Counterexamples to a Conjecture of Harris on Hall Ratio

Abstract

In this work, we refute a conjecture of Harris by constructing various graphs whose fractional chromatic number grows much faster than their Hall ratio. The Hall ratio of a graph G is the maximum value of the ratio between the number of vertices and the independence number taken over all non-null subgraphs of G. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number.