On Even Rainbow or Nontriangular Directed Cycles

TL;DR

This paper extends rainbow cycle existence results to even rainbow cycles and directed cycles in large graphs, establishing minimum degree conditions that guarantee their presence, and employs the stability method for proofs.

Contribution

It introduces new minimum degree conditions for the existence of even rainbow cycles and directed cycles in large graphs, generalizing previous results for rainbow triangles.

Findings

Minimum degree condition for even rainbow cycles is $(n+5)/3$ for large $n$.

Large oriented graphs with minimum outdegree $(n+1)/3$ contain directed cycles of fixed length.

Results are optimal when cycle length is not divisible by 3.

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 establish a corresponding result for fixed even rainbow $\ell$-cycles $C_{\ell}$: if every vertex $v \in V$ is incident to at least $(n+5)/3$ distinctly colored edges, where $n \geq n_0(\ell)$ is sufficiently large, then $G$ admits an even rainbow $\ell$-cycle $C_{\ell}$. This result is best possible whenever $\ell \not\equiv 0$ (mod 3). Correspondingly, we also show that for a fixed (even or odd) integer $\ell \geq 4$, every large $n$-vertex oriented graph $\vec{G} = (V, \vec{E})$ with minimum outdegree at least $(n+1)/3$ admits a (consistently) directed $\ell$-cycle $\vec{C}_{\ell}$. Our latter result relates to one of Kelly, K\"uhn, and Osthus, who proved a similar statement for oriented graphs with large semi-degree. Our proofs are based on the stability method.