Proportional Choosability of Complete Bipartite Graphs

Abstract

Proportional choosability is a list analogue of equitable coloring that was introduced in 2019. The smallest k for which a graph G is proportionally k-choosable is the proportional choice number of G, and it is denoted χ pc (G). In the first ever paper on proportional choosability, it was shown that when 2 ≤ n ≤ m, max{n + 1, 1 + ⌈m/2⌉} ≤ χ pc (K n,m ) ≤ n + m − 1. In this note we improve on this result by showing that max{n + 1, ⌈n/2⌉ + ⌈m/2⌉} ≤ χ pc (K n,m ) ≤ n + m − 1 − ⌊m/3⌋. In the process, we prove some new lower bounds on the proportional choice number of complete multipartite graphs. We also present several interesting open questions.