Monochromatic-degree conditions for properly colored cycles in edge-colored complete graphs☆
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 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 . Let denote the minimum value of over all vertices , 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 . Let denote the maximum value of over all vertices , 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 , then contains a PC Hamiltonian cycle.
Li et al. in [9] studied long PC cycles in and proved that if , then contains a PC cycle of length at least . Later on, Wang et al. in [15] improved the bound on the lengths of PC cycles.
Theorem 2 [15] If , then contains a PC cycle of length at least .
In this paper, we obtain a bound on the lengths of PC cycles under monochromatic-degree conditions.
Theorem 3 If , then contains a PC cycle of length at least .
Remark Theorem 3 can be seen as a generalization of Theorem 2, since from , we have and then . The main idea is the rotation-extension technique of Pósa [12], which was used on edge-colored graphs in [10], [15]. Since , we can get the following corollary.
Corollary 4 If , then contains a PC cycle of length at least .
Thus we completely solve the problem “Does every edge-colored complete graph with contain a PC cycle of length at least ?”, 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 Let G be an edge-colored graph containing no PC cycles. Then G contains a vertex v such that no component of is joined to v with edges of more than one color.[6], [16]
Li et al. [8] observed that in an edge-colored complete graph G, for any PC cycle C, each vertex is contained in
Proof of Theorem 3
If , then the result follows from Theorem 6. Then, we may assume . Suppose, to the contrary, that each PC cycle in is of length at most . Let P be a longest PC path in , and for simplicity, we label the vertices of P by and . According to Lemma 3, we know that and do exist. For convenience, we use instead. Without loss of generality, assume that P is a longest PC path satisfying that 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 is sufficient for the existence of a PC Hamiltonian cycle. Up to 2016, Lo [11] showed that for any , there exists an integer such that every edge-colored complete graph with and contains a PC Hamiltonian cycle, which implies a result obtained by Alon and Gutin [1] that for every
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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