Game Domatic Number in Graph Theory

• Game domatic number is a game-theoretic concept that refines the classical domatic number by having players alternate vertex coloring to ensure every color class dominates the graph.
• It employs strategic methods such as perfect matchings and local obstruction patterns, with parity and degree conditions critically influencing the outcome.
• The invariant establishes novel bounds, reveals first-player asymmetries, and connects hypergraph reformulations with Maker–Breaker strategies for robust analysis.

The game domatic number is a game-theoretic refinement of the classical domatic number. In the domatic number game, introduced by Hartnell and Rall, Alice and Bob alternately color previously uncolored vertices of a graph from a palette $[k]=\{1,2,\dots,k\}$; when all vertices are colored, Alice wins if every color class is a dominating set, while Bob wins if at least one color class fails to dominate. The largest palette size for which Alice has a winning strategy defines the corresponding game domatic invariant, with two versions according to whether Alice or Bob moves first (Hartnell et al., 14 Aug 2025). Subsequent work reformulated these invariants as $\operatorname{dom_g}(G,A)$ and $\operatorname{dom_g}(G,B)$, established general degree-based lower bounds, introduced a score variant, and proved monotonicity with respect to palette size (English et al., 13 Mar 2026).

1. Formal definition and notation

Let $G=(V,E)$ be a finite simple graph. A set $D\subseteq V(G)$ is a dominating set if every vertex $x\in V(G)$ satisfies $D\cap N[x]\neq\emptyset$, where $N[x]$ denotes the closed neighborhood of $x$. A domatic partition of size $k$ is a partition

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

into pairwise disjoint dominating sets, and the classical domatic number is the maximum such $\operatorname{dom_g}(G,A)$1, denoted $\operatorname{dom_g}(G,A)$2 in one formulation and $\operatorname{dom_g}(G,A)$3 in another (Hartnell et al., 14 Aug 2025).

In the domatic number game with palette $\operatorname{dom_g}(G,A)$4, Alice and Bob alternate moves. On each move, the current player selects an uncolored vertex and assigns it a color from $\operatorname{dom_g}(G,A)$5. If $\operatorname{dom_g}(G,A)$6 denotes the final color class of color $\operatorname{dom_g}(G,A)$7, then Alice wins exactly when each $\operatorname{dom_g}(G,A)$8 is a dominating set of $\operatorname{dom_g}(G,A)$9; equivalently, every closed neighborhood $\operatorname{dom_g}(G,B)$0 contains all $\operatorname{dom_g}(G,B)$1 colors. Bob wins if some $\operatorname{dom_g}(G,B)$2 contains fewer than $\operatorname{dom_g}(G,B)$3 colors, so that at least one color class is not dominating (Hartnell et al., 14 Aug 2025).

Two first-player variants are standard. In the $\operatorname{dom_g}(G,B)$4-game, Alice moves first, and the maximum palette size for which she has a winning strategy is the game domatic number

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

In the $\operatorname{dom_g}(G,B)$6-game, Bob moves first, and the corresponding invariant is the delayed game domatic number

$\operatorname{dom_g}(G,B)$7

Later work packages these as $\operatorname{dom_g}(G,B)$8 with $\operatorname{dom_g}(G,B)$9, where $G=(V,E)$0 identifies the first player (English et al., 13 Mar 2026).

A basic monotonicity theorem resolves the palette-size question: if Bob wins the $G=(V,E)$1-game, then Bob also wins the $G=(V,E)$2-game for every $G=(V,E)$3. Consequently, $G=(V,E)$4 is well defined as the largest palette size for which Alice wins (English et al., 13 Mar 2026).

2. Position within domination theory

The game domatic number belongs to domination theory but is distinct from the game domination number. In the domination game, Dominator and Staller alternately choose vertices, each chosen vertex must dominate at least one new vertex, and the key invariant is the number of moves needed to build a single dominating set under optimal play, denoted $G=(V,E)$5 or $G=(V,E)$6 depending on the first player (Tananyan, 2014). By contrast, the domatic number game colors the entire vertex set and asks whether all color classes are dominating simultaneously (Hartnell et al., 14 Aug 2025).

This distinction is structural. The classical domination number $G=(V,E)$7 measures the size of one minimum dominating set, whereas the classical domatic number $G=(V,E)$8 measures how many pairwise disjoint dominating sets can be packed into a partition of $G=(V,E)$9. The domatic number game inherits the partition-based objective, not the move-length objective of the domination game (Tananyan, 2014).

The immediate inequality is

$D\subseteq V(G)$0

because every Alice-winning play produces a domatic partition of size $D\subseteq V(G)$1 (Hartnell et al., 14 Aug 2025). Ore’s theorem, as used in the game-domatic literature, implies that every graph without isolated vertices has domatic number at least $D\subseteq V(G)$2, but the game variants can still collapse to $D\subseteq V(G)$3 under adversarial play (Hartnell et al., 14 Aug 2025).

3. General bounds and extremal separations

The first general upper bounds sharpen the trivial estimate $D\subseteq V(G)$4. If $D\subseteq V(G)$5 has no isolated vertices, then

$D\subseteq V(G)$6

If $D\subseteq V(G)$7 is regular of odd degree, then the Alice-first bound improves to

$D\subseteq V(G)$8

These estimates are sharp on complete graphs (Hartnell et al., 14 Aug 2025).

Several lower bounds are also available. If $D\subseteq V(G)$9 has a perfect matching, then

$x\in V(G)$0

and if $x\in V(G)$1 has $x\in V(G)$2 universal vertices, then both first-player variants satisfy

$x\in V(G)$3

Later work established the asymptotic lower bound

$x\in V(G)$4

for every graph $x\in V(G)$5 of order $x\in V(G)$6 and both $x\in V(G)$7, and also proved the upper bound

$x\in V(G)$8

in terms of the classical domination number (English et al., 13 Mar 2026).

The most striking phenomena are separation results. For every $x\in V(G)$9, there exists a graph $D\cap N[x]\neq\emptyset$0 with

$D\cap N[x]\neq\emptyset$1

and there exists a graph $D\cap N[x]\neq\emptyset$2 with

$D\cap N[x]\neq\emptyset$3

for both first-player choices (English et al., 13 Mar 2026). Hence neither high minimum degree nor large classical domatic number forces a large game domatic number.

4. Exact values on standard graph classes

A substantial part of the theory consists of exact computations for canonical graph families (Hartnell et al., 14 Aug 2025).

Graph class	$D\cap N[x]\neq\emptyset$4	$D\cap N[x]\neq\emptyset$5
Tree $D\cap N[x]\neq\emptyset$6 of order at least $D\cap N[x]\neq\emptyset$7	$D\cap N[x]\neq\emptyset$8	$D\cap N[x]\neq\emptyset$9 iff $N[x]$0 has a perfect matching, else $N[x]$1
Path $N[x]$2	$N[x]$3	$N[x]$4 if $N[x]$5 is odd, $N[x]$6 if $N[x]$7 is even
Cycle $N[x]$8	$N[x]$9 if $x$0, else $x$1	$x$2 if $x$3 or $x$4 is even, else $x$5
Complete graph $x$6	$x$7	$x$8 if $x$9 is odd, $k$0 if $k$1 is even

For complete bipartite graphs $k$2 with $k$3, parity governs both invariants. The exact formulas are

$k$4

and

$k$5

These formulas show that the smaller bipartition class controls the palette size, while parity determines which player can force compression of the available colors (Hartnell et al., 14 Aug 2025).

For Cartesian grids $k$6, the Bob-first game is often favorable to Alice: if at least one of $k$7 or $k$8 is even, then

$k$9

By contrast, the Alice-first two-row case collapses: $\operatorname{dom_g}(G,A)$00 Subdivision graphs are also fragile: if $\operatorname{dom_g}(G,A)$01 contains two edge-disjoint cycles sharing a single vertex, then

$\operatorname{dom_g}(G,A)$02

and in particular

$\operatorname{dom_g}(G,A)$03

(Hartnell et al., 14 Aug 2025).

Later work added another exact family: if $\operatorname{dom_g}(G,A)$04 is a $\operatorname{dom_g}(G,A)$05-tree, meaning a graph formed by repeatedly gluing copies of $\operatorname{dom_g}(G,A)$06 at single vertices, then

$\operatorname{dom_g}(G,A)$07

(English et al., 13 Mar 2026).

5. Strategic methods and auxiliary formulations

The 2025 treatment is largely combinatorial and local. Perfect matchings yield immediate Alice strategies in the Bob-first game: if Bob colors one endpoint of a matching edge, Alice colors the other endpoint with the opposite color, so both colors appear in every matched pair (Hartnell et al., 14 Aug 2025). Conversely, local obstructions drive Bob’s wins. If a graph has a strong support vertex, then both game-domatic invariants are $\operatorname{dom_g}(G,A)$08. On cycles and grids, Bob exploits degree-2 path patterns of the form $\operatorname{dom_g}(G,A)$09, where two consecutive internal vertices already carry the same color $\operatorname{dom_g}(G,A)$10; from such a configuration he can force a monochromatic closed neighborhood and thereby kill at least one color class (Hartnell et al., 14 Aug 2025).

The 2026 paper adds two major methodological layers. First, it introduces the score variant $\operatorname{dom_g}(G,A)$11, defined as the number of colors whose color classes are dominating under optimal play with palette size $\operatorname{dom_g}(G,A)$12. Alice wins the ordinary domatic game with palette $\operatorname{dom_g}(G,A)$13 exactly when

$\operatorname{dom_g}(G,A)$14

If $\operatorname{dom_g}(G,A)$15, then $\operatorname{dom_g}(G,A)$16. If $\operatorname{dom_g}(G,A)$17, then

$\operatorname{dom_g}(G,A)$18

and if $\operatorname{dom_g}(G,A)$19 and $\operatorname{dom_g}(G,A)$20, then

$\operatorname{dom_g}(G,A)$21

This score formalism is used to compare first-player variants and to control the effect of larger palettes (English et al., 13 Mar 2026).

Second, the lower bound $\operatorname{dom_g}(G,A)$22 is proved through a hypergraph reformulation and Maker–Breaker methods. Closed neighborhoods become hyperedges, a uniformization step produces edges of controlled size, and an Erdős–Selfridge-type criterion is then used to build Alice strategies that guarantee all colors meet all closed neighborhoods (English et al., 13 Mar 2026).

Graph operations exhibit mixed behavior. For edge deletion,

$\operatorname{dom_g}(G,A)$23

and for the score variant,

$\operatorname{dom_g}(G,A)$24

Moreover, the score can drop by at most $\operatorname{dom_g}(G,A)$25 under deletion of a single edge. Vertex deletion is explicitly non-monotone for $\operatorname{dom_g}(G,A)$26 (English et al., 13 Mar 2026).

6. First-player asymmetry and open problems

The first-player effect is substantial but appears constrained. Complete graphs already show that the Bob-first invariant can exceed the Alice-first invariant: $\operatorname{dom_g}(G,A)$27 (Hartnell et al., 14 Aug 2025). Later work proved the general comparison inequalities

$\operatorname{dom_g}(G,A)$28

and

$\operatorname{dom_g}(G,A)$29

It also proposed the sharper conjecture

$\operatorname{dom_g}(G,A)$30

The known examples fit this conjecture exactly: for $\operatorname{dom_g}(G,A)$31,

$\operatorname{dom_g}(G,A)$32

while for the graph obtained by attaching a pendant vertex to a vertex of $\operatorname{dom_g}(G,A)$33,

$\operatorname{dom_g}(G,A)$34

(English et al., 13 Mar 2026).

One early open problem asked whether increasing the palette size could convert a Bob-win into an Alice-win. This has now been answered negatively: Bob-win monotonicity in the palette size implies that once Bob wins at $\operatorname{dom_g}(G,A)$35, he wins at every larger palette (Hartnell et al., 14 Aug 2025, English et al., 13 Mar 2026).

Several problems remain open. The asymptotic lower bound $\operatorname{dom_g}(G,A)$36 has no matching general upper bound in the same scale. The behavior of $\operatorname{dom_g}(G,A)$37 above $\operatorname{dom_g}(G,A)$38 is not fully understood. The effect of deleting a single vertex or edge on $\operatorname{dom_g}(G,A)$39 remains only partially controlled. Product and union formulas are also incomplete: the 2025 paper explicitly asks for bounds on $\operatorname{dom_g}(G,A)$40 and $\operatorname{dom_g}(G,A)$41, and for expressions for $\operatorname{dom_g}(G,A)$42 and $\operatorname{dom_g}(G,A)$43 in terms of the component invariants (Hartnell et al., 14 Aug 2025, English et al., 13 Mar 2026).

The resulting picture is that the game domatic number is neither a minor perturbation of the classical domatic number nor a direct analogue of the game domination number. It is a distinct adversarial partition parameter, controlled by dominating color classes, highly sensitive to parity and local obstructions, yet also governed by global degree-based phenomena and hypergraph methods.