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Rainbow vertex pair-pancyclicity of strongly edge-colored graphs

Peixue Zhao, Fei Huang

Abstract

An edge-colored graph is \emph{rainbow }if no two edges of the graph have the same color. An edge-colored graph $G^c$ is called \emph{properly colored} if every two adjacent edges of $G^c$ receive distinct colors in $G^c$. A \emph{strongly edge-colored} graph is a proper edge-colored graph such that every path of length $3$ is rainbow. We call an edge-colored graph $G^c$ \emph{rainbow vertex pair-pancyclic} if any two vertices in $G^c$ are contained in a rainbow cycle of length $\ell$ for each $\ell$ with $3 \leq \ell \leq n$. In this paper, we show that every strongly edge-colored graph $G^c$ of order $n$ with minimum degree $δ\geq \frac{2n}{3}+1$ is rainbow vertex pair-pancyclicity.

Rainbow vertex pair-pancyclicity of strongly edge-colored graphs

Abstract

An edge-colored graph is \emph{rainbow }if no two edges of the graph have the same color. An edge-colored graph is called \emph{properly colored} if every two adjacent edges of receive distinct colors in . A \emph{strongly edge-colored} graph is a proper edge-colored graph such that every path of length is rainbow. We call an edge-colored graph \emph{rainbow vertex pair-pancyclic} if any two vertices in are contained in a rainbow cycle of length for each with . In this paper, we show that every strongly edge-colored graph of order with minimum degree is rainbow vertex pair-pancyclicity.
Paper Structure (2 sections, 4 theorems, 35 equations)

This paper contains 2 sections, 4 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thmme']}

Key Result

Theorem 1.1

For any $\varepsilon >0$, there exists an integer $n_0$ such that every edge-colored graph $G^c$ with $n$ vertices and $\delta ^c (G) \geq (\frac{2}{3}+\varepsilon)n$ and $n \geq n_0$ contains a properly edge-colored cycle of length $l$ for all $3 \leq l \leq n$, where $\delta ^c (G)$ is the minimum

Theorems & Definitions (12)

  • Theorem 1.1: 10
  • Conjecture 1.2: 12
  • Conjecture 1.3: 12
  • Theorem 1.4: 12
  • Theorem 1.5: 14
  • Theorem 1.6
  • Claim 1
  • Claim 2
  • Claim 3
  • Claim 4
  • ...and 2 more