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.

比例choosability是均匀染色的列表类似物，其是在2019年引入的最小ķ针对其图表ģ按比例ķ -choosable是成比例的选择号码的ģ，它被表示为\（\志_ {PC}（G ）\）。在关于比例选择性的第一篇论文中，表明当\（2 \ le n \ le m \）时，\（\ max \ {n + 1，1 + \ lceil m / 2 \ rceil \} \ le \ chi _ {pc}（K_ {n，m}）\ le n + m -1 \）。在本说明中，我们通过显示\（\ 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 \）。在这个过程中，我们证明了完整的多部分图的比例选择数的一些新的下界。我们还提出了几个有趣的开放性问题。