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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

1Introduction

1.1Motivation 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.

1.2Terminology 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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2.2Upper 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

2.3For 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