Problems on Hall \(t\)-Chromaticity

Abstract

We find a family of graphs each of which is not Hall \(t\)-chromatic for all \(t \geq 3\), and use this to prove that the same holds for the Kneser graphs \(K_{a,b}\) when \(a/b \geq 3\) and \(b\) is sufficiently large (depending on \(3 – (a/b)\)). We also make some progress on the problem of characterizing the graphs that are Hall \(t\)-chromatic for all \(t\).