Turn Skipping and the Game Coloring Number

TL;DR

This paper explores the game coloring number of graphs, introducing a preordered variant to analyze turn skipping, and establishes bounds relating the coloring numbers of graphs and their subgraphs.

Contribution

It introduces the preordered game coloring number to study turn skipping effects and derives bounds connecting graph and subgraph coloring numbers.

Findings

Turn skipping does not improve player performance.

Bounds relate the coloring number of a graph to its subgraphs.

Results provide tight bounds for induced subgraphs.

Abstract

The game coloring number gcol($G$) of a graph $G$ is a two player competitive variant of the coloring number. We introduce the preordered game coloring number to study the consequences of either player skipping any number of turns. In particular, we show that neither player can improve their performance by doing so. We use this result to show that for any induced subgraph $H \subset G$, $|G-H| = k$ implies the tight bound gcol($H$) $\leq$ gcol($G$) $-$ $2k$ and if $d_G(x)$ $\leq$ gcol($G-x$), then gcol($G-x$) $\leq$ gcol($G$) $-$ $1$.