The domatic number game played on graphs

The domatic number of a graph is the maximum number of pairwise disjoint dominating sets admitted by the graph. We introduce a game based around this graph invariant. The domatic number game is played on a graph $G$ by two players, Alice and Bob, who take turns selecting a vertex and placing it into one of $k$ sets. Alice is trying to make each of these sets into a dominating set of $G$ while Bob's goal is to prevent this from being accomplished. The maximum $k$ for which Alice can achieve her goal when both players are playing optimal strategies, is called the game domatic number of $G$. There are two versions of the game and two resulting invariants depending on whether Alice or Bob is the first to play. We prove several upper bounds on these game domatic numbers of arbitrary graphs and find the exact values for several classes of graphs including trees, complete bipartite graphs, cycles and some narrow grid graphs. We pose several open problems concerning the effect of standard graph operations on the game domatic number as well as a vexing question related to the monotonicity of the number of sets available to Alice.