Towards an edge-coloured Corr\'adi--Hajnal theorem

TL;DR

This paper extends the classical Corrádi-Hajnal theorem to edge-coloured graphs, establishing conditions under which a rainbow triangle tiling exists based on minimum colour degree, and explores related directed graph problems.

Contribution

It introduces a new minimum colour degree condition for rainbow triangle tilings in edge-coloured graphs, generalizing a well-known graph tiling theorem.

Findings

If elta^c(G) rac{5}{6} + \u03b5) n, then G contains a rainbow triangle tiling.

The bound for elta^c(G) is at least 5n/7, showing near-optimality.

Discussion of related tiling problems in directed graphs.

Abstract

A classical result of Corr\'adi and Hajnal states that every graph $G$ on $n$ vertices with $n\in 3\mathbb{N}$ and $\delta(G) \ge 2n/3$ contains a perfect triangle-tiling, i.e.,\ a spanning set of vertex-disjoint triangles. We explore a generalisation of this result to edge-coloured graphs. Let $G$ be an edge-coloured graph on $n$ vertices. The minimum colour degree $\delta^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v \in V(G)$, there are at least $k$ distinct colours on edges incident to $v$. We show that if $\delta^c(G) \ge (5/6 + \varepsilon) n$, then $G$ has a spanning set of vertex-disjoint rainbow triangles. On the other hand, we find an example showing the bound should be at least $5n/7$. We also discuss a related tiling problems on digraphs, which may be of independent interest.