Bounds for Partial List Colourings

Abstract

Let \(G\) be a simple graph on \(n\) vertices with list chromatic number \(\chi_\ell = s\). If each vertex of \(G\) is assigned a list of \(t\) colours, Albertson, Grossman, and Haas [1] asked how many of the vertices, \(\lambda_{t,s}\), are necessarily colourable from these lists? They conjectured that \(\lambda_{t,s} \geq \frac{tn}{s}\). Their work was extended by Chappell [2]. We improve the known lower bounds for \(\lambda_{t,s}\).