On the Domatic Game

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.