A note on the connected game coloring number

TL;DR

This paper investigates the connected game coloring number, a graph parameter related to a two-player coloring game, focusing on graphs of bounded treedepth and k-trees, and provides new bounds and examples.

Contribution

It introduces new bounds for the connected game coloring number on specific graph classes and constructs an outerplanar 2-tree with a high connected game coloring number.

Findings

Existence of an outerplanar 2-tree with connected game coloring number 5

Bounded the connected game coloring number for graphs of bounded treedepth

Analyzed the parameter for k-trees and answered an open question

Abstract

We consider the \emph{connected game coloring number} of a graph, introduced by Charpentier et al. as a game theoretic graph parameter that measures the degeneracy of a graph with respect to a certain two-player game played with an uncooperative adversary. We consider the connected game coloring number of graphs of bounded treedepth and of $k$-trees. In particular, we show that there exists an outerplanar $2$-tree with connected game coloring number of $5$, which answers a question from [C. Charpentier, H. Hocquard, E. Sopena, and X. Zhu. A connected version of the graph coloring game. \textit{Discrete Appl. Math.}, 2020].