EDGe: Edge-Distinguishing Coloring Game

• EDGe is a finite impartial game played on graphs using λ(G) colors, enforcing that induced edge-colors remain unique throughout play.
• It employs strategic mechanisms like repeated-color freezing, parity reduction, and symmetry exploitation to secure winning positions across different graph families.
• Analytical and computational studies on graphs such as paths, cycles, and complete bipartite graphs reveal intricate interactions between the minimal palette and game dynamics.

Edge-Distinguishing Game (EDGe) is a finite impartial normal-play graph-coloring game in which two players alternately color vertices of a graph $G$ using exactly $\lambda(G)$ colors, where $\lambda(G)$ is the edge-distinguishing chromatic number of $G$, subject to the constraint that the induced edge-colors on the currently colored subgraph remain pairwise distinct. A move is legal only if the resulting partial coloring is still edge-distinguishing, and the player making the last legal move wins. Introduced in 2025, EDGe places a dynamic game on top of the static theory of edge-distinguishing colorings and analyzes it through $P$- and $N$-positions, explicit strategy proofs, and exhaustive game-tree computation (Benjamin et al., 28 Aug 2025).

1. Formal definition and underlying parameter

Let $[k]=\{1,2,\dots,k\}$. A $k$-coloring of a graph $G$ is a map

$c:V(G)\to [k].$

This induces an edge-coloring

$\lambda(G)$0

where the notation is used in the multiset sense, so colors such as $\lambda(G)$1 are allowed. A coloring is edge-distinguishing if $\lambda(G)$2 is injective. The edge-distinguishing chromatic number is

$\lambda(G)$3

EDGe is played with exactly $\lambda(G)$4 colors (Benjamin et al., 28 Aug 2025).

The game state is a partial $\lambda(G)$5-coloring

$\lambda(G)$6

and legality is tested only on the induced subgraph $\lambda(G)$7. A move consists of choosing an uncolored vertex and a color so that the extended partial coloring still induces an injective edge-coloring on $\lambda(G)$8. If an uncolored vertex cannot be assigned any color without violating injectivity, it is unmarkable. A position is terminating if every vertex is colored or every uncolored vertex is unmarkable (Benjamin et al., 28 Aug 2025).

The choice of $\lambda(G)$9 is structural rather than cosmetic. The paper proves that on $\lambda(G)$0, Player 1 wins when the game is played with $\lambda(G)$1 colors, but Player 2 wins if a fourth color is added. In this sense, EDGe is not merely a game on a graph; it is a game on a graph together with the minimal edge-distinguishing palette (Benjamin et al., 28 Aug 2025).

2. Impartial-game structure and computational model

EDGe is treated as a finite impartial game with no draws. The standard recursive classification is used: terminal positions are $\lambda(G)$2-positions; a position is an $\lambda(G)$3-position if it has at least one move to a $\lambda(G)$4-position; and it is a $\lambda(G)$5-position if every legal move goes to an $\lambda(G)$6-position. The paper invokes Zermelo’s theorem to justify that one of the two players has a winning strategy from every initial position (Benjamin et al., 28 Aug 2025).

The computational approach adapts a Python solver previously used for the Game of Cycles. The directed game tree has legal partial colorings as nodes and legal moves as edges. Terminal states are labeled first, after which backward induction classifies all predecessors as $\lambda(G)$7 or $\lambda(G)$8. The empty board is then winning for Player 1 iff it is an $\lambda(G)$9-position. This is a direct minimax evaluation on the reachable legal state space (Benjamin et al., 28 Aug 2025).

A central symmetry principle is that if two boards are isomorphic as partially colored graphs, they play identically. The paper also proposes further reductions: quotienting by color permutations, quotienting by graph isomorphisms, and recognizing when a partially colored board becomes equivalent to a position on a smaller or different graph after some vertices become unmarkable. One explicit example is a state on $G$0 that becomes equivalent to a state on $G$1 once a same-colored configuration forces an unmarkable vertex (Benjamin et al., 28 Aug 2025).

3. Strategic mechanisms

A recurring tactic is repeated-color freezing. If an uncolored vertex is adjacent to two already colored vertices with the same color, then coloring that vertex often creates duplicate induced edge-colors immediately. This turns local same-color placements into a mechanism for making large sets of vertices unmarkable. The complete-graph, wheel, and triangular-book proofs all exploit this phenomenon (Benjamin et al., 28 Aug 2025).

A second motif is parity reduction. In many analytically solved families, play reaches a phase in which no further unmarkables can be created, the remaining legal moves are forced one-by-one, and the winner is determined by the parity of the remaining markable vertices. This is the backbone of the complete bipartite and wheel classifications, and it also appears in path and cycle endgames (Benjamin et al., 28 Aug 2025).

A third motif is geometric or reflective symmetry on sparse graphs. For $G$2, the winning strategy is centered at the midpoint and then mirrors the opponent’s move across the center. For $G$3, Player 1 starts at $G$4 and tries to create a second copy of the same color at $G$5, constraining all remaining edge-colors incident to that color class. For the Moser spindle, the graph is decomposed into two symmetric wings, and a response in one wing reduces the residual game to a smaller favorable subgraph (Benjamin et al., 28 Aug 2025).

These mechanisms show that EDGe strategy is neither purely local nor purely palette-driven. Legal play depends on the global injectivity of induced edge-color multisets, but the decisive tactics are often small certificates: a repeated color at the right pair of vertices, a symmetry-forcing reply, or a move that converts the board to a solved residual instance.

4. Solved graph families and exact outcomes

The paper gives exact analytical results for several families and computer-determined outcomes for others. The following summary consolidates the stated results (Benjamin et al., 28 Aug 2025).

Graph or family	Winner under EDGe	Basis
$G$6, $G$7	Player 2	analytic
$G$8	Player 1 iff $G$9 is odd; Player 2 iff $P$0 is even	analytic
$P$1, $P$2	Player 1 if $P$3 is odd; Player 2 if $P$4 is even	analytic
$P$5, $P$6	Player 2	analytic
Paths	$P$7: Player 1; $P$8: Player 2; $P$9: Player 1; $N$0: Player 2	mixed
Cycles	$N$1: Player 2; $N$2: Player 2; $N$3: Player 1; $N$4: Player 1; $N$5: Player 2; $N$6: Player 2	mixed
Triangular chorded cycles $N$7	$N$8: Player 2; $N$9: Player 1; $[k]=\{1,2,\dots,k\}$0: Player 1; $[k]=\{1,2,\dots,k\}$1: Player 2; $[k]=\{1,2,\dots,k\}$2: Player 1	computational
Other graphs	Cube, octahedron, $[k]=\{1,2,\dots,k\}$3: Player 2; Petersen, Moser spindle, envelope: Player 1	mixed

The analytic proofs rely on explicit values of the static parameter, including

$[k]=\{1,2,\dots,k\}$4

$[k]=\{1,2,\dots,k\}$5

together with previously known values such as $[k]=\{1,2,\dots,k\}$6, $[k]=\{1,2,\dots,k\}$7, $[k]=\{1,2,\dots,k\}$8, $[k]=\{1,2,\dots,k\}$9, $k$0, $k$1, $k$2, and $k$3 (Benjamin et al., 28 Aug 2025).

Beyond simple graphs, the paper also proves a parity classification for the looped complete graph $k$4. For $k$5, Player 1 wins if $k$6 is odd and Player 2 wins if $k$7 is even. The proof uses the fact that once a loop produces the edge-color $k$8, all subsequent legal play is forced to use distinct new colors, so the game collapses to a parity count on the vertices (Benjamin et al., 28 Aug 2025).

5. Relation to static edge-distinguishing theory

EDGe is built on a pre-existing static optimization problem: the edge-distinguishing chromatic number. Earlier work determined exact values of this parameter for petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs, and characterized feasibility in terms of embeddings into the looped complete graph $k$9 (Fickes et al., 2021). That static theory supplies the palette size $G$0 and, more importantly, a structural language for describing edge-distinguishing colorings independently of gameplay.

The older embedding viewpoint is especially relevant because it reframes an edge-distinguishing coloring as an edge-injective homomorphism into $G$1. A plausible implication is that future EDGe analyses on petal graphs, chorded cycles, and spider graphs may be organized around the same embedding templates, now interpreted dynamically as partial embeddings that must remain extendable after each move (Fickes et al., 2021).

EDGe is also adjacent to, but distinct from, edge-sum distinguishing labeling. In that setting, vertices receive injective integer labels and each edge gets the sum of its endpoint labels; the objective is uniqueness of sums rather than uniqueness of unordered endpoint-color multisets. That literature includes a Maker-Breaker style labeling game and structural theorems for paths, trees, cycles, complete bipartite graphs, grids, and complete graphs (Bok et al., 2018). The relation is conceptual rather than definitional: both subjects are vertex-labeling games with edge-level injectivity constraints, but the induced invariants are different.

6. Distinctions from adjacent graph games and current research directions

EDGe is not interchangeable with other recent edge-based graph games. The edge-colouring games studied by Erdős, and by Bensmail and Mc Inerney, are played by alternately coloring edges of a clique red and blue and then scoring clique number, maximum degree, or local-majority vertex capture; they do not use injective induced edge-colors on partially colored vertex sets as the legality condition (Cambie et al., 6 May 2025). The undirected edge geography game on grids is an impartial edge-deletion token game with a moving root and normal-play terminal condition, again structurally different from EDGe despite its edge-centric move mechanism (Sereekiatdilok et al., 16 Apr 2025). Likewise, edge-degeneracy arises from an edge-blocking cops-and-robber game and yields width parameters and decomposition theorems rather than a distinguishing coloring game (Limnios et al., 2020).

Within EDGe itself, the paper leaves several families unresolved and formulates a broad program rather than a finished classification. Open questions include general winner patterns for paths, cycles, triangular chorded cycles, spider graphs, generic chorded cycles, and petal graphs; whether Player 2 always wins on triangular ladders $G$2 for $G$3; whether there is a general strategy principle of “winning by winning on subgraphs”; how graph operations $G$4 interact with winnability; how loops affect play more generally; when changing the number of colors changes the winner; and how to scale the exhaustive solver through stronger symmetry reduction and state compression (Benjamin et al., 28 Aug 2025).

The present state of the subject is therefore sharply bifurcated. On one side, EDGe already has exact parity or family-wise classifications for complete graphs, complete bipartite graphs, wheels, triangular books, several paths and cycles, the Moser spindle, and $G$5. On the other side, its broader theory is still being assembled, with the most conspicuous unresolved theme being how the static parameter $G$6 interacts with dynamic obstruction creation in larger structured families.