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 $2\le n\le m$, $\max \{n+1,1+\lceil m/2\rceil \}\le {\chi }_{pc}(K_{n,m})\le n+m-1$. In this note we improve on this result by showing that $\max \{n+1,\lceil n/2\rceil +\lceil m/2\rceil \}\le {\chi }_{pc}(K_{n,m})\le n+m-1-\lfloor m/3\rfloor$. 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.

11 pages