Fractional chromatic number vs. Hall ratio

TL;DR

This paper investigates the relationship between fractional chromatic number and Hall ratio in graphs, nearly resolving the growth rate of their ratio and constructing graphs with specific properties related to these parameters.

Contribution

It proves that the maximum ratio of fractional chromatic number to Hall ratio grows roughly as a power of log n and constructs graphs with bounded Hall ratio and large fractional chromatic number with specific subgraph properties.

Findings

The ratio g(n) is approximately (log n)^{1-o(1)}.

Existence of graphs with bounded Hall ratio and large fractional chromatic number with dense subgraphs.

Resolved two open problems posed by Dvořák et al.

Abstract

Given a graph $G$, its Hall ratio $\rho(G)=\max_{H\subseteq G}\frac{|V(H)|}{\alpha(H)}$ forms a natural lower bound on its fractional chromatic number $\chi_f(G)$. A recent line of research studied the fundamental question of whether $\chi_f(G)$ can be bounded in terms of a (linear) function of $\rho(G)$. In a breakthrough-result, Dvo\v{r}\'{a}k, Ossona de Mendez and Wu gave a strong negative answer by proving the existence of graphs with bounded Hall ratio and arbitrarily large fractional chromatic number. In this paper, we solve two natural follow-up problems that were raised by Dvo\v{r}\'{a}k et al. The first problem concerns determining the growth of $g(n)$, defined as the maximum ratio $\frac{\chi_f(G)}{\rho(G)}$ among all $n$-vertex graphs. Dvo\v{r}\'{a}k et al. obtained the bounds $\Omega(\log\log n) \le g(n)\le O(\log n)$, leaving an exponential gap between the lower and upper bound. We almost fully resolve this problem by proving that the truth is close to the upper bound, i.e., $g(n)=(\log n)^{1-o(1)}$. The second problem posed by Dvo\v{r}\'{a}k et al. asks for the existence of graphs with bounded Hall ratio, arbitrarily large fractional chromatic number and such that every subgraph contains an independent set that touches a constant fraction of its edges. We affirmatively solve this second problem by showing that such graphs indeed exist.