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arXiv:1001.0287v1 [math.CO] 2 Jan 2010

Upper bounds for the rainbow connection

numbers of line graphs∗

Xueliang Li, Yuefang Sun

Center for Combinatorics and LPMC-TJKLC

Nankai University, Tianjin 300071, P.R. China

E-mail: lxl@nankai.edu.cn; syf@cfc.nankai.edu.cn

Abstract

A path in an edge-colored graph G, where adjacent edges may be colored

the same, is called a rainbow path if no two edges of it are colored the same.

A nontrivial connected graph G is rainbow connected if for any two vertices of

G there is a rainbow path connecting them. The rainbow connection number

of G, denoted by rc(G), is defined as the smallest number of colors by using

which there is a coloring such that G is rainbow connected. In this paper,

we mainly study the rainbow connection number of the line graph of a graph

which contains triangles and get two sharp upper bounds for rc(L(G)), in

terms of the number of edge-disjoint triangles of G where L(G) is the line

graph of G. We also give results on the iterated line graphs.

Keywords: rainbow path, rainbow connection number, (iterated) line graph,

edge-disjoint triangles.

AMS Subject Classification 2000: 05C15, 05C40

1 Introduction

All graphs in this paper are simple, finite and undirected. Let G be a nontrivial

connected graph with an edge coloring c : E(G) → {1,2,··· ,k}, k ∈ N, where

adjacent edges may be colored the same. A path of G is called rainbow if no two

edges of it are colored the same. An edge-colored graph G is rainbow connected if

for any two vertices there is a rainbow path connecting them. 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, i.e., the coloring such that

each edge has a distinct color. Thus, we define the rainbow connection number

∗Supported by NSFC, PCSIRT and the “973” program.

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of a connected graph G, denoted by rc(G), as the smallest number of colors for

which there is an edge coloring of G such that G is rainbow connected. An easy

observation is that if G has n vertices then rc(G) ≤ n − 1, since one may color the

edges of a spanning tree with distinct colors, and color the remaining edges with

one of the colors already used. Generally, if G1is a connected spanning subgraph of

G, then rc(G) ≤ rc(G1). We notice the trivial fact that rc(G) = 1 if and only if G

is complete, and the fact that rc(G) = n−1 if and only if G is a tree, as well as the

easy observation that a cycle with k > 3 vertices has rainbow connection number

⌈k

2⌉ ([2]). Since a Hamiltonian graph G has a Hamiltonian cycle which contains all

n vertices, then G has rainbow connection number at most ⌈n

clearly, rc(G) ≥ diam(G) where diam(G) denotes the diameter of G.

2⌉. Also notice that,

Chartrand et al. in [2] determined that the rainbow connection numbers of

some graphs including trees, cycles, wheels, complete bipartite graphs and complete

multipartite graphs. Caro et al. [3] gave some results on general graphs in terms

of some graph parameters, such as the order or the minimum degree of a graph.

They observed that rc(G) can be bounded by a function of δ(G), the minimum

degree of G. They proved that if δ(G) ≥ 3 then rc(G) ≤ αn where α < 1 is

a constant and n = |V (G)|. They conjectured that α = 3/4 suffices and proved

that α < 5/6. Specifically, it was proved in [3] that if δ = δ(G) then rc(G) ≤

min{lnδ

δ

}. Some special 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 [3] the authors got a very good upper bound for the rainbow connection

number of a 2-connected graph according to their ear-decomposition. And in [6],

we studied the rainbow connection numbers of line graphs of triangle-free graphs

in the light of particular properties of line graphs of triangle-free graphs shown

in [4], and particularly, of 2-connected triangle-free graphs according to their ear

decompositions. However, we did not get bounds of the rainbow connection numbers

for line graphs that do contain triangles. In this paper, we aim to investigate the

remaining case, i.e., line graphs that do contain triangles, and give two sharp upper

bounds in terms of the number of edge-disjoint triangles.

δn(1 + oδ(1)),n4lnδ+3

We use V (G), E(G) for the sets of vertices and 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 E1 of E(G), let G[E1] denote the

subgraph induced by E1. Let G be a set of graphs, then V (G) =

E(G) =?

of G, and a maximal clique is a clique that is not contained in any larger clique.

The clique graph K(G) of G is the intersection graph of the maximal cliques of

G–that is, the vertices of K(G) correspond to the maximal cliques of G, and two of

these vertices are joined by an edge if and only if the corresponding maximal cliques

intersect. Let [n] = {1,··· ,n} denote the set of the first n natural numbers. For a

set S, |S| denotes the cardinality of S. We follow the notations and terminology of

[1] for those not defined here.

?

G∈GV (G),

G∈GE(G). We define a clique in a graph G to be a complete subgraph

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2 Some basic observations

We first list two observations which were given in [6] and will be used in the

sequel.

Observation 2.1 ([6]) 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] and rc(Gi) = ci, then

rc(G) ≤

t

?

i=1

ci.

Let G be a connected graph, and X a proper subset of V (G). To shrink X is to

delete all the edges between vertices of X and then identify the vertices of X into a

single vertex, namely w. We denote the resulting graph by G/X.

Observation 2.2 ([6]) Let G′and G be two connected graphs, where G′is obtained

from G by shrinking a proper subset X of V (G), that is, G′= G/X, such that any

two vertices of X have no common adjacent vertex in V \ X. Then

rc(G′) ≤ rc(G).

Now we introduce two graph operations and two corresponding results which

will be used later.

Operation 1: As shown in Figure 2.1, for any edge e = uv ∈ G with min{degG(u),

degG(v)} ≥ 2, we first subdivide e, then replace the new vertex with two new vertices

ue,vewith degG′(ue) = degG′(ve) = 1 where G′is the new graph.

Operation 1

u

v

e

u

v

ue

ve

G

G′

Figure 2.1 G′is obtained from G by doing Operation 1 to edge e.

Since degG′(ue) = degG′(ve) = 1 in G′, and by the definition of a line graph, it is

easy to show that L(G) can be obtained from L(G′) by shrinking a vertex set of two

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nonadjacent vertices (these two vertices correspond to edges uue, vve, and belong to

cliques ?S(u)?, ?S(v)?, respectively, in L(G′)). Recall that the line graph of a graph

G is the graph L(G) whose vertex set V (L(G)) = E(G) and two vertices e1, e2of

L(G) are adjacent if and only if they are adjacent in G. So by Observation 2.2, we

have

Observation 2.3 If graph G′is obtained from a connected graph G by doing Oper-

ation 1 at some edge e ∈ G, then

rc(L(G)) ≤ rc(L(G′)).

Operation 2. As shown in Figure 2.2, v is a common vertex of a set of edge-disjoint

triangles in G. We replace v by two nonadjacent vertices v′and v′′such that v′is the

common vertex of some triangles, and v′′is the common vertex of the rest triangles.

G

v

v′′

v′

G′

Operation 2

Figure 2.2 Figure of Operation 2.

Since during this procedure, the number of edges does not change, the order of

the line graph L(G) is equal to that of L(G′). Furthermore, by the definition of a

line graph, L(G′) is a spanning subgraph of L(G). So we have

Observation 2.4 If a connected graph G′is obtained from a connected graph G by

doing Operation 2 at some vertex v ∈ G, then

rc(L(G)) ≤ rc(L(G′)).

3Main results

3.1 A sharp upper bound

Recall that the star, S(v), at a vertex v of graph G, is the set of all edges incident

to v. A clique decomposition of G is a collection C of cliques such that each edge

of G occurs in exactly one clique in C.

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We now introduce a new terminology. For a connected graph G, we call G a

clique-tree-structure, if it satisfies the following condition:

T1. 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 condition T1, 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 the clique-tree(forest)-structure is the number of its maximal cliques. An

example of clique-forest-structure is shown in Figure 3.1.

Figure 3.1 A clique-forest-structure with size 6 and 2 components.

If each block of a 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 triangle-forest-structure. A clique-tree-structure G is called a clique-

path-structure if the clique graph K(G) is a path.

For a connected graph G, we call G a clique-cycle-structure, if it satisfies the

following three conditions:

C1. G has at least three maximal cliques;

C2. Each edge belongs to exactly one maximal clique;

C3. The clique graph is a cycle. (In particular, if each maximal clique is a

triangle, then it is a triangle-cycle-structure. An example of triangle-cycle-structure

is shown in Figure 3.2.)

Figure 3.2 An example of triangle-cycle-structure.

An inner vertex of a graph is a vertex with degree at least two. For a graph G,

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we use V2to denote the set of all inner vertices of G. Let n1= |{v : degG(v) = 1}|,

n2= |V2|. ?S(v)? is the subgraph of L(G) induced by S(v), clearly it is a clique

of L(G). Let K0= {?S(v)? : v ∈ V (G)}, K = {?S(v)? : v ∈ V2}. It is easy to

show that K0is a clique decomposition of L(G) ([5]) and each vertex of the line

graph belongs to at most two elements of K0. We know that each element ?S(v)?

of K0\ K , a single vertex of L(G), is contained in the clique induced by u that is

adjacent to v in G. So K is a clique decomposition of L(G).

Let X and Y be sets of vertices of a graph G. We denote by E[X,Y ] the set of

the edges of G with one end in X and the other end in Y . If Y = X, we simply

write E(X) for E[X,X]. When Y = V \X, the set E[X,Y ] is called the edge cut of

G associated with X, and is denoted by ∂(X).

Theorem 3.1 For any set T of t edge-disjoint triangles of a connected graph G, if

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.

Proof.

be a set of t edge-disjoint triangles of G such that the subgraph of G, G[E(Ti)],

induced by each E(Ti) is a component of the subgraph G[E(T )], that is, a triangle-

tree-structure of size ti.

Let T =?c

i=1Ti=?c

i=1{Ti,jiis a triangle of G: 1 ≤ ji≤ ti}(?c

i=1ti= t)

In G, for each 1 ≤ i ≤ c, let Gi= G[E(Ti)], Vi = V (Gi), Ei= E(Ti); E0

E(Vi) ∪ ∂(Vi) ⊇ Ei, and G0

Operation 1 at each edge e ∈ E(Vi)\Eifor 1 ≤ i ≤ c, and we denote by G′

new subgraph (of G′) corresponding to G0

we have rc(L(G)) ≤ rc(L(G′)).

i=

i= G[E0

i]. We obtain a new graph G′from G by doing

ithe

i. Applying Observation 2.3 repeatedly,

Next we will show rc(L(G′)) ≤ n2− t. By previous discussion, we know that

K = {?S(v)? : v ∈ V2} =

c?

i=1

{?S(v)? : v ∈ Vi}

?

{?S(v)? : v ∈ V2\

c?

i=1

Vi}

is a clique partition of L(G). So

{E(?S(v)?) : v ∈ Vi}c

i=1

?

{E(?S(v)?) : v ∈ V2\

c?

i=1

Vi},

that is,

{E(L(G0

i))}c

i=1

?

{E(?S(v)?) : v ∈ V2\

c?

i=1

Vi}

is an edge partition of L(G). So

{E(L(G′

i))}c

i=1

?

{E(?S(v)?) : v ∈ V2\

c?

i=1

Vi}

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is an edge partition of L(G′). By Observation 2.1, we have

rc(L(G′)) ≤

c

?

i=1

rc(L(G′

i)) +

?

v∈V2\?c

i=1Vi

rc(?S(v)?).

We know |Vi| = 2ti+ 1, since the triangle-tree-structure Gihas size ti. So

rc(L(G′)) ≤

c

?

i=1

rc(L(G′

i)) + (n2− 2t − c).

In order to get rc(L(G′)) ≤ n2− t, we need to show rc(L(G′

i)) ≤ ti+ 1.

Claim. rc(L(G′

i)) ≤ ti+ 1.

Proof of the Claim. Since the graph G′

1 at each edge e ∈ E(Vi)\Ei, G′

ti}. We will show that there is a (ti+ 1)-rainbow coloring of L(G′

on ti. For ti = 1, G′

2-rainbow coloring as shown in Figure 3.3. We give color 1 to the edges of ?S(u)?

incident with vertex e1, edges of ?S(v)? incident with vertex e2, and edges of ?S(w)?

incident with vertex e3; We then give color 2 to the edges of ?S(u)? incident with

vertex e3, edges of ?S(v)? incident with vertex e1, and edges of ?S(w)? incident with

vertex e2; Finally, give color 2 to the rest of the edges. It is easy to show that this is

a rainbow coloring. So, the above conclusion holds for the case ti= 1. We assume

iis obtained from Giby doing Operation

icontains exactly titriangles: {Ti,ji∈ Ti: 1 ≤ ji≤

i) by induction

icontains exactly one triangle, and we give its line graph a

u

v

w

e1

e2

e3

1

1

1

2

2

2

Figure 3.3 2-rainbow coloring of line graph of graph with exactly one triangle.

that the conclusion holds for the case ti= h − 1, and now show that it holds for

the case ti= h. By the definition of triangle-tree-structure, there must exist one

triangle: T = {u,v,w}, which has exactly one common vertex, namely u, with other

triangles and v,w do not belong to any other triangle. We now obtain a new graph

G′

be obtained from L(G′

into two vertices {e′

hypothesis, rc(L(H1)) ≤ h. So the subgraph L(G′

a rainbow h-coloring, and we now color the edges of ?S(v)? and ?S(w)? in the graph

L(G′

ifrom G′

iby doing Operation 1 at edges e1 and e2. It is clear that L(G′

i) by subdividing vertex e1into two vertices {e′

2,e′′

i) can

1,e′′

1} and e2

2}. Since H1has h−1 (edge-disjoint) triangles, by induction

i)\{?S(v)?,?S(w)?}∼= L(H1) has

i) as follows: give a new color to the edges of ?S(w)? incident with vertex e1, and

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edges of ?S(v)? incident with vertex e2. Let e3= vw, we then give any one color, say

c1, of the former h colors to the edges of ?S(w)? incident with vertex e3, and give

a distinct color, say c2, of the former h colors to the edges of ?S(v)? incident with

vertex e3. It is easy to show that, with above coloring, L(G′

and so L(G′

i) is rainbow connected,

i) ≤ ti+ 1 holds for ti= h.

u

v

w

e1

e2

u

v

w

G′

i

G′i

H1

e′

1

e′′

1

e′

2

e′′

2

H2

Figure 3.4 G′

iis obtained from Giby doing Operation 1 to edges e1and e2.

For the sharpness of the upper bound, see Example 3.3.

We call a set of triangles independent if any two of them are vertex-disjoint. Since

each single triangle is a triangle-tree-structure, applying Theorem 3.1, we have the

following corollary, and for the sharpness of the upper bound, see Example 3.3:

Corollary 3.2 If G is a connected graph with t′independent triangles, then

rc(L(G)) ≤ n2− t′.

Moreover, the bound is sharp.

Example 3.3 As shown in Figure 3.5, G consists of t (independent) triangles and

t − 1 edges which do not belong to any triangles. Since G has 3t inner vertices, by

Theorem 3.1(Corollary 3.2), we know rc(L(G)) ≤ 2t; on the other hand, it is easy to

show that the diameter of the line graph L(G) is 2t, and so we have rc(L(G)) = 2t.

3.2Another upper bound from Theorem 3.1

Now we use the notation similar to that of Theorem 3.1. Let T =?c

i=1{Ti,jiis a triangle of G: 1 ≤ ji≤ ti}(?c

triangles of G; the subgraph of G, G[E(Ti)], induced by each E(Ti) is a connected

i=1Ti=

?c

i=1ti= t) be a set of t edge-disjoint

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G

Figure 3.5 Figure of Example 3.3.

component of the subgraph G[E(T )] which may not be a triangle-tree-structure,

that is, it may contain a triangle-cycle-structure as a subgraph.

In G, for each 1 ≤ i ≤ c, let Gi= G[E(Ti)], Vi = V (Gi), Ei= E(Ti); E0

E(Vi) ∪ ∂(Vi) ⊇ Ei, and G0

Operation 1 at each edge e ∈ E(Vi)\Eifor 1 ≤ i ≤ c, and we denote by G′

new subgraph (of G′) corresponding to G0

we have rc(L(G)) ≤ rc(L(G′)). Now each G′

where 1 ≤ ji≤ ti. We now obtain a new graph G′′from G′by doing Operation 2 to

those G′

corresponding to G′

times of doing Operation 2 needed during above procedure.

op(G[E(T )]) (minimum times of doing Operation 2 needed for G[E(T )] such that

the resulting graph contains no triangle-cycle-structure). Since there are op(G′)

new inner vertices totally produced, and by Observation 2.4 and the discussion of

Theorem 3.1, we have

i=

i= G[E0

i]. We obtain a new graph G′from G by doing

ithe

i. Applying Observation 2.3 repeatedly,

icontains exactly ti triangles: Ti,ji

is which contain triangle-cycle-structures such that each subgraph (of G′′) G′′

icontains no triangle-cycle-structure. Let op(G′) be the minimum

i

Clearly, op(G′) =

Lemma 3.4 For any set T of t edge-disjoint triangles of a connected graph G with

n2inner vertices, we have

rc(L(G)) ≤ n2+ op(G[E(T )]) − t.

We know that a triangle-forest-structure of size ℓ contains 2ℓ+c (inner) vertices

where c is the number of components of it. Operation 2 does not change the number

of edge-disjoint triangles, but the number of inner vertices increases 1 after we did

Operation 2 once. Then it is easy to show that after doing Operation 2 op(G[E(T )])

times, the number of inner vertices of the new graph is op(G[E(T )]) + n2=2t+1+n′

where n′

disjoint triangles. So, by Lemma 3.4, we have the following result and for the

sharpness of the bound see Example 3.6.

2

2denotes the number of inner vertices not covered by the original t edge-

Theorem 3.5 If G is a connected graph, T is a set of t edge-disjoint triangles

that cover all but n′

subgraph G[E(T )], then

rc(L(G)) ≤ t + n′

2inner vertices of G and c is the number of components of the

2+ c.

Moreover, the bound is sharp.

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Example 3.6 Let G be a graph shown in Figure 3.6. The set T = {ui,vi,ui+1}k−1

is a set of k−1 edge-disjoint triangles, n′

rc(L(G)) ≤ k + 1; on the other hand, it is easy to show that the diameter of L(G)

is k + 1, and so rc(L(G)) = k + 1.

i=1

2= 1 and c = 1. By Theorem 3.5, we have

x

y

u1

u2

u3

uk

v1

v2

v3

G

vk

Figure 3.6 Figure of Example 3.6.

4 Bounds for iterated line graphs

Recall that the iterated line graph of a graph G, denoted by L2(G), is the line

graph of the graph L(G). The following corollary deduced from Theorem 3.4 is an

upper bound of the rainbow connection numbers of iterated line graphs of connected

cubic graphs.

Corollary 4.1 If G is a connected cubic graph with n vertices, then

rc(L2(G)) ≤ n + 1.

Proof.

clique ?S(v)? in L(G) corresponding to each vertex v is a triangle. We know that

K = {?S(v)? : v ∈ V2} = {?S(v)? : v ∈ V } is a clique decomposition of L(G). Let

T = K . Then T is a set of n edge-disjoint triangles that cover all vertices of L(G)

and L(G) = L(G)[E(T )]. So n′

holds.

Since G is a connected cubic graph, each vertex is an inner vertex and the

2= 0 and c = 1, and by Theorem 3.5, the conclusion

In graph G, we call a path of length k a pendent k-length path if one of its end

vertex has degree 1 and all inner vertices has degree 2. By definition, a pendent

k-length path contains a pendent ℓ-length path(1 ≤ ℓ ≤ k). A pendent 1-length

path is a pendent edge.

Theorem 4.2 Let G be a connected graph with m edges and m1pendent 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.

Proof.

has m − m1 inner vertices. By the discussion before Theorem 3.1, we give each

Now L(G) is a graph with m vertices and m1 pendent edges. Then it

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clique ?S(v)? in L2(G) a fresh color, where v is an inner vertex of L(G). It is

easy to show that this gives a rainbow (m − m1)-edge-coloring of L2(G), and so

rc(L2(G)) ≤ m − m1.

If G is a path of length at least 3, then the equality clearly holds.

If G contains a cycle, then L(G) clearly contains a minimal cycle C : v1,··· ,vℓ.

So L2(G) contains a clique-cycle-structure of size ℓ which is formed by cliques

{?S(v)?}ℓ

colors to make sure that it is rainbow connected. The rest part of L2(G) is formed by

cliques {?S(v)? : v ∈ V2\{vi}ℓ

give each ?S(v)? a fresh color, by Observation 2.1, rc(L2(G) ≤ ⌈ℓ+1

m − m1.

i=1. By a theorem in [6], we know that this structure needs at most ⌈ℓ+1

2⌉

i=1} where V2is the set of inner vertices of L(G). We

2⌉+(m−m1−ℓ) <

If G is a tree with a vertex of degree at least 3, then L(G) contains a cycle, a

similar argument will show rc(L2(G)) < m − m1.

So G is a tree with maximum degree 2, and hence it must be a path of length at

least 3.

References

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[2] G. Chartrand, G.L. Johns, K.A. McKeon, P. Zhang, Rainbow connection in graphs,

Math. Bohem. 133(2008), 85-98.

[3] Y. Caro, A. Lev, Y. Roditty, Z. Tuza, R. Yuster, On rainbow connection, Electron.

J. Combin. 15 (2008), R57.

[4] S.T. Hedetniemi, P.J. Slater, Line graphs of triangleless graphs and iterated clique

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MR49#151.

[5] Bo-Jr Li, G.J. Chang, Clique coverings and partitions of line graphs, Discrete

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11

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