size ℓ and c components. We denote n2
the number of inner vertices (degrees at least 2) of a graph.
Theorem 2.35 [43] For any set T of t edge-disjoint triangles of a connected graph G, if
Page 16
the subgraph induced by the edge set E(T ) is a triangle-forest-structure, then
rc(L(G)) ≤ n2− t.
Moreover, the bound is sharp.
Theorem 2.36 [43] If G is a connected graph, T is a set of t edge-disjoint triangles that
cover all but n′
G[E(T )], then
rc(L(G)) ≤ t + n′
Moreover, the bound is sharp.
2inner vertices of G and c is the number of components of the subgraph
2+ c.
Theorem 2.37 [43] Let G be a connected graph with m edges and m1 pendant 2-length
paths. Then
rc(L2(G)) ≤ m − m1.
The equality holds if and only if G is a path of length at least 3.
In the proofs of the above three theorems, Li and Sun used the particular structure of
line graphs and the observation: If G is a connected graph and {Ei}i∈[t]is a partition of the
edge set of G into connected subgraphs Gi= G[Ei], then rc(G) ≤?t
The above three theorems give upper bounds for rainbow connection number of Lk(G)(k =
1,2) according to some parameters of G. One may consider the relation between rc(G) and
rc(L(G)).
i=1rc(Gi) (see [42]).
Problem 2.38 Determine the relationship between rc(G) and rc(L(G)), is there an upper
bound for one of these parameters in terms of the other?
One also can consider the rainbow connection number of the general iterated line graph
Lk(G) when k is sufficiently large.
Problem 2.39 Consider the value of rc(Lk(G)) as k → ∞, is it bounded by a constant?
or, does it convergence to a function of some graph parameters, such as the order n of G?
For Problem 2.39, we know if G is a cycle Cn(n ≥ 4), then Lk(G) = G, so rc(Lk(G)) =
2⌉. But for many graphs, we know, as k grows, Lk(G) will become more dense, and
rc(Lk(G)) may decrease.
⌈n
An intersection graph of a family of sets F, is a graph whose vertices can be mapped
to the sets in F such that there is an edge between two vertices in the graph if and only
if the corresponding two sets in F have a non-empty intersection. An interval graph is
an intersection graph of intervals on the real line. A unit interval graph is an intersection
Page 17
graph of unit length intervals on the real line. A circular arc graph is an intersection
graph of arcs on a circle. An independant triple of vertices x,y,z in a graph G is an
asteroidal triple (AT), if between every pair of vertices in the triple, there is a path that
does not contain any neighbour of the third. A graph without asteroidal triples is called
an AT-free graph [20]. A graph G is a threshold graph, if there exists a weight function
w : V (G) → R and a real constant t such that two vertices u,v ∈ V (G) are adjacent if and
only if w(u) + w(v) ≥ t. A bipartite graph G(A,B) is called a chain graph if the vertices
of A can be ordered as A = (a1,a2,··· ,ak) such that N(a1) ⊆ N(a2) ⊆ ··· ⊆ N(ak) [56].
In [11], Chandran, Das, Rajendraprasad and Varma investigated the rainbow connection
numbers of these special graph classes. They first showed a result concerning the connected
two-way dominating sets.
Theorem 2.40 [11] If D is a connected two-way dominating set in a graph G, then
rc(G) ≤ rc(G[D]) + 3.
They also proved that every connected graph G of order n and minimum degree δ has a
connected two-step dominating set D of size at most
3(n−|N2(D)|)
δ+1
− 2.
From Theorem 2.40, the following result can be derived.
Theorem 2.41 [11] Let G be a connected graph with δ(G) ≥ 2. Then
(i) if G is an interval graph, diam(G) ≤ rc(G) ≤ diam(G) + 1, in particular, if G is a unit
interval graph, then rc(G) = diam(G);
(ii) if G is AT-free, diam(G) ≤ rc(G) ≤ diam(G) + 3;
(iii) if G is a threshold graph, diam(G) ≤ rc(G) ≤ 3;
(iv) if G is a chain graph, diam(G) ≤ rc(G) ≤ 4;
(v) if G is a circular arc graph, diam(G) ≤ rc(G) ≤ diam(G) + 4.
Moreover, there exist threshold graphs and chain graphs with minimum degree at least 2 and
rainbow connection number equal to the corresponding upper bound above. There exists an
AT-free graph G with minimum degree at least 2 and rc(G) = diam(G) + 2, which is 1 less
than the upper bound above.
Recall that the concept of rainbow connection number is of great use in transferring
information of high security in multicomputer networks. Cayley graphs are very good models
that have been used in communication networks. So, it is of significance to study the rainbow
connection numbers of Cayley graphs. Li, Li and Liu [38] investigated the rainbow connection
numbers of cayley graphs on Abelian groups.
Let Γ be a group, and let a ∈ Γ be an element. We use ?a? to denote the cyclic subgroup
of Γ generated by a. The number of elements of ?a? is called the order of a, denoted by |a|.
A pair of elements a and b in a group commutes if ab = ba. A group is Abelian if every
pair of its elements commutes. A Cayley graph of Γ with respect to S is the graph C(Γ,S)
with vertex set Γ in which two vertices x and y are adjacent if and only if xy−1∈ S (or
equivalently, yx−1∈ S), where S ⊆ Γ \ {1} is closed under taking inverse [52].
Page 18
Theorem 2.42 [38] Given an Abelian group Γ and an inverse closed set S ⊆ Γ \ {1}, we
have the following results:
(i) rc(C(Γ,S)) ≤ min{?
(ii) If S is an inverse closed minimal generating set of Γ, then
a∈S∗⌈|a|/2⌉ | S∗⊆ S is a minimal generating set of Γ}.
?
a∈S∗
⌊|a|/2⌋ ≤ rc(C(Γ,S)) ≤ src(C(Γ,S)) ≤
?
a∈S∗
⌈|a|/2⌉,
where S∗⊆ S is a minimal generating set of Γ.
Moreover, if every element a ∈ S has an even order, then
rc(C(Γ,S)) = src(C(Γ,S)) =
?
a∈S∗
|a|/2.
They also investigated the rainbow connection numbers of recursive circulants (see [50]
for an introduction to recursive circulants).
Let G be an r-regular graph with n vertices. G is said to be strongly regular and denoted
by SRG(v,k,λ,µ) if there are also integers λ and µ such that every two adjacent vertices
have λ common neighbours and every two nonadjacent vertices have µ common neighbours.
Clearly, a strongly regular graph with parameters (v,k,λ,µ) is connected if and only if µ ≥ 1.
In [1], Ahadi and Dehghan derived the following result: For every connected strongly regular
graph G, rc(G) ≤ 600. As each strongly regular graph is a graph with diam(G) = 2, from
our next subsection (Theorem 2.51), we know rc(G) ≤ 5 if G is a strongly regular graph
with parameters (v,k,λ,µ), other than a star [39]. But 5 may not be the optimal upper
bound, so one may consider the following question.
Question 2.43 [1] Determine max{rc(G)| G is an SRG}.
There are other results on some special graph classes. In [14], Chartrand, Johns, McK-
eon and Zhang investigated the rainbow connection numbers of cages, and in [33], Johns,
Okamoto and Zhang investigated the rainbow connection numbers of small cubic graphs.
The details are omitted.
2.4 For dense and sparse graphs
For any given graph G, we know 1 ≤ rc(G) ≤ src(G) ≤ m. Here a graph G is called
a dense graph if its (strong) rainbow connection number is small, especially it is close to
1; while G is called a sparse graph if its (strong) rainbow connection number is large,
especially it is close to m. By Proposition 2.1, the cases that rc(G) = 1,src(G) = 1 and
rc(G) = m,src(G) = m are clear. So we want to investigate other cases.
Page 19
In [9], Caro, Lev, Roditty, Tuza and Yuster investigated the graphs with small rainbow
connection numbers, and they gave a sufficient condition that guarantees rc(G) = 2.
Theorem 2.44 [9] Any non-complete graph with δ(G) ≥ n/2 + logn has rc(G) = 2.
We know that having diameter 2 is a necessary requirement for having rc(G) = 2, al-
though certainly not sufficient (e.g., consider a star). Clearly, if δ(G) ≥ n/2 then diam(G) =
2, but we do not know if this guarantees rc(G) = 2. The above theorem shows that by slightly
increasing the minimum degree assumption, rc(G) = 2 follows.
Kemnitz and Schiermeyer [34] gave a sufficient condition to guarantee rc(G) = 2 accord-
ing to the number of edges m.
Theorem 2.45 [34] Let G be a connected graph of order n and size m. If?n−1
?n
2
?+1 ≤ m ≤
2
?− 1, then rc(G) = 2.
Let G = G(n,p) denote, as usual [3], the random graph with n vertices and edge proba-
bility p. For a graph property A and for a function p = p(n), we say that G(n,p) satisfies A
almost surely if the probability that G(n,p(n)) satisfies A tends to 1 as n tends to infinity.
We say that a function f(n) is a sharp threshold function for the property A if there are
two positive constants c and C so that G(n,cf(n)) almost surely does not satisfy A and
G(n,p) satisfies A almost surely for all p ≥ Cf(n). It is well known that all monotone graph
properties have a sharp threshold function (see [6] and [27]). Since having rc(G) ≤ 2 is
a monotone graph property (adding edges does not destroy this property), it has a sharp
threshold function. The following theorem establishes it.
Theorem 2.46 [9] p =
rc(G(n,p)) ≤ 2.
?logn/n is a sharp threshold function for the graph property
Theorem 2.44 asserts that minimum degree n/2 + logn guarantees rc(G) = 2. Clearly,
minimum degree n/2 − 1 does not, as there are connected graphs with minimum degree
n/2 − 1 and diameter 3 (just take two vertex-disjoint cliques of order n/2 each and connect
them by a single edge. It is therefore interesting to raise:
Problem 2.47 [9] Determine the minimum degree threshold that guarantees rc(G) = 2.
By Proposition 2.1, we know that the problem of considering graphs with rc(G) = 2 is
equivalent to that of considering graphs with src(G) = 2.
A bipartite graph which is not complete has diameter at least 3. A proof similar to that
of Theorem 2.46 gives the following result.
Page 20
Theorem 2.48 [9] Let c = 1/log(9/7). If G is a non-complete bipartite graph with n vertices
and any two vertices in the same vertex class have at least 2clogn common neighbors in the
other vertex class, then rc(G) = 3.
The following theorem asserts however that having diameter 2 and only logarithmic
minimum degree suffices to guarantee rainbow connection 3.
Theorem 2.49 [10] If G is an n-vertex graph with diameter 2 and minimum degree at least
8logn, then rc(G) ≤ 3.
Since a graph with minimum degree n/2 is connected and has diameter 2, we have as
an immediate result [10]: If G is an n-vertex graph with minimum degree at least n/2 then
rc(G) ≤ 3. We know that any graph G with rc(G) = 2 must have diam(G) = 2, so graphs
with rc(G) = 2 belong to the graph class with diam(G) = 2. By Corollary 2.28, we know
that for a bridgeless graph with diam(G) = 2, rc(G) ≤ r(r+2) ≤ 8. As rc(G) is at least the
number of bridges of G, so it may be very large if the number of bridges of G is sufficiently
large by Observation 2.11. So there is an interesting problem:
Problem 2.50 For any bridgeless graph G with diam(G) = 2, determine the smallest con-
stant c such that rc(G) ≤ c.
Recently, Li, Li and Liu [39] derived that c ≤ 5 by showing the following result:
Theorem 2.51 [39] rc(G) ≤ 5 if G is a bridgeless graph with diameter 2; and that rc(G) ≤
k + 2 if G is a connected graph with diameter 2 and k bridges, where k ≥ 1.
In the proof, Li, Li and Liu derived that if G is a bridgeless graph with order n and
diameter 2, then it is either 2-connected, or it has only one cut vertex v, furthermore, v
is the center of G with radius 1. They showed that 5 is almost best possible as there are
infinity many bridgeless graphs with diameter 2 whose rainbow connection numbers are 4,
however they have not found examples of such graphs with rc(G) = 5. The bound k + 2 is
sharp as there are infinity graphs with diameter 2 and k bridges whose rainbow connection
numbers attain this bound [39].
In [46] and [47], Li and Sun investigated the graphs with large rainbow connection num-
bers and strong rainbow connection numbers, respectively. They derived the two following
results. Note that each path Pjin the member of graph class Gi(1 ≤ i ≤ 4) of Figure 2.3
may be trivial.
Theorem 2.52 [46] For a connected graph G with m edges, we have rc(G) ?= m − 1; and
rc(G) = m − 2 if and only if G is a 5-cycle or belongs to one of four graph classes Gi’s
(1 ≤ i ≤ 4) shown in Figure 2.3.
Page 21
P2
P4
P3
P5 P6
G1
G2
P1
P2
G3
P1
P2
P3
P1
P1
P2
G4
P4
Figure 2.3 The figure for the four graph classes.
We now introduce two graph classes. Let C be the cycle of a unicyclic graph G, V (C) =
{v1,··· ,vk} and TG= {Ti: 1 ≤ i ≤ k} where Tiis the unique tree containing vertex viin
subgraph G\E(C). We say Tiand Tjare adjacent (nonadjacent) if viand vjare adjacent
(nonadjacent) in cycle C. Then let
G5= {G : G is a unicyclic graph, k = 3, TGcontains at most two nontrivial elements},
G6= {G : G is a unicyclic graph, k = 4, TGcontains two nonadjacent trivial elements and
the other two (nonadjacent) elements are paths}.
Theorem 2.53 [47] For a connected graph G with m edges, src(G) ?= m−1; src(G) = m−2
if and only if G is a 5-cycle or belongs to one of the Gi’s (i = 5,6).
By Proposition 2.1, Theorems 2.52 and 2.53, we investigated the graphs with rc(G) ≥
m − 2 (src(G) ≥ m − 2). Furthermore, we have the following interesting problem.
Problem 2.54 Give a sufficient condition to guarantee rc(G) ≥ αm (src(G) ≥ αm), where
0 < α < 1.
2.5For graph operations
Products of graphs occur naturally in discrete mathematics as tools in combinatorial
constructions, they give rise to important classes of graphs and deep structural problems.
The extensive literature on products that has evolved over the years presents a wealth
of profound and beautiful results [32]. In [46], Li and Sun obtained some results on the
rainbow connection numbers of products of graphs, including Cartesian product, composition
(lexicographic product), union of graphs, etc. Actually, we know that the line graph of a
graph is also a graph operation.
We first introduce the rainbow connection number of the Cartesian product of some
graphs. The Cartesian product of graphs G and H is the graph G?H whose vertex set
is V (G) × V (H) and whose edge set is the set of all pairs (u1,v1)(u2,v2) such that either
u1u2∈ E(G) and v1= v2, or v1v2∈ E(H) and u1= u2. The strong product of G and H is
the graph G⊠H whose vertex set is V (G)×V (H) and whose edge set is the set of all pairs
Page 22
(u1,v1)(u2,v2) such that either u1u2∈ E(G) and v1= v2, or v1v2∈ E(H) and u1= u2, or
u1u2∈ E(G) and v1v2∈ E(H). By definition, the graph G?H is the spanning subgraph of
the graph G ⊠ H. By using the definition and structure of Cartesian product, Li and Sun
derived the following.
Theorem 2.55 [46] Let G∗= G1?G2?···?Gk (k ≥ 2), where each Gi (1 ≤ i ≤ k) is
connected. Then we have
rc(G∗) ≤
k
?
i=1
rc(Gi).
Moreover, if diam(Gi) = rc(Gi) for each Gi, then the equality holds.
We know that there are infinity many graph with diam(G) = rc(G), such as the unit
interval graphs shown in Theorem 2.41. The following problem could be interesting but
maybe difficult:
Problem 2.56 Characterize the graphs G with rc(G) = diam(G), or give some sufficient
conditions to guarantee that rc(G) = diam(G).
Similar problem for src(G) can be considered.
Let G∗= G1⊠G2⊠···⊠Gk(k ≥ 2), where each Gi(1 ≤ i ≤ k) is connected. Since the
Cartesian product of any two graphs is a spanning subgraph of their strong product, G∗is
the spanning subgraph of G∗, then we have the following result.
Corollary 2.57 [46] Let G∗= G1⊠ G2⊠ ··· ⊠ Gk(k ≥ 2), where each Gi(1 ≤ i ≤ k) is
connected. Then we have
rc(G∗) ≤
k
?
i=1
rc(Gi).
For i = 1,2,··· ,r, let mi≥ 2 be given integers. Consider the graph G whose vertices
are the r-tuples b1b2···brwith bi∈ {0,1,··· ,mi−1}, and let two vertices be adjacent if the
corresponding tuples differ in precisely one place. Such a graph is called a Hamming graph.
Clearly, a graph G is a Hamming graph if and only if it can be written in the form G =
Km1?Km2···?Kmrfor some r ≥ 1, where mi≥ 2 for all i. So we call G a Hamming graph
with r factors. A Hamming graph is a hypercube (or r-cube) [28], denoted by Qr, if and only
mi= 2 for all i. The concept of Hamming graph is useful in communication networks [32].
Corollary 2.58 [46] If G is a Hamming graph with r factors, then rc(G) = r. In particular,
rc(Qr) = r.
The composition (lexicographic product) of two graphs G and H is the simple graph
G[H] with vertex set V (G)×V (H) in which (u,v) is adjacent to (u′,v′) if and only if either
Page 23
uu′∈ E(G) or u = u′and vv′∈ E(H). By definition, G[H] can be obtained from G by
substituting a copy Hvof H for every vertex v of G and by joining all vertices of Hvwith
all vertices of Huif uv ∈ E(G). Note that G[H] is connected if and only if G is connected.
By definition, it is easy to show: If G is complete, then diam(G[H]) = 1 if H is complete
(as now G[H] is complete), diam(G[H]) = 2 if H is not complete; If G is not complete, then
diam(G[H]) = diam(G). Then we have the following result.
Theorem 2.59 [46] If G and H are two graphs and G is connected, then we have
(1) if H is complete, then
rc(G[H]) ≤ rc(G).
In particular, if diam(G) = rc(G), then rc(G[H]) = rc(G).
(2) if H is not complete, then
rc(G[H]) ≤ rc(G) + 1.
In particular, if diam(G) = rc(G), then diam(G[H]) = 2 if G is complete and rc(G) ≤
diam(G[H]) ≤ rc(G) + 1 if G is not complete.
In [46], Li and Sun also investigated other graph operations, such as the union of graphs
which we will not introduce here.
2.6 An upper bound for strong rainbow connection number
The topic of rainbow connection number is fairly interesting and recently a series papers
have been published about it. The strong rainbow connection number is also interesting, and
by definition, the investigation of it is more challenging than that of the rainbow connection
number. However, there are very few papers that have been published about it. In [12],
Chartrand, Johns, McKeon and Zhang determined the precise strong rainbow connection
numbers for some special graph classes including trees, complete graphs, wheels and complete
bipartite (multipartite) graphs as shown in Subsection 2.1. However, for a general graph G,
it is almost impossible to give the precise value for src(G), so we aim to give upper bounds
for it according to some graph parameters. Li and Sun [47] derived a sharp upper bound for
src(G) according to the number of edge-disjoint triangles (if exist) in a graph G, and give
a necessary and sufficient condition for the sharpness. We need to introduce a new graph
class.
Recall that a block of a connected graph G is a maximal connected subgraph without a
cut vertex. Thus, every block of G is either a maximal 2-connected subgraph or a bridge.
We now introduce a new graph class. For a connected graph G, we say G ∈ Gt, if it satisfies
the following conditions:
C1. Each block of G is a bridge or a triangle;
C2. G contains exactly t triangles;
Page 24
C3. Each triangle contains at least one vertex of degree two in G.
By the definition, each graph G ∈ Gtis formed by (edge-disjoint) triangles and paths
(may be trivial), these triangles and paths fit together in a treelike structure, and G contains
no cycles but the t (edge-disjoint) triangles. For example, see Figure 2.4, here t = 2, and
u2
u1
u3
T1
T2
u4
u5
u6
u7
u8
G
TG
u1
u3
u4
u5
u7
u8
Figure 2.4 An example of G ∈ Gtwith t = 2.
u1, u2and u6are vertices of degree 2 in G. If a tree is obtained from a graph G ∈ Gtby
deleting one vertex of degree 2 from each triangle, then we call this tree is a D2-tree of G,
denoted by TG. For example, in Figure 2.4, TGis a D2-tree of G. Clearly, the D2-tree is
not unique, since in this example, we can obtain another D2-tree by deleting the vertex u1
instead of u2. On the other hand, we can say that any element of Gtcan be obtained from
a tree by adding t new vertices of degree 2. It is easy to show that the number of edges of
TGis m − 2t where m is the number of edges of G.
They derived the following result.
Theorem 2.60 [47] If G is a graph with m edges and t edge-disjoint triangles, then
src(G) ≤ m − 2t,
the equality holds if and only if G ∈ Gt.
In [1], Ahadi and Dehghan also derived an upper bound for strongly regular graph: if G
is an SRG(n,r,λ,µ), then src(G) ≤ ⌈(e(4µr −4µλ−6µ+1))
whether it is sharp.
1
µ⌉. However, we do not know
Unlike rainbow connection number, which is a monotone graph property (adding edges
never increases the rainbow connection number), this is not the case for the strong rainbow
connection number (see Figure 2.1 for an example). The investigation of strong rainbow
connection number is much harder than that of rainbow connection number. Chakraborty,
Fischer, Matsliah and Yuster gave the following conjecture.
Conjecture 2.61 [10] If G is a connected graph with minimum degree at least ǫn, then it
has a bounded strong rainbow connection number.
Page 25
3 Rainbow k-connectivity
In this section, we survey the results on rainbow k-connectivity. By its definition, we know
that it is difficult to derive exact value or a nice upper bound of the rainbow k-connectivity
for a general graph. Chartrand, Johns, McKeon and Zhang [13] did some basic research
on the rainbow k-connectivity of two special graph classes. They first studied the rainbow
k-connectivity of the complete graph Knfor various pairs k,n of integers, and derived the
following result:
Theorem 3.1 [13] For every integer k ≥ 2, there exists an integer f(k) such that if n ≥
f(k), then rck(Kn) = 2.
They obtained an upper bound (k + 1)2for f(k), namely f(k) ≤ (k + 1)2. Li and Sun
[44] continued their investigation, and the following result is derived:
Theorem 3.2 [44] For every integer k ≥ 2, there exists an integer f(k) = ck
c is a constant and C(k) = o(k
3
2+C(k) where
3
2) such that if n ≥ f(k), then rck(Kn) = 2.
From Theorem 3.2, we can obtain an upper bound ck
constant and C(k) = o(k
to O(k
best possible upper bound 2k, which is linear in k (see Theorem 3.3). However, the proof
of Theorem 3.2 is more structural or constructive, and informative. In the argument of
[21], Dellamonica, Magnant and Martin put forward a new concept, the rainbow (k,l)-
connectivity.
3
2 + C(k) for f(k), where c is a
3
2), that is, they improved the upper bound of f(k) from O(k2)
Dellamonica, Magnant and Martin [21] got the
3
2), a considerable improvement.
Given an edge-colored simple graph G, let l ≤ k be integers. Suppose the edges of G
are k-colored. For a,b ∈ V (G), denote by p(a,b) the maximum number of internally dis-
joint rainbow paths of length l having endpoints a and b. The rainbow (k,l)-connectivity
of G is the minimum p(a,b) among all distinct a,b ∈ V (G). Note that this new defined
rainbow (k,l)-connectivity computes the number of internally disjoint paths with the same
length l (this is distinct from the rainbow k-connectivity which, as mentioned above, com-
putes the number of colors); and by definition, for different edge-colorings, the values of
rainbow (k,l)-connectivity could be different.
From Theorem 3.1, we know that for any r, there exists an explicit 2-coloring of Krin
which the number of bi-chromatic paths of length 2 between any pair of vertices is at least
⌊√r −1⌋. Using the above definition, it is a statement about the rainbow (2,2)-connectivity
of a given 2-coloring of the edges of Kr. In [21], Dellamonica, Magnant and Martin greatly
improve and generalize the above lower bound for graphs of sufficiently large order by provid-
ing a different constructive coloring. Their construction attains asymptotically the maximum
rainbow connectivity possible.
Page 26
Theorem 3.3 [21] For any k ≥ 2 and r ≥ r0= r0(k) there exists an explicit k-coloring of
the edges of Krhaving rainbow (k,2)-connectivity
(k − 1
k
− o(1))r.
More generally, They considered the problem of finding longer rainbow paths.
Theorem 3.4 [21] For any 3 ≤ l ≤ k, there exists r0= r0(k) such that for every r ≥ r0,
there is an explicit k-coloring of the edges of Krhaving rainbow (k,2)-connectivity
(1 − o(1))
r
l − 1.
This result is also asymptotically best possible, since any collection of internally disjoint
paths of length l can contain at most
through due to Bourgain [8, 51], which consists of a powerful explicit extractor. Roughly
speaking, an (explicit) extractor is a polynomial time algorithm used to convert some spe-
cial probability distributions into uniform distributions. See [54] for a good but somewhat
outdated survey on extractors.)
r
l−1paths. Their proof employed a very recent break-
Chartrand, Johns, McKeon and Zhang [13] also investigated the rainbow k-connectivity
of r-regular complete bipartite graphs for some pairs k,r of integers with 2 ≤ k ≤ r, and
they showed:
Theorem 3.5 [13] For every integer k ≥ 2, there exists an integer r such that rck(Kr,r) = 3.
However, they could not show a similar result for complete graphs, and therefore they
left an open question: For every integer k ≥ 2, determine an integer (function) g(k), for
which rck(Kr,r) = 3 for every integer r ≥ g(k), that is, the rainbow k-connectivity of the
complete bipartite graph Kr,ris essentially 3. In [45], Li and Sun solved this question using
a similar but more complicated method to that of Theorem 3.5, and they proved:
Theorem 3.6 [45] For every integer k ≥ 2, there exists an integer g(k) = 2k⌈k
rck(Kr,r) = 3 for any r ≥ g(k).
2⌉ such that
More generally, we have the following question.
Question 3.7 [13] Does the following hold? For each integer k ≥ 2, there exists an integer
h(k) such that for every two integers s and t with h(k) ≤ s ≤ t, we have rck(Ks,t) = 3.
Recently, He and Liang [29] investigated the rainbow k-connectivity in the setting of
random graphs. They determined a sharp threshold function for the property rck(G(n,p)) ≤
d for every fixed integer d ≥ 2. This substantially generalizes a result due to Caro, Lev,
Roditty, Tuza and Yuster (see Theorem 2.46). Their main result is as follows.
Page 27
Theorem 3.8 [29] Let d ≥ 2 be a fixed integer and k = k(n) ≤ O(logn). Then p =
is a sharp threshold function for the property rck(G(n,p)) ≤ d.
(logn)1/d
n(d−1)/d
The key ingredient of their proof is the following result: With probability at least 1 −
n−Ω(1), every two different vertices of G(n,C(logn)1/d
internally disjoint paths of length exactly d.
n(d−1)/d ) are connected by at least 210dc0logn
So far, results on the rainbow k-connectivity are just those for two special graph classes,
and a sharp threshold function for the property rck(G(n,p)) ≤ d where d ≥ 2 is a fixed
integer and k = k(n) ≤ O(logn). Clearly, there are a lot of things one can do further on
this concept. As mentioned above, it is difficult to derive the precise value or a nice upper
bound for rck(G) of a κ-connected graph G, where 2 ≤ k ≤ κ. So one may consider the
following problem.
Problem 3.9 Derive a sharp upper bound for rc2(G), where G is a κ-connected graph with
κ ≥ 2. Does rc2(G) ≤ αn, where 0 < α < 1 is independent of n?
4k-rainbow index
The k-rainbow coloring as defined above, is another generalization of the rainbow coloring.
In [15], Chartrand, Okamoto and Zhang did some basic research on this topic. There is a
rather simple upper bound for rxk(G) in terms of the order of G, regardless the value of k.
Proposition 4.1 [15] Let G be a nontrivial connected graph of order n ≥ 3. For each
integer k with 3 ≤ k ≤ n − 1, rxk(G) ≤ n − 1 while rxn(G) = n − 1.
There is a class of graphs of order n ≥ 3 whose k-rainbow index attains the upper bound
of Proposition 4.1, it is an immediate consequence of Observation 2.11.
Proposition 4.2 [15] Let T be a tree of order n ≥ 3. For each integer k with 3 ≤ k ≤ n,
rxk(T) = n − 1.
There is also a rather obvious lower bound for the k-rainbow index of a connected graph
G of order n, where 3 ≤ k ≤ n. The Steiner distance d(S) of a set S of vertices in G is the
minimum size of a tree in G containing S. Such a tree is called a Steiner S-tree or simply a
Steiner tree. The k-Steiner diameter, say sdiamk(G) of G is the maximum Steiner distance
of S among all sets S with k vertices in G. Thus if k = 2 and S = {u,v}, then d(S) = d(u,v)
and the 2-Steiner diameter sdiam2(G) = diam(G). The k-Steiner diameter then provides a
lower bound for the k-rainbow index of G: for every connected graph G of order n ≥ 3 and
each integer k with 3 ≤ k ≤ n, rxk(G) ≥ sdiamk(G) ≥ k − 1.
Page 28
Theorem 4.3 [15] If G is a unicyclic graph of order n ≥ 3 and girth g ≥ 3, then
?n − 2 if k = 3 and g ≥ 4,
n − 1 if g = 3 or 4 ≤ k ≤ n.
rxk(G) =
The investigation of rxk(G) for a general k and a general graph G is rather difficult. So
one may consider rxk(G) for a special graph class, or, for a general graph and small k, such
as k = 3.
Problem 4.4 Derive a sharp upper bound for rx3(G).
There is also a generalization of the k-rainbow index, say (k,ℓ)-rainbow index rxk,ℓ, of
G which is mentioned in [15] and we will not introduce it here.
5Rainbow vertex-connection number
The above several parameters are all defined on edge-colored graphs. Here we will intro-
duce a new graph parameter which is defined on vertex-colored graphs. It is, as mentioned
above, a vertex-version of the rainbow connection number. Krivelevich and Yuster [36] put
forward this new concept and proved a theorem analogous to Theorem 2.17.
Theorem 5.1 [36] A connected graph G with n vertices has rvc(G) <
11n
δ(G).
The argument of this theorem used the concept of k-strong two-step dominating sets.
They proved that if H is a connected graph with n vertices and minimum degree δ, then it
contains aδ
an edge-coloring for G according to its connected
they showed that, with positive probability, their coloring yields a rainbow connected graph
by the Lov´ asz Local Lemma (see [3]).
2-strong two-step dominating set S whose size is at most
2n
δ+2. Then they derived
δ
2-strong two-step dominating set. And
Motivated by the method of Theorem 5.1, Li and Shi derived an improved result.
Theorem 5.2 [40] A connected graph G of order n with minimum degree δ has rvc(G) ≤
3n/(δ + 1) + 5 for δ ≥√n − 1 − 1,n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5 for 16 ≤ δ ≤
√n − 1−2, rvc(G) ≤ 4n/(δ+1)+C(δ) for 6 ≤ δ ≤ 16 where C(δ) = e
rvc(G) ≤ n/2 − 2 for δ = 5, rvc(G) ≤ 3n/5 − 8/5 for δ = 4,rvc(G) ≤ 3n/4 − 2 for δ = 3.
Moreover, an example shows that when when δ ≥√n − 1−1, and δ = 3,4,5 the bounds are
seen to be tight up to additive factors.
3 log(δ3+2δ2+3)−3(log 3−1)
δ−3
−2,
Motivated by the method of Theorem 5.1, Dong and Li [22] also proved a theorem
analogous to Theorem 2.21 for the rainbow vertex-connection version according to the degree
sum condition σ2, which is stated as the following theorem.
Page 29
Theorem 5.3 [22] For a connected graph G of order n, rvc(G) ≤ 8n−2
and σ2 ≥ 28, while for 7 ≤ σ2 ≤ 8 and 16 ≤ σ2 ≤ 27, rvc(G) ≤
9 ≤ σ2≤ 15, rvc(G) ≤10n−16
σ2+2+10 for 2 ≤ σ2≤ 6
10n−16
σ2+2+ 10, and for
σ2+2+ A(σ2), where A(σ2) = 63,41,27,20,16,13,11, respectively.
Note that by the definition of σ2, we know σ2≥ 2δ, so we have 8n−2
and hence the bound of Theorem 5.3 is an improvement to that of Theorem 5.2 in the case
16 ≤ δ ≤√n − 1 − 2.
σ2+2+10 ≤
4(n−2)
δ+1+10,
6Algorithms and computational complexity
At the end of [9], Caro, Lev, Roditty, Tuza and Yuster gave two conjectures (see Conjec-
ture 4.1 and Conjecture 4.2 in [9]) on the complexity of determining the rainbow connection
numbers of graphs. Chakraborty, Fischer, Matsliah and Yuster [10] solved these two conjec-
tures by the following theorem.
Theorem 6.1 [10] Given a graph G, deciding if rc(G) = 2 is NP-Complete. In particular,
computing rc(G) is NP-Hard.
Chakraborty, Fischer, Matsliah and Yuster divided the proof of Theorem 6.1 into three
steps: the first step is showing the computational equivalence of the problem of rainbow
connection number 2, that asks for a red-blue edge coloring in which all vertex pairs have
a rainbow path connecting them, to the problem of subset rainbow connection number 2,
that asks for a red-blue coloring in which every pair of vertices in a given subset of pairs has
a rainbow path connecting them. In the second step, they reduced the problem of extending
to rainbow connection number 2, asking whether a given partial red-blue coloring can be
completed to obtain a rainbow connected graph, to the problem of subset rainbow connection
number 2. Finally, the proof of Theorem 6.1 is completed by reducing 3-SAT to the problem
of extending to rainbow connection number 2.
Chakraborty, Fischer, Matsliah and Yuster [10] also raised the following problem.
Problem 6.2 [10] Suppose that we are given a graph G for which we are told that rc(G) = 2.
Can we rainbow-color it in polynomial time with o(n) colors ?
For the usual coloring problem, this version has been well studied. It is known that if
a graph is 3-colorable (in the usual sense), then there is a polynomial time algorithm that
colors it with˜O(n3/14) colors [5].
Suppose we are given an edge coloring of the graph. Is it then easier to verify whether
the colored graph is rainbow connected? Clearly, if the number of colors is constant then this
problem becomes easy. However, if the coloring is arbitrary (with an unbounded number of
colors), the problem becomes NP-Complete.
Page 30
Theorem 6.3 [10] The following problem is NP-Complete: Given an edge-colored graph G,
check whether the given coloring makes G rainbow connected.
For the proof of Theorem 6.3, Chakraborty, Fischer, Matsliah and Yuster first showed
that the s −t version of the problem is NP-Complete. That is, given two vertices s and t of
an edge-colored graph, decide whether there is a rainbow path connecting them. Then they
reduced the problem of Theorem 6.3 from it.
More generally it has been shown in [37], that for any fixed k ≥ 2, deciding if rc(G) = k
is NP-complete.
In [10], Chakraborty, Fischer, Matsliah and Yuster also derived some positive algorithmic
results. Parts of the following two results were shown in Theorems 2.16 and 2.48. They
proved:
Theorem 6.4 [10] For every ǫ > 0 there is a constant C = C(ǫ) such that if G is a connected
graph with n vertices and minimum degree at least ǫn, then rc(G) ≤ C. Furthermore, there
is a polynomial time algorithm that constructs a corresponding coloring for a fixed ǫ.
As mentioned above, Theorem 6.4 is based upon a modified degree-form version of Sze-
mer´ edis Regularity Lemma that they proved and that may be useful in other applications.
From their algorithm it is also not hard to find a probabilistic polynomial time algorithm
for finding this coloring with high probability (using on the way the algorithmic version of
the Regularity Lemma from [2] or [26]).
Theorem 6.5 [10] If G is an n-vertex graph with diameter 2 and minimum degree at least
8logn, then rc(G) ≤ 3. Furthermore, such a coloring is given with high probability by a
uniformly random 3-edge-coloring of the graph G, and can also be found by a polynomial
time deterministic algorithm.
As mentioned, He and Liang [29] investigated the rainbow k-connectivity in the setting
of random graphs. They also investigated rainbow k-connectivity from the algorithmic point
of view. The NP-hardness of determining rc(G) was shown by Chakraborty et al. as shown
above. They showed that the problem (even the search version) becomes easy in random
graphs.
Theorem 6.6 [29] For any constant ǫ ∈ [0,1), p = n−ǫ(1±o(1))and k ≤ O(logn), there
is a randomized polynomial time algorithm that, with probability 1 − o(1), makes G(n,p)
rainbow-k-connected using at most one more than the optimal number of colors.
Since almost all natural edge probability functions p encountered in various scenarios have
such form, their result is quite strong. Note that G(n,n−ǫ) is almost surely disconnected
Page 31
when ǫ > 1 [24], which makes the problem become trivial. Therefore they ignored these
cases.
Recall that the values of rainbow (k,l)-connectivity may be different for distinct edge-
colorings. In [21], Dellamonica, Magnant and Martin derived that a random k-coloring
of a sufficiently large complete graph has asymptotically optimal rainbow rainbow (k,l)-
connectivity (see Theorems 3.3 and 3.4). They obtained an explicit edge-coloring. By
explicit, we mean that they gave a polynomial time algorithm to compute such an edge-
coloring.
Recently, the computational complexity of rainbow vertex-connection numbers has been
determined by Chen, Li and Shi [18].
Motivated by the proofs of Theorems 6.1 and 6.3, they derived two corresponding results
to the rainbow vertex-connection.
Theorem 6.7 [18] Given a graph G, deciding if rvc(G) = 2 is NP-Complete. In particular,
computing rvc(G) is NP-Hard.
Theorem 6.8 [18] The following problem is NP-Complete: given a vertex-colored graph G,
check whether the given coloring makes G rainbow vertex-connected.
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