Monochromatic-degree conditions for properly colored cycles in edge-colored complete graphs

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Abstract

Let G be an edge-colored graph and v be a vertex of G. Define the monochromatic-degree dmon(v) of v to be the maximum number of edges with the same color incident with v in G, and the maximum monochromatic-degree Δmon(G) of G to be the maximum value of dmon(v) over all vertices v of G. A cycle (path) in G is called properly colored if any two adjacent edges of the cycle (path) have distinct colors. Wang et al. in 2014 showed that an edge-colored complete graph Knc with Δmon(Knc)<n2 contains a properly colored cycle of length at least n2+2. In this paper, we obtain a generalization of their result that an edge-colored complete graph Knc of order n with Δmon(Knc)=dn2 contains a properly colored cycle of length at least nd+1.

Introduction

An edge-coloring of a graph is an assignment of colors to the edges of the graph. An edge-colored graph is a graph with an edge-coloring. Let Knc denote an edge-colored complete graph with an edge-coloring c. A cycle (path) in an edge-colored graph G is properly colored, or PC for short, if any two adjacent edges of the cycle (path) have distinct colors. For other notation and terminology not defined here, we refer to [4].

In an edge-colored graph G, the color-degree of a vertex v of G is the number of colors on the edges incident with v in G, denoted by dc(v). Let δc(G) denote the minimum value of dc(v) over all vertices vV(G), called the minimum color-degree of G. Actually, there are many results on the color-degree conditions for the existence of PC cycles, for which we refer the reader to [2], [9], [10], [14].

In this paper, we consider the monochromatic-degree conditions for the existence of PC cycles. The monochromatic-degree of a vertex v of G is the maximum number of edges with the same color incident with v in G, denoted by dmon(v). Let Δmon(G) denote the maximum value of dmon(v) over all vertices vV(G), called the maximum monochromatic-degree of G. In recent years, many people have worked on the conditions for the existence of a PC Hamilton cycle in an edge-colored graph. In 1976, Bollobás and Erdős in [3] posed the following famous conjecture.

Conjecture 1 [3]

If Δmon(Knc)<n2, then Knc contains a PC Hamiltonian cycle.

Li et al. in [9] studied long PC cycles in Knc and proved that if Δmon(Knc)<n2, then Knc contains a PC cycle of length at least n+23+1. Later on, Wang et al. in [15] improved the bound on the lengths of PC cycles.

Theorem 2 [15]

If Δmon(Knc)<n2, then Knc contains a PC cycle of length at least n2+2.

In this paper, we obtain a bound on the lengths of PC cycles under monochromatic-degree conditions.

Theorem 3

If Δmon(Knc)=dn2, then Knc contains a PC cycle of length at least nd+1.

Remark

Theorem 3 can be seen as a generalization of Theorem 2, since from Δmon(Knc)=d<n2, we haved{n32n is odd;n22n is even, and then nd+1n2+2.

The main idea is the rotation-extension technique of Pósa [12], which was used on edge-colored graphs in [10], [15].

Since Δmon(Knc)+δc(Knc)n, we can get the following corollary.

Corollary 4

If δc(Knc)2, then Knc contains a PC cycle of length at least δc(Knc)+1.

Thus we completely solve the problem “Does every edge-colored complete graph Knc with δc(Knc)2 contain a PC cycle of length at least δc(Knc)?”, which was posed by Li et al. in [7].

The paper is organized as follows. In Section 2, we give some notation and tools. In Section 3 we prove our main result Theorem 3. In Section 4, we give a remark concerning the lengths of PC cycles in Theorem 3 and pose two conjectures.

Section snippets

Preliminaries

Grossman and Häggkvist in [6] gave a condition for the exitance of a PC cycle in an edge-colored graph with two colors, and later on, Yeo in [16] extended the result to an edge-colored graph with any number of colors.

Theorem 5

[6], [16]

Let G be an edge-colored graph containing no PC cycles. Then G contains a vertex v such that no component of Gv is joined to v with edges of more than one color.

Li et al. [8] observed that in an edge-colored complete graph G, for any PC cycle C, each vertex vV(C) is contained in

Proof of Theorem 3

If d=n2, then the result follows from Theorem 6. Then, we may assume dn3. Suppose, to the contrary, that each PC cycle in Knc is of length at most nd. Let P be a longest PC path in Knc, and for simplicity, we label the vertices of P by (1,2,,) and P=(,1,,1). According to Lemma 3, we know that r(P),s(P),t(P),u(P) and w(P) do exist. For convenience, we use r,s,t,u,w instead. Without loss of generality, assume that P is a longest PC path satisfying that |S(P)| is maximum over all the

Concluding remarks

There have been many researchers working on Conjecture 1, which implies that the bound on the length of a PC cycle in Theorem 3 is not sharp. The author in [13] showed that Δmon(Knc)n7 is sufficient for the existence of a PC Hamiltonian cycle. Up to 2016, Lo [11] showed that for any ε>0, there exists an integer n0 such that every edge-colored complete graph Knc with Δmon(Knc)<(12ε)n and nn0 contains a PC Hamiltonian cycle, which implies a result obtained by Alon and Gutin [1] that for every ε

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors are very grateful to the reviewers and the editor for their very useful suggestions and comments, which helped to improving the presentation of the paper greatly.

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Supported by NSFC No. 12131013, 11871034 and 12161141006.

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