The minimum color degree and a large rainbow cycle in an edge-colored graph

The minimum color degree and a large rainbow cycle in an edge-colored graph

Abstract

Let be an edge-colored graph with vertices. A subgraph of is called a rainbow subgraph of if the colors of each pair of the edges in are distinct. We define the minimum color degree of to be the smallest number of the colors of the edges that are incident to a vertex , for all . Suppose that contains no rainbow-cycle subgraph of length four. We show that if the minimum color degree of is at least , then contains a rainbow-cycle subgraph of length at least , where . Moreover, if the condition of is restricted to a triangle-free graph that contains a rainbow path of length at least , then the lower bound of the minimum color degree of that guarantees an existence of a rainbow-cycle subgraph of length to at least can be reduced to .

1 Introduction

For a finite simple undirected graph , where and , we define an edge coloring of to be a function, where . For a subgraph of , the edge coloring of is the restriction of to . If the colors of each pair of the edges in are distinct, then is called a rainbow subgraph or a heterochromatic subgraph of . The works related to rainbow subgraphs in various types including paths, trees and cycles appear in the survey by M. Kano and X. Li [4].

In this paper, we are interested in a condition of an existence of a large rainbow-cycle subgraph in . In 2005, J.J. Montellano-Ballesteros and V. Neumann-Lara [6] solved the conjecture of Erdős, Simonovits and Sós on the rainbow cycle. They gave a condition of the number of the colors in a complete graph that guarantees an existence of a rainbow-cycle subgraph. Let be the set of the colors of the edges appearing in . They showed that if , then contains a rainbow cycle of length at least . Meanwhile, H. J. Broesma et. al. [1] showed that if , then contains a rainbow cycle of length at least .

In 2012, H. Li and G. Wang [5] took a different approach and studied the existence of the rainbow-cycle subgraph of by considering its minimum color degree of , which is the smallest number of all distinct colors of the edges that are incident to a vertex , for all . In Theorem 1.1, H. Li and G. Wang showed that, for a triangle-free graph with at least eight vertices, if , then contains a rainbow cycle of length at least . The largest lower bound of the guaranteed length of the rainbow-cycle subgraph of in Theorem 1.1 is at most .

Theorem 1.1.

[5] Let be a triangle-free graph with vertices, where . If , then contains a rainbow cycle of length at least .

In 2016, R. Čada, A. Kaneko and Z. Ryjáček [7] gave a sufficient condition of the minimum color degree of so that contains a rainbow-cycle subgraph of length at least four in Theorem 1.2. We adopt the notations used by Čada et. al. [7]. For a pair of vertices , where , let be the color of the edge . For a subgraph of , the notation is the set of the colors of the edges joining the vertex and a vertex in . Let be a path and let be a subpath of that starts at and ends at , where . Throughout this paper, we let be an edge-colored graph with an edge-coloring and .

Theorem 1.2.

[7] If , then contains a rainbow cycle of length at least four.

Lemma 1.3 and 1.4 are used to prove Theorem 1.2. We will later use these lemmas to prove the main theorem.

Lemma 1.3.

[7] For a graph , let be the longest rainbow path of . If contains no rainbow cycle of length at least , where , then for any color and vertex , where , there is an edge such that .

Lemma 1.4.

[7] For a graph , let be the longest rainbow path of . If contains no rainbow cycle of length at least , where , then for any positive integers , such that ,

 |c(u1,ukPup−(t−1))∩c(up,usPup−(k−1))|≤1.

From Theorem 1.2, Čada et. gave the following conjecture.

Conjecture 1.5.

If , then contains a rainbow-cycle subgraph of lenght at least .

In this work, by using the method appearing in Theorem 1.2, we give a progress toward Conjecture 1.5. In our main theorem, we showed that if does not contain a rainbow cycle of length four and , then contains a rainbow cycle of length at least . In Section 3, we will discuss the result in the main theorem in comparison with Theorem 1.1. We showed that by restricting the condition of Theorem 1.1 to be rainbow--free, we can ignore the condition that the graph is triangle-free and the length of the guaranteed rainbow-cycle subgraph can be at least larger than in Theorem 1.1.

In order to apply Theorem 2.1, we require an existence of a long rainbow path. In 2014, A. Das, S. V. Subrahmanya and P. Suresh [3] gave a lower bound of the length of the longest rainbow path in the term of the lower bound of the minimum color degree in Theorem 1.6.

Theorem 1.6.

[3] Let be an edge-colored graph, where and . The maximum length of the rainbow paths in is at least .

In 2016, H. Chen and X. Li gave a larger lower bound of the length of the longest rainbow path with respect to the lower bound of .

Theorem 1.7.

[2] Let be an edge-colored graph. If , then contains a rainbow path of length at least .

2 Main Results

Theorem 2.1 is obtained by using the method in Theorem 1.2 with some generalization. As a result, Theorem 2.1 gives a lower bound of guaranteeing a rainbow-cycle subgraph of length at least , where , in a graph containing no rainbow-cycle subgraph of length four. The largest length of the guaranteed rainbow-cycle subgraph of in Theorem 1.1 is at most , while, it is in Theorem 2.1.

Theorem 2.1.

Let be a graph with no rainbow-cycle subgraph of length four. If , then contains a rainbow-cycle subgraph of length at least , where .

Proof.

For a fixed . Suppose that contains no rainbow-cycle subgraph of length at least . Since , it follows that . Let be the longest rainbow path in . By Theorem 1.7, it follows that

 p≥⌈(n+3k−4)3⌉+1≥2k.

Let be such that and . So and . Let and . By Lemma 1.4, we have . Let be the subgraph of induced by and let

 C0=(c(u1,PC)∖c(u1,P))∩(c(up,PC)∖c(up,P)).

So . By Lemma 1.3, if , then . If , then ; otherwise, there exists a longer rainbow path.

By Lemma 1.3, if , then there exists an edge such that ; hence, the path is not rainbow which is a contradiction. So, and, similarly, . Since is rainbow and contains no rainbow-cycle subgraph of length at least , it follows that if , then . Let

 ϵ1 ={1, if c(u1u2)∉A,0, otherwise, ϵ2 ={1, if c(up−1up)∉B,0, otherwise, ϵ′1 ={1, if c(u1up)∉A∪c(u1u2),0, otherwise, ϵ′2 ={1, if c(u1up)∉B∪c(upup−1),0, otherwise.

So,

 |V(P)| =|E(P)|+1 ≥|c(E(P))|+1 ≥|A∪B|+|C0|+ϵ1+ϵ2+ϵ′1ϵ′2+1 ≥|A|+|B|+|C0|+ϵ1+ϵ2+ϵ′1ϵ′2. (1)

Let and . For each , we have ; otherwise, there exists a longer rainbow path. By the construction of and , it follows that

 C1∩C2=((c(u1,PC)∖c(u1,P))∩(c(u1,PC)∖c(up,P)))∖C0=∅.

Suppose and . Let be a subset of , where , for all . Similarly, let be a subset of , where , for all . Next, we choose one vertex from each and one vertex from , where and . We have and . Since and contains no rainbow-cycle subgraph of length four, it follows that

 |{x1,…,x|C1|}∩{y1,…,y|C2|}|≤1.

So,

 |V(PC)|≥|C0|+|C1|+|C2|−1. (2)

We note that and . Suppose and . It follows that the number of the colors in is also less than by at least . So,

 |c(u1,P)|≤(p−1)−l′=t+k−4+l.

Hence,

 |c(u1,P)∖A∪{c(u1u2)}| =|c(u1,P)|−|A∪{c(u1u2)}| ≤(t+k−4+l)−l =t+k−4. (3)

Analogously, since , it follows that

 |c(up,P)∖(B∪{c(up−1up})|≤s+k−3. (4)

Next, we consider the number of the colors of the edges that are incident to . Since, and are all disjoint, it follows that

 |A|+|C0|+|C1|+ϵ1+ϵ′1+(k+t−4)≥dc(u1)≥δc(G). (5)

So, if we omit , then

 |A|+|C0|+|C1|+ϵ1≥δc(G)−k−t+3. (6)

Similarly,

 |B|+|C0|+|C2|+ϵ2+ϵ′2+k+s−3≥dc(up)≥δc(G).

Hence, if we omit , then

 |B|+|C0|+|C2|+ϵ2≥δc(G)−k−s+2. (7)

By (1), (2), (6) and (7),

 |V(P)|+|V(PC)| ≥(|A|+|C0|+|C1|+ϵ1)+(|B|+|C0|+|C2|+ϵ2)−1+ϵ′1ϵ′2 ≥(δc−k−t+3)+(δc−k−s+2)−1 =2δc−2k−t−s+4 =2δc−3k+4>n,

which is a contradiction. Therefore, contains a rainbow-cycle subgraph of length at least . ∎

If is triangle-free, then the bound of sizes of , and can be reduced to

• .

Hence, (2) and (4) can be reduced to

 |c(u1,P)∖A∪{c(u1u2)}|≤t+k−22 (8)

and

 |c(up,P)∖(B∪{c(up−1up})|≤s+k−12. (9)

Thus, if is triangle-free with no rainbow-cycle subgraph of length four, then the lower bound of can be reduced to ; however, we need a condition of the existence of a rainbow path of length at least in .

Theorem 2.2.

Let be a triangle-free graph with no rainbow-cycle subgraph of length four. If contains a rainbow path of length at least and , then contains a rainbow cycle of length at least , where .

We note that if we omit the condition of the length of the longest rainbow path in Theorem 2.2, then the condition of only the minimum degree is not able to guarantee the existence of the needed rainbow path. To omit such condition, we combine Theorem 1.7 and Theorem 2.1 which result to Corollary 2.3 as follows.

Corollary 2.3.

If and , then contains a rainbow cycle of length at least .

Proof.

By Theorem 1.7, there exists a path , where

 p≥⌈2n+3k−16⌉+1>3k2.

Hence, by Theorem 2.1, there exists a rainbow-cycle subgraph of length at least . ∎

3 Discussion

In this section, we compare the length of the rainbow-cycle subgraphs guaranteed by Theorem 1.1 and Theorem 2.1. We consider a graph satisfying the conditions in both theorems. The maximum length of the guaranteed rainbow-cycle subgraph in Theorem 1.1 is at most , whereas, the guaranteed length of the rainbow-cycle subgraph of in Theorem 2.1 can be up to . Let be such that . Theorem 2.1 implies that contains a rainbow-cycle subgraph of length at least . Next, we show that the guaranteed length of the rainbow-cycle subgraph obtained by Theorem 1.1 of such graph is less than . Since , it follows that

 δc(G)=n+3k−12 or δc(G)=n+3k−22.

So,

 δc(G)=(3n4+1)+6k−n−64 or δc(G)=(3n4+1)+6k−n−84.

We note that if , then Theorem 1.1 is not applicable. We consider case . By Theorem 1.1, contains a rainbow cycle of length at least , which is either

 k+(k2−n+64) or k+(k2−n+44),

with respect to . In order to guarantee a larger length of a rainbow cycle in , the value of in Theorem 1.1 has to be larger than , and hence, which is not possible, because the maximum guaranteed length of the rainbow cycle from Theorem 1.1 is at most . Therefore, Theorem 2.1 guarantees an existence of a larger length of a rainbow-cycle subgraph in by at least . However, in order to apply Theorem 2.1, the graph cannot contain a rainbow cycle of length four, whereas, this condition is not necessary in Theorem 1.1.

The result in Theorem 2.1 is a progress toward the Conjecture 1.5 given by R. Čada, A.Kaneko and Z. Ryjáček [7]. However, the lower bound of is still larger than the conjectured bound. We also note that Theorem 2.1 is not a generalization of Theorem 1.2 because of the exclusion of the rainbow-cycle subgraph of length four.

4 Acknowledgement

I would like to thank Kittikorn Nakprasit for a suggestion on this paper and Chokchai Viriyapong for helping with LaTeX technicality. This research was financially supported by National Science and Technology Development Agency of Thailand with grant number: SCH-NR2016-531.

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