Rainbow vertex-pancyclicity of strongly edge-colored graphs
Introduction
We consider only finite simple graphs. For a graph , we denote by and the vertex set, edge set and minimum degree of , respectively. An edge-coloring of is a mapping , where is a set of colors. A graph is called edge-colored if it is equipped with an edge-coloring. A graph is called properly edge-colored if it is edge-colored so that no two adjacent edges have the same color. A subgraph of an edge-colored graph is rainbow if no two edges in have the same color. In particular, an edge-colored graph is called rainbow Hamiltonian if contains a rainbow Hamiltonian cycle and rainbow pancyclic (resp., rainbow vertex-pancyclic) if contains (resp., every vertex in is contained in) a rainbow cycle of length for each with . A graph is called strongly edge-colored if it is edge-colored so that every path of length is rainbow, or equivalently, every monochromatic subgraph is an induced matching.
The existence of cycles in graphs with certain length considerations is a central concern in the literature. A classical result may date back to the one given by Dirac [8], which states that every graph with vertices and minimum degree contains a Hamiltonian cycle. Further, Hendry [11] showed that every graph with minimum degree is vertex-pancyclic. For an edge-colored graph, a natural generalization on this problem is to give a minimum degree threshold for the graph to be rainbow Hamiltonian or rainbow (vertex-)pancyclic. However, as pointed out in [12], such thresholds do not exist for all . Even so, instead of rainbow cycles, Lo [12] proved the following asymptotic version of Dirac’s theorem for the existence of properly edge-colored cycles:
Theorem 1.1 [12] For any , there exists an integer such that every edge-colored graph with vertices and and contains a properly edge-colored cycle of length for all , where is the minimum number of distinct colors of edges incident with a vertex in .
Recently, the condition ‘’ in Theorem 1.1 was improved by Lo [13] to for the existence of ‘long properly colored cycles’.
On the other hand, the interests on the existence of rainbow cycles (or rainbow path) also focus on strengthening the edge-colored graphs to the properly or strongly edge-colored graphs, and the -bounded edge-colored graphs (see [5] for details). In fact, as early as 1980, Hahn [10] conjectured that every properly edge-colored complete graph has a rainbow Hamiltonian path. Although this conjecture was later disproved by Maamoun and Meyniel [14], it is widely believed that such a coloring always contains a rainbow cycle of length at least . For more details on Hahn’s conjecture, we refer to [1], [2], [3], [6], [9] for examples. Recently, inspired by Hahn’s conjecture, Cheng et al. [7] considered the existence of rainbow Hamiltonian cycles in strongly edge-colored graph and proposed the following two conjectures.
Conjecture 1.2 [7] Every strongly edge-colored graph with vertices and degree at least has a rainbow Hamiltonian cycle.
Conjecture 1.3 [7] Every strongly edge-colored graph with vertices and degree at least has a rainbow Hamiltonian path.
We note that Conjecture 1.2 implies Conjecture 1.3 . Indeed, for a graph that satisfies the condition of Conjecture 1.3 , we may construct a graph obtained from by adding a new vertex and new edges joining to each vertex of with distinct new colors. Obviously, satisfies the condition of Conjecture 1.2 and, therefore, contains a rainbow Hamiltonian cycle . Hence, is a rainbow Hamiltonian path in .
To support the above two conjectures, Cheng et al. proved the following theorem:
Theorem 1.4 [7] Let be a strongly edge-colored graph with vertices and minimum degree . If , then has a rainbow Hamiltonian cycle.
For Hamiltonicity and pancyclicity of a graph, in the early 1970s Bondy [4] stated his well-known meta-conjecture that almost any condition that implies a graph being Hamiltonian will imply the graph being pancyclic, possibly with a well defined class of exceptional graphs. It is natural to ask for a rainbow version of Bondy’s meta-conjecture: Is it true that almost any condition that implies a strongly edge-colored graph being rainbow Hamiltonian will imply the graph being rainbow (vertex-)pancyclic?
Inspired by Theorem 1.1, Theorem 1.4, in this paper we prove the following result.
Theorem 1.5 Let be a strongly edge-colored graph with vertices and minimum degree . If , then is rainbow vertex-pancyclic.
Section snippets
Proof of Theorem 1.5
Let be a strongly edge-colored graph with edge coloring . For two vertices and in , we denote by the edge joining and . For a set of edges and a subgraph of , let and . Given a rainbow cycle in , we call a color a -color (resp., -color) if (resp., ). Correspondingly, an edge is called a -color edge (resp., -color edge) if (resp., ), and two adjacent vertices and are called -adjacent (resp., -
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.
Acknowledgments
The authors would like to thank the anonymous reviewer for her/his constructive suggestions. This work was supported by the National Natural Science Foundation of China [Grant numbers, 11971406, 11561058].
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