The Slow-coloring Game on Outerplanar, Planar, and $k$-degenerate Graphs

Grzegorz Gutowski, Tomasz Krawczyk, Douglas B. West, Michał Zając, Xuding Zhu

Published 2018-01-21Version 1

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a non-empty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. In this paper we provide strategies for Painter on outerplanar, planar, and $k$-degenerate graphs. On an $n$-vertex graph $G$, we show that Painter can keep the score to at most $(1+3k/4)n$ when $G$ is $k$-degenerate, $7n/3$ when $G$ is outerplanar, and $3.9857n$ when $G$ is planar.