A note on heterochromatic cycles of length 4 in edge-colored graphs

TL;DR

This paper proves that large edge-colored graphs with certain color neighborhood conditions necessarily contain heterochromatic cycles of length 4, extending previous results on shorter cycles.

Contribution

It extends existing theorems by establishing conditions for the existence of heterochromatic 4-cycles in large graphs.

Findings

Graphs with at least 60 vertices and specific color neighborhood unions contain heterochromatic 4-cycles.

The result generalizes earlier work on heterochromatic cycles of length 3 or 4.

Provides a new sufficient condition for the existence of heterochromatic cycles of length 4.

Abstract

Let $G$ be an edge-colored graph. A heterochromatic cycle of $G$ is one in which every two edges have different colors. For a vertex $v\in V(G)$, let $CN(v)$ denote the set of colors which are assigned to the edges incident to $v$. In this note we prove that $G$ contains a heterochromatic cycle of length 4 if $G$ has $n\geq 60$ vertices and $|CN(u)\cup CN(v)|\geq n-1$ for every pair of vertices $u$ and $v$ of $G$. This extends a result of Broersma et al. on the existence of heterochromatic cycles of length 3 or 4.