Chromatic number and regular subgraphs

Abstract

In 1992, Erdős and Hajnal posed the following natural problem: Does there exist, for every $r\in N$, an integer $F\left(r\right)$ such that every graph with chromatic number at least $F\left(r\right)$ contains $r$ edge-disjoint cycles on the same vertex set? We solve this problem in a strong form, by showing that there exist $n$-vertex graphs with fractional chromatic number $\Omega \left(\frac{\log \log n}{\log \log \log n}\right)$ that do not even contain a 4-regular subgraph. This implies that no such number $F\left(r\right)$ exists for $r⩾2$. We show that, assuming a conjecture of Harris, the bound on the fractional chromatic number in our result cannot be improved. (After our paper was written, Harris' conjecture was proved by Martinsson.)