Rainbow triangles and the Caccetta-H\"aggkvist conjecture

TL;DR

This paper explores a rainbow triangle problem in edge-colored graphs, generalizing a famous digraph conjecture, and determines extremal configurations for large graphs with bounded color class sizes.

Contribution

It proves the existence of rainbow triangles under certain conditions and characterizes extremal graphs avoiding rainbow triangles with bounded color class sizes.

Findings

Existence of rainbow triangles under strengthened conditions.

Maximum edges in large graphs without rainbow triangles with bounded color classes.

Complete characterization of extremal graphs for the problem.

Abstract

A famous conjecture of Caccetta and H\"aggkvist is that in a digraph on $n$ vertices and minimum out-degree at least $\frac{n}{r}$ there is a directed cycle of length $r$ or less. We consider the following generalization: in an undirected graph on $n$ vertices, any collection of $n$ disjoint sets of edges, each of size at least $\frac{n}{r}$, has a rainbow cycle of length $r$ or less. We focus on the case $r=3$, and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. For any fixed $k$ and large enough $n$, we determine the maximum number of edges in an $n$-vertex edge-coloured graph where all colour classes have size at most $k$ and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.