On the Domatic Game

TL;DR

This paper investigates the domatic game played on graphs, establishing bounds on the game number related to minimum degree, and explores how game dynamics change with different players and subgraph considerations.

Contribution

It introduces bounds on the domatic game number based on graph properties, constructs specific graphs with particular game outcomes, and explores variations of the game including a score-based version.

Findings

The domatic game number is at least proportional to the minimum degree divided by log n.

Existence of graphs with minimum degree k but game number 1.

Existence of graphs with game number 1 but domatic number k.

Abstract

The domatic game with pallete size $k$ is a $2$-player game played on a graph $G$ recently introduced by Hartnell and Rall. Players Alice and Bob take turns choosing an uncolored vertex from $G$, and coloring it a color from $\{1,2,\dots,k\}$. The game ends once all vertices in $G$ have been assigned a color. Alice wins if all $k$ colors induce a dominating set of $G$, and otherwise Bob wins. The domatic game number, $\operatorname{dom_g}(G,X)$ is the the largest pallete size $k$ such that Alice wins the domatic game when player $X$ goes first (where $X$ is either Alice or Bob). We prove for any graph $G$ of order $n$, \[ \operatorname{dom_g}(G,X)=\Omega\left(\frac{\delta(G)}{\log n}\right). \] In addition, we show that for any $k$ there exists a graph $G$ with minimum degree $\delta(G)=k$ and $\operatorname{dom_g}(G,X)=1$, and there exists a graph $G'$ with $\operatorname{dom_g}(G',X)=1$ while having (non-game) domatic number $\operatorname{dom}(G')=k$. We explore how the domatic game number changes when changing who goes first, and when considering subgraphs of $G$. We also introduce a score variant of the domatic game, and use this to get bounds on the original domatic game.