Fractional Chromatic Number Vs. Hall Ratio

Abstract

Given a graph G, its Hall ratio \rho (G)=\max _{H\subseteq G}\frac{|V(H)|}{\alpha (H)}$\rho (G)=\max _{H\subseteq G}\frac{|V(H)|}{\alpha (H)}$ forms a natural lower bound for its fractional chromatic number \chi _f(G)$\chi _f(G)$. A recent line of research studied the question whether \chi _f(G)$\chi _f(G)$ can be bounded in terms of a (linear) function of \rho (G)$\rho (G)$. Dvořák, Ossona de Mendez and Wu [6, Combinatorica, 2020] gave a negative answer by proving the existence of graphs with bounded Hall ratio and arbitrarily large fractional chromatic number. In this paper, we solve two follow-up problems that were raised by Dvořák et al. The first problem concerns determining the growth of g(n), defined as the maximum ratio \frac{\chi _f(G)}{\rho (G)}$\frac{\chi _f(G)}{\rho (G)}$ among all n-vertex graphs. Dvořák et al. obtained the bounds \Omega (\log \log n) \le g(n)\le O(\log n)$\Omega (\log \log n) \le g(n)\le O(\log n)$. We show that the true value is close to the upper bound: g(n)=(\log n)^{1-o(1)}$g(n)=(\log n)^{1-o(1)}$. The second problem posed by Dvořák 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 show that such graphs indeed exist.