Chip games and multipartite graph paintability

TL;DR

This paper investigates the online paintability of complete multipartite graphs using chip games, establishing bounds and relations to hypergraph coloring parameters through computational and theoretical methods.

Contribution

It introduces new bounds on the paintability of multipartite graphs and relates these to hypergraph coloring parameters using chip game models.

Findings

Established upper bounds on paintability for small multipartite graphs.

Improved upper bounds on the online hypergraph parameter p_{OL}(n, m) for large n and m ≥ 3.

Derived lower bounds on the paintability of multipartite graphs with equal-sized parts.

Abstract

We study the paintability, an on-line version of choosability, of complete multipartite graphs. We do this by considering an equivalent chip game introduced by Duraj, Gutowski, and Kozik. We consider complete multipartite graphs with $ n $ parts of size at most 3. Using a computational approach, we establish upper bounds on the paintability of such graphs for small values of $ n. $ The choosability of complete multipartite graphs is closely related to value $ p(n, m) $, the minimum number of edges in a $n$-uniform hypergraph with no panchromatic $m$-coloring. We consider an online variant of this parameter $ p_{OL}(n, m), $ introduced by Khuzieva et al. using a symmetric chip game. With this symmetric chip game, we find an improved upper bound for $ p_{OL}(n, m)$ when $m \geq 3$ and $n$ is large. Our method also implies a lower bound on the paintability of complete multipartite graphs with $m \geq 3$ parts of equal size.