On Odd Rainbow Cycles in Edge-Colored Graphs

TL;DR

This paper proves that in sufficiently large edge-colored graphs with high minimum color degree, one can find rainbow cycles of any odd length, extending previous results for even cycles.

Contribution

It establishes the existence of rainbow odd cycles under a high minimum color degree condition, generalizing earlier work on even cycles.

Findings

Rainbow odd cycles exist when n ≥ 432ℓ under the given conditions.

The result is sharp for all odd ℓ ≥ 3.

Extends previous results from even to odd cycle lengths.

Abstract

Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis ensures a rainbow $\ell$-cycle $C_{\ell}$ whenever $n \ge 432 \ell$. This result is sharp for all odd integers $\ell \geq 3$, and extends earlier work of the authors for when $\ell$ is even.