Chromatic number and regular subgraphs

TL;DR

This paper disproves a long-standing conjecture by constructing graphs with high fractional chromatic number that lack certain regular subgraphs, showing that the proposed universal number for containing multiple cycles does not exist.

Contribution

The authors resolve a 1992 problem by demonstrating that no universal number guarantees multiple edge-disjoint cycles in high chromatic graphs, and they connect their results to a conjecture of Harris.

Findings

Existence of graphs with high fractional chromatic number without 4-regular subgraphs.

No universal chromatic number ensures multiple cycles for all graphs.

Conditional bounds based on Harris's conjecture.

Abstract

In 1992, Erd\H{o}s and Hajnal posed the following natural problem: Does there exist, for every $r\in \mathbb{N}$, an integer $F(r)$ such that every graph with chromatic number at least $F(r)$ 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 $F(r)$ exists for $r\ge 2$. We show that assuming a conjecture of Harris, the bound on the fractional chromatic number in our result cannot be improved.