Rainbow vertex-pancyclicity of strongly edge-colored graphs

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Abstract

A strongly edge-colored graph is an edge-colored graph in which every path of length 3 is rainbow or, equivalently, every monochromatic subgraph is an induced matching. In this paper, we show that every strongly edge-colored graph with n vertices and minimum degree δ2n3 is rainbow vertex-pancyclic. This extends a recent result given by Cheng et al., that is, every strongly edge-colored graph with n vertices and minimum degree δ2n3 is rainbow Hamiltonian.

Introduction

We consider only finite simple graphs. For a graph G, we denote by V(G),E(G) and δ(G) the vertex set, edge set and minimum degree of G, respectively. An edge-coloring of G is a mapping f:E(G)S, where S 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 H of an edge-colored graph is rainbow if no two edges in H have the same color. In particular, an edge-colored graph G is called rainbow Hamiltonian if G contains a rainbow Hamiltonian cycle and rainbow pancyclic (resp., rainbow vertex-pancyclic) if G contains (resp., every vertex in G is contained in) a rainbow cycle of length l for each l with 3ln. A graph is called strongly edge-colored if it is edge-colored so that every path of length 3 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 G with n3 vertices and minimum degree δ(G)n2 contains a Hamiltonian cycle. Further, Hendry [11] showed that every graph G with minimum degree δ(G)n+12 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 n. 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 ε>0, there exists an integer n0 such that every edge-colored graph G with n vertices and δc(G)(23+ε)n and nn0 contains a properly edge-colored cycle of length l for all 3ln, where δc(G) is the minimum number of distinct colors of edges incident with a vertex in G.

Recently, the condition ‘δc(G)(23+ε)n’ in Theorem 1.1 was improved by Lo [13] to δc(G)(12+ε)n 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 cn-bounded edge-colored graphs (see [5] for details). In fact, as early as 1980, Hahn [10] conjectured that every properly edge-colored complete graph Kn 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 n2. 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 G with n vertices and degree at least n+12 has a rainbow Hamiltonian cycle.

Conjecture 1.3 [7]

Every strongly edge-colored graph G with n vertices and degree at least n2 has a rainbow Hamiltonian path.

We note that Conjecture 1.2 implies Conjecture 1.3 . Indeed, for a graph G that satisfies the condition of Conjecture 1.3 , we may construct a graph G obtained from G by adding a new vertex w and new edges joining w to each vertex of G with distinct new colors. Obviously, G satisfies the condition of Conjecture 1.2 and, therefore, contains a rainbow Hamiltonian cycle C. Hence, Cw is a rainbow Hamiltonian path in G.

To support the above two conjectures, Cheng et al. proved the following theorem:

Theorem 1.4 [7]

Let G be a strongly edge-colored graph with n vertices and minimum degree δ. If δ2n3, then G 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 G be a strongly edge-colored graph with n vertices and minimum degree δ. If δ2n3, then G is rainbow vertex-pancyclic.

Section snippets

Proof of Theorem 1.5

Let G be a strongly edge-colored graph with edge coloring f. For two vertices u and v in G, we denote by uv the edge joining u and v. For a set F of edges and a subgraph H of G, let f(F)={f(e):eF} and f(H)=f(E(H)). Given a rainbow cycle C in G, we call a color s a C-color (resp., C˜-color) if sf(C) (resp., sf(C)). Correspondingly, an edge e is called a C-color edge (resp., C˜-color edge) if f(e)f(C) (resp., f(e)f(C)), and two adjacent vertices u and v are called C-adjacent (resp., C˜-

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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