The Slow-coloring Game on Sparse Graphs: $k$-Degenerate, Planar, and Outerplanar

TL;DR

This paper studies the slow-coloring game on various sparse graph classes, providing strategies for Painter to limit the total score, with specific bounds for $k$-degenerate, planar, and outerplanar graphs.

Contribution

It introduces new Painter strategies for sparse graph classes, achieving explicit bounds on the game score, extending previous work on graph coloring games.

Findings

Painter can limit the score to at most $rac{3k+4}{4}n$ on $k$-degenerate graphs.

On outerplanar graphs, the score can be kept at $rac{7}{3}n$.

Specific bounds are provided for planar graphs with Hamiltonian duals and 4-colorable graphs.

Abstract

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 nonempty 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. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $\frac73n$ when $G$ is outerplanar.