Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture

TL;DR

This paper improves bounds on the conditions under which a rainbow triangle must exist in edge-colored graphs, extending previous results and providing infinitely many new parameter pairs.

Contribution

It presents improved bounds for the triangular property in rainbow cycles, advancing the understanding of rainbow cycle existence in edge-colored graphs.

Findings

Improved bounds: (1.1077,1/3) and (1,0.3988) are triangular.

Extended results to infinitely many pairs, including those with smaller .

New pair: (1.3481,1/4) is triangular.

Abstract

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size at least $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. Let us call $(\alpha, \beta)$ \emph{triangular} if every simple edge-colored graph on $n$ vertices with at least $\alpha n$ color classes, each with at least $\beta n$ edges, has a rainbow triangle. Aharoni, Holzman, and DeVos showed the following: $(9/8,1/3)$ is triangular; $(1,2/5)$ is triangular. In this paper, we improve those bounds, showing the following: $(1.1077,1/3)$ is triangular; $(1,0.3988)$ is triangular. Our methods give results for infinitely many pairs $(\alpha, \beta)$, including $\beta < 1/3$; we show that $(1.3481,1/4)$ is triangular.