Domatic Number Game: Strategies & Graph Bounds

• The domatic number game is a two-player graph game where players alternately color vertices aiming for each color class to form a dominating set.
• The game introduces start-order invariants and sharper upper bounds compared to classical domatic numbers, highlighting how move order affects strategy.
• Recent studies incorporate score variants and subgraph analyses, exposing non-monotonic behavior and offering open problems in graph domination theory.

The domatic number game is a two-player graph game built around the classical domatic number, the maximum number of pairwise disjoint dominating sets admitted by a graph. Hartnell and Rall introduced the game by fixing a palette of $k$ colors and letting Alice and Bob alternately color previously unchosen vertices of a graph $G$; Alice wins precisely when every color class is a dominating set of $G$, and Bob wins otherwise (Hartnell et al., 14 Aug 2025). The resulting game invariants depend on who moves first, and subsequent work reformulated them as domatic game numbers $\operatorname{dom_g}(G,A)$ and $\operatorname{dom_g}(G,B)$, established general lower bounds, introduced a score variant, and analyzed how the parameter behaves under subgraphs and change of first player (English et al., 13 Mar 2026).

1. Classical parameter and foundational definitions

For a finite, simple graph $G=(V,E)$, a dominating set is a set $D\subseteq V$ such that every vertex in $V\setminus D$ has a neighbor in $D$. Equivalently, every vertex of $G$ is either in $G$0 or adjacent to a vertex in $G$1. The domatic number $G$2, also denoted $G$3 in later work, is the largest number $G$4 such that $G$5 can be partitioned into $G$6 dominating sets (Hartnell et al., 14 Aug 2025).

In the formulation used by Hartnell and Rall, the partition may be treated weakly, with some parts empty for technical convenience. Formally,

$G$7

A standard upper bound is

$G$8

where $G$9 is the minimum degree (Hartnell et al., 14 Aug 2025).

This classical invariant supplies the ambient ceiling for the game versions. Later work explicitly records

$G$0

for either first player $G$1 (English et al., 13 Mar 2026).

2. Game model and the two start-order invariants

The domatic game with palette size $G$2 is played on a graph $G$3 with colors $G$4. On each turn, a player selects an uncolored vertex and assigns it one of the $G$5 colors. Play continues until all vertices are colored (Hartnell et al., 14 Aug 2025).

If $G$6 denotes the set of vertices receiving color $G$7, then $G$8 is the game induced partition. Alice’s objective is to ensure that for every color $G$9 and every vertex $\operatorname{dom_g}(G,A)$0, some vertex in the closed neighborhood $\operatorname{dom_g}(G,A)$1 receives color $\operatorname{dom_g}(G,A)$2. Equivalently, each color class $\operatorname{dom_g}(G,A)$3 must be a dominating set of $\operatorname{dom_g}(G,A)$4. Bob’s objective is to prevent this, that is, to ensure that at least one color is absent from the closed neighborhood of some vertex (Hartnell et al., 14 Aug 2025).

There are two standard start-order variants. In Hartnell and Rall’s notation,

$\operatorname{dom_g}(G,A)$5

and

$\operatorname{dom_g}(G,A)$6

Later work denotes the same two quantities by $\operatorname{dom_g}(G,A)$7 and $\operatorname{dom_g}(G,A)$8, respectively (English et al., 13 Mar 2026).

The distinction between the two invariants is substantive rather than notational. The literature treats move order as an independent parameter and develops separate bounds and examples for the two cases.

3. General bounds and extremal phenomena

Hartnell and Rall proved game-specific upper bounds sharper than the classical domatic bound. For graphs with minimum degree $\operatorname{dom_g}(G,A)$9,

$\operatorname{dom_g}(G,B)$0

For regular graphs of odd degree $\operatorname{dom_g}(G,B)$1, this improves in the Alice-first game to

$\operatorname{dom_g}(G,B)$2

They also showed that if $\operatorname{dom_g}(G,B)$3 has a perfect matching, then $\operatorname{dom_g}(G,B)$4 (Hartnell et al., 14 Aug 2025).

Subsequent work proved the first substantial general lower bound for the game parameter: $\operatorname{dom_g}(G,B)$5 for any graph $\operatorname{dom_g}(G,B)$6 of order $\operatorname{dom_g}(G,B)$7 and either choice of first player $\operatorname{dom_g}(G,B)$8. The same paper also gives the bounds

$\operatorname{dom_g}(G,B)$9

and

$G=(V,E)$0

where $G=(V,E)$1 is the domination number (English et al., 13 Mar 2026).

The lower-bound proof adapts probabilistic partition methods and Maker-Breaker games, specifically invoking Erdős–Selfridge-type thresholds through a hypergraph reduction in which a part size of $G=(V,E)$2 is sufficient for the argument (English et al., 13 Mar 2026). The same work shows that the $G=(V,E)$3 estimate is tight up to the logarithmic factor: for every $G=(V,E)$4, there exists a graph $G=(V,E)$5 with $G=(V,E)$6 and $G=(V,E)$7, and there exists a graph $G=(V,E)$8 with $G=(V,E)$9 but $D\subseteq V$0 (English et al., 13 Mar 2026). These constructions demonstrate that neither large minimum degree nor large classical domatic number forces a large game domatic number.

4. Exact values for standard graph classes

Hartnell and Rall determined exact game domatic numbers for several basic families (Hartnell et al., 14 Aug 2025).

Graph class	$D\subseteq V$1	$D\subseteq V$2
Tree $D\subseteq V$3 of order at least $D\subseteq V$4	$D\subseteq V$5 iff $D\subseteq V$6 has a perfect matching; otherwise $D\subseteq V$7	$D\subseteq V$8
Path $D\subseteq V$9	$V\setminus D$0	$V\setminus D$1 if $V\setminus D$2 even; $V\setminus D$3 if $V\setminus D$4 odd
Cycle $V\setminus D$5	$V\setminus D$6 if $V\setminus D$7; $V\setminus D$8 if $V\setminus D$9	$D$0 if $D$1 or $D$2 even; $D$3 if $D$4 odd
Complete graph $D$5	$D$6	$D$7 if $D$8 odd; $D$9 if $G$0 even
Complete bipartite graph $G$1, $G$2	$G$3 if $G$4 even; $G$5 otherwise	$G$6 if $G$7 even and $G$8 odd; $G$9 otherwise

For trees, the classical domatic number remains $G$00 for every tree of order at least $G$01, but the game parameter is stricter: $G$02 holds if and only if the tree has a perfect matching, equivalently in the statement given by Hartnell and Rall, when the order is even and there are no strong support vertices. In contrast, $G$03 for all trees (Hartnell et al., 14 Aug 2025).

The same paper also gives exact or sharp game values for additional constructions. For subdivision graphs $G$04, if $G$05 contains two edge-disjoint cycles sharing a single vertex, or four such cycles in a certain configuration, then $G$06. For grid subdivisions, if $G$07, then

$G$08

(Hartnell et al., 14 Aug 2025).

5. Score variant, subgraphs, and decomposition tools

Later work introduced a score version of the domatic game in which the terminal objective is not merely win or loss. With palette size $G$09 and first player $G$10, $G$11 is the number of colors that induce dominating sets at the end under optimal play, with Alice maximizing and Bob minimizing this value (English et al., 13 Mar 2026).

This variant is used to derive and transfer bounds. If $G$12, then

$G$13

If $G$14, then

$G$15

and the bound is tight: there exist graphs for which increasing the palette by one decreases the score by exactly $G$16. On the other hand, for sufficiently large $G$17,

$G$18

(English et al., 13 Mar 2026).

The same paper analyzes graph operations. For edge deletion,

$G$19

so deleting an edge cannot increase the domatic game number. In the score game, for any $G$20,

$G$21

By contrast, vertex deletion is not monotone and can either increase or decrease the domatic game number by large amounts in special cases (English et al., 13 Mar 2026).

For composite constructions, the paper gives technical gluing and union lemmas, named BobGood and AliceGood, that bound domatic game numbers when graphs are joined over small separators. It also identifies disjoint unions of cliques as an infinite family witnessing sharpness of the score-drop phenomenon (English et al., 13 Mar 2026).

6. Move order, palette monotonicity, and open problems

Move order affects the invariant quantitatively. Later work proves that the Alice-first and Bob-first parameters are mutually bounded: $G$22 and symmetrically with the roles of $G$23 and $G$24 reversed. The same work states the conjecture, based on all known examples, that

$G$25

and notes that examples exist in both directions (English et al., 13 Mar 2026).

Palette-size monotonicity appears in two distinct forms. Hartnell and Rall posed a vexing question about monotonicity in the number of sets available to Alice (Hartnell et al., 14 Aug 2025). Subsequent work resolves the win/loss version in Bob’s favor: adding more colors does not help Alice. If Bob has a winning strategy with $G$26 colors, then he also has a winning strategy with $G$27 colors, and therefore if $G$28, Alice cannot win for any $G$29 (English et al., 13 Mar 2026). An open question remains for the score setting: whether new dominating sets can ever be gained for $G$30 (English et al., 13 Mar 2026).

Other open problems recorded in the two papers include improving the lower bound to $G$31; determining whether the maximal effect of changing the first player is exactly $G$32; understanding the maximal drop in domatic game number under edge or vertex deletion; finding formulas for $G$33 and $G$34; and obtaining precise or sharp bounds for $G$35 and $G$36 (Hartnell et al., 14 Aug 2025). The papers also emphasize that the game invariants lack a trivial lower bound in terms of minimum degree alone, since it is possible for $G$37 even when $G$38 is large (Hartnell et al., 14 Aug 2025).

Taken together, these results place the domatic number game at the intersection of domination theory, positional games, and structural graph decomposition. The exact formulas for classical graph families coexist with extremal constructions showing that the game parameter can be much smaller than the ordinary domatic number, while the score formulation and subgraph lemmas supply a framework for further analysis (Hartnell et al., 14 Aug 2025).