Proportional Choosability of Complete Bipartite Graphs

TL;DR

This paper investigates the proportional choice number of complete bipartite graphs, providing improved bounds and new lower bounds, advancing understanding of proportional choosability in graph theory.

Contribution

It refines bounds on the proportional choice number of complete bipartite graphs and introduces new lower bounds for complete multipartite graphs.

Findings

Improved lower bounds for _{pc}(K_{n,m})

Enhanced upper bounds for _{pc}(K_{n,m})

New lower bounds for complete multipartite 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.