Rainbow vertex pair-pancyclicity of strongly edge-colored graphs

TL;DR

This paper proves that strongly edge-colored graphs with high minimum degree are guaranteed to contain rainbow cycles of all lengths between 3 and n passing through any pair of vertices.

Contribution

It establishes a sufficient minimum degree condition for strongly edge-colored graphs to be rainbow vertex pair-pancyclic.

Findings

Every strongly edge-colored graph with minimum degree at least 2n/3+1 is rainbow vertex pair-pancyclic.

The result generalizes previous conditions for rainbow cycles in edge-colored graphs.

Provides a new characterization linking minimum degree and rainbow cycle existence.

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 $\delta \geq \frac{2n}{3}+1$ is rainbow vertex pair-pancyclicity.