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 $\chi_{pc}(G)$. In the first ever paper on proportional choosability, it was shown that when $2 \leq n \leq m$, $ \max{ n + 1, 1 + \lceil m / 2 \rceil} \leq \chi_{pc}(K_{n,m}) \leq n + m - 1$. In this note we improve on this result by showing that $ \max{ n + 1, \lceil n / 2 \rceil + \lceil m / 2 \rceil} \leq \chi_{pc}(K_{n,m}) \leq n + m -1- \lfloor m/3 \rfloor$. 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.