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.
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