论文标题
完整两分图的比例可选性
Proportional Choosability of Complete Bipartite Graphs
论文作者
论文摘要
比例可选性是2019年推出的公平着色的列表类似物。图$ g $是成比例的$ k $ -choosable的最小$ k $是$ g $的比例选择数，并且它表示为$χ_{pc {pc}（g）$。在有史以来可选性的第一篇论文中，结果表明，当$ 2 \ leq n \ leq m $，$ \ max \ {n + 1，1 + \ lceil m / 2 \ rceil \ rceil \} \ leq uq uq uq uq uqχ_{pc}（k_ {n，m}）\ leq n + m -m -1 $。在此注释中，我们通过证明$ \ max \ {n + 1，\ lceil n / 2 \ rceil + \ lceil m / 2 \ rceil \} \ leq uqχ_{pc}（k_ {k_ {n，m}）在此过程中，我们证明了对完整多部分图的比例选择数量的一些新的下限。我们还提出了几个有趣的开放问题。
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 $2 \leq n \leq m$, $ \max\{ n + 1, 1 + \lceil m / 2 \rceil\} \leq χ_{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 χ_{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.
