Graph Coloring Nonlocal Game

• Graph coloring nonlocal game is a two-player one-round game where vertices are assigned colors following proper coloring rules and graph homomorphism constraints.
• It incorporates synchronous strategies using classical, quantum, and non-signalling models, with a formulation based on operator-algebraic projections.
• The universality of the 3-coloring reduction connects arbitrary synchronous games to graph coloring, enabling analyses in approximate strategies and undecidability.

A graph coloring nonlocal game is a two-player one-round game in which a referee sends vertices of a graph to two separated players, the players reply with colors, and the winning condition encodes the constraints of a proper coloring. In its standard form, for a graph $G=(V,E)$ and an integer $k$, the referee sends $v,w\in V$, each player answers a color in $\{1,\dots,k\}$, and the players must give the same color on equal questions and different colors on adjacent vertices. Equivalently, the game is the graph homomorphism game $\mathrm{Hom}(G,K_k)$, so perfect classical strategies coincide with proper $k$-colorings of $G$ (Botteron et al., 2024). Recent work places this game at the center of a broader theory of synchronous nonlocal games: every synchronous game can be encoded as a $3$-coloring game on an explicit graph, and the same encoding extends to approximate strategies, undecidability phenomena, and operator-algebraic formulations of quantum graph coloring (Harris, 2023).

1. Standard formulation and homomorphism viewpoint

The standard graph coloring nonlocal game is specified by a graph $G=(V,E)$ and a color set $\{1,\dots,k\}$. Alice receives a vertex $k$0, Bob receives a vertex $k$1, and they answer $k$2. They win precisely when the graph-coloring constraints hold: if $k$3, then $k$4; if $k$5, then $k$6 (Botteron et al., 2024).

This is the nonlocal reformulation of ordinary graph coloring. A proper coloring is a map

$k$7

such that $k$8 whenever $k$9, and a perfect deterministic strategy exists iff the graph is $v,w\in V$0-colorable in the usual classical sense (Botteron et al., 2024). The same correspondence is commonly written as a graph homomorphism

$v,w\in V$1

which explains the notation $v,w\in V$2 used throughout the graph-homomorphism literature (Botteron et al., 2024).

The game is naturally synchronous. In the more general language of synchronous nonlocal games $v,w\in V$3, synchrony means

$v,w\in V$4

The graph coloring game is the special case in which $v,w\in V$5, $v,w\in V$6, and the rule function forbids unequal answers on equal inputs and equal answers on adjacent inputs (Harris, 2023). This places graph coloring within the general framework of synchronous correlations

$v,w\in V$7

with perfectness expressed by algebraic zero constraints on the disallowed answer pairs (Botteron et al., 2024).

2. Strategy models and algebraic constraints

A strategy for a graph coloring nonlocal game is a conditional distribution

$v,w\in V$8

and the standard classes considered in the synchronous literature are loc, q, qa, and qc, corresponding respectively to classical/local, finite-dimensional quantum, closure of quantum, and quantum commuting models (Harris, 2023). In each model, a strategy is perfect when all losing events have zero probability.

For the coloring game, the perfectness constraints take the explicit form

$v,w\in V$9

and, for every edge $\{1,\dots,k\}$0,

$\{1,\dots,k\}$1

(Botteron et al., 2024). In the non-signalling setting, one additionally requires that

$\{1,\dots,k\}$2

does not depend on $\{1,\dots,k\}$3, and that

$\{1,\dots,k\}$4

does not depend on $\{1,\dots,k\}$5 (Botteron et al., 2024).

In the synchronous quantum framework, perfect strategies are encoded by projections $\{1,\dots,k\}$6 satisfying

$\{1,\dots,k\}$7

so a coloring game becomes a system of projection-valued constraints indexed by vertices and colors (Harris, 2023). This operator-algebraic formulation is fundamental because it allows one to compare classical, finite-dimensional, closure, commuting, $\{1,\dots,k\}$8, hereditary, and algebraic notions of colorability within a single formalism (Harris, 2023).

A further extension replaces classical questions by quantum questions. In the quantum-to-classical graph homomorphism game, the inputs are quantum states in

$\{1,\dots,k\}$9

while the outputs remain classical labels, and specialization to a complete target graph $\mathrm{Hom}(G,K_k)$0 yields a coloring problem for quantum graphs (Brannan et al., 2020). In that setting, the quantum chromatic number is defined by

$\mathrm{Hom}(G,K_k)$1

providing a noncommutative analogue of graph coloring (Brannan et al., 2020).

3. Universality of the $\mathrm{Hom}(G,K_k)$2-coloring game

A central structural result is that graph coloring nonlocal games are not merely examples of synchronous games; the $\mathrm{Hom}(G,K_k)$3-coloring game is universal for all synchronous nonlocal games. Given any synchronous game $\mathrm{Hom}(G,K_k)$4 with $\mathrm{Hom}(G,K_k)$5 and $\mathrm{Hom}(G,K_k)$6, there is a graph $\mathrm{Hom}(G,K_k)$7 such that the $\mathrm{Hom}(G,K_k)$8-coloring game $\mathrm{Hom}(G,K_k)$9 is weakly $k$0-equivalent to $k$1 (Harris, 2023).

The graph has at most

$k$2

vertices (Harris, 2023). The construction is built from a control triangle $k$3 together with gadgets encoding answer structure and forbidden tuples. The summary identifies three ingredients used in the proof: the $k$4 rook graph $k$5, triangular prism gadgets, and commutation facts special to $k$6-colorings (Harris, 2023). This yields an explicit reduction from arbitrary synchronous constraints to a graph $k$7-coloring instance.

The operational content of weak $k$8-equivalence is that the existence of perfect winning strategies is preserved across models $k$9 (Harris, 2023). In particular, if one of the two games has a winning strategy in such a model, then so does the other. A direct corollary is a quantum version of Lovász’s reduction from $G$0-coloring to $G$1-coloring: if $G$2 has $G$3 vertices and $G$4 edges, then the reduction produces a graph with at most

$G$5

vertices (Harris, 2023).

The same paper shows that the “graph of the game” $G$6, with vertex set $G$7 and edges determined by forbidden tuples, gives an independence-number encoding of the original game. Specifically,

$G$8

is hereditarily $G$9-equivalent to $3$0, and for $3$1,

$3$2

(Harris, 2023). Thus the quantum chromatic number, quantum independence number, and quantum clique number encode winning strategies for all synchronous games.

4. Approximate colorings, gap transfer, and undecidability

The universality result extends beyond perfect play. For every synchronous non-local game $3$3 with $3$4 and $3$5, there is an associated graph $3$6 such that approximate winning strategies for $3$7 and for the $3$8-coloring game $3$9 are preserved (Harris, 2024).

The associated graph again begins with a control triangle

$G=(V,E)$0

contains gadgets $G=(V,E)$1 that are copies of $G=(V,E)$2, triangular prism gadgets $G=(V,E)$3, and gadgets $G=(V,E)$4 for tuples in $G=(V,E)$5 (Harris, 2024). The game $G=(V,E)$6 is analyzed under the uniform edge distribution

$G=(V,E)$7

while the original synchronous game is evaluated under the uniform distribution on $G=(V,E)$8 (Harris, 2024).

In the forward direction, if a synchronous $G=(V,E)$9-correlation for $\{1,\dots,k\}$0 wins with probability $\{1,\dots,k\}$1, then the induced synchronous $\{1,\dots,k\}$2-coloring strategy on $\{1,\dots,k\}$3 wins with probability at least

$\{1,\dots,k\}$4

for $\{1,\dots,k\}$5 (Harris, 2024). In the reverse direction, if a synchronous $\{1,\dots,k\}$6-correlation for $\{1,\dots,k\}$7 wins with probability $\{1,\dots,k\}$8, then there exists a synchronous $\{1,\dots,k\}$9-correlation $k$00 for $k$01 such that

$k$02

for a polynomial $k$03 in $k$04 and $k$05 (Harris, 2024).

This quantitative transfer yields a gapped undecidability result for quantum $k$06-coloring. There exists $k$07 such that, for $k$08, the gapped $k$09-promise problem for the synchronous $k$10-value of the $k$11-coloring game is undecidable (Harris, 2024). The same framework implies that there exists $k$12 such that deciding whether

$k$13

is undecidable, under the stated promise (Harris, 2024). This places graph coloring nonlocal games directly inside the undecidability theory of synchronous quantum games.

5. Non-signalling strategies and communication complexity

Graph coloring nonlocal games also serve as a testbed for separations between classical, quantum, and non-signalling correlations. In the standard coloring game, classical perfect strategies correspond exactly to proper colorings, but the non-signalling model is more permissive (Botteron et al., 2024).

A recent analysis of graph isomorphism, graph coloring, and vertex distance games studies the graph coloring game as one of three central graph-theoretic nonlocal games and proves that perfect non-signalling strategies can collapse communication complexity under favorable conditions (Botteron et al., 2024). The result is presented jointly for the three games, but graph coloring is explicitly included in the scope of the theorem-level statement. The paper also emphasizes the standard hierarchy in which classical strategies are the most restrictive, quantum strategies are more powerful, and non-signalling strategies are the most general (Botteron et al., 2024).

The non-signalling constraints for coloring are the usual marginal-independence conditions, together with the zero constraints for equal and adjacent inputs (Botteron et al., 2024). This makes the graph coloring game a clean vehicle for comparing correlation models. A plausible implication is that its importance is not limited to colorability itself; the game also acts as a probe for principles such as no-collapse of communication complexity, which are used to distinguish physically motivated correlation sets from broader non-signalling ones (Botteron et al., 2024).

6. Operator-algebraic and quantum-graph generalizations

The graph coloring nonlocal game admits a broad operator-algebraic generalization in which the source object is a quantum graph rather than a classical graph. In the quantum-to-classical homomorphism game, the questions are quantum and the answers remain classical, producing a “quantum-classical game” that extends the usual graph homomorphism game (Brannan et al., 2020).

The corresponding game algebra is denoted

$k$14

and is generated by projection matrices $k$15 subject to projection, off-diagonal graph, and diagonal synchronous constraints (Brannan et al., 2020). The paper states the dictionary between perfect strategies and algebraic realizations: classical perfect strategies correspond to unital $k$16-homomorphisms into $k$17, finite-dimensional quantum strategies to unital $k$18-homomorphisms into matrix algebras, commuting strategies to homomorphisms into tracial algebras, and algebraic strategies to nonzero game algebras (Brannan et al., 2020).

When the target is a complete graph $k$19, the homomorphism game becomes a coloring game for quantum graphs. The framework yields quantum chromatic numbers

$k$20

together with the chain

$k$21

For quantum complete graphs $k$22, the paper proves that for the $k$23, $k$24, $k$25, $k$26, and hereditary models,

$k$27

(Brannan et al., 2020).

A particularly striking algebraic statement is that every quantum graph is $k$28-colorable in the algebraic model, i.e.

$k$29

and equivalently the game algebra of the $k$30-coloring game is always non-trivial: $k$31 (Brannan et al., 2020). This does not assert finite-dimensional or commuting realizability in general, but it shows that the coloring game formalism extends far beyond classical graphs.

7. Distinction from classical sequential coloring games

The phrase “graph coloring game” is also used for several classical perfect-information combinatorial games in which players alternate coloring vertices of a graph. Those games are not nonlocal games. In the Bodlaender-style game studied on $k$32 grids, Alice and Bob alternately color an uncolored vertex, Alice wins if all vertices are eventually colored, and the parameter of interest is the game chromatic number $k$33 (Brosse et al., 2024). Likewise, several start/pass and connected variants are PSPACE-complete as decision problems (Marcilon et al., 2019).

These classical coloring games differ formally from graph coloring nonlocal games in every essential respect: they are sequential rather than one-round, there is no referee-generated pair of questions, there is no correlation $k$34, and there are no classical/quantum/non-signalling strategy classes (Brosse et al., 2024). The same distinction applies to other impartial coloring games derived from proper, oriented, weak, $k$35-distance, or sequential coloring rulesets (Beaulieu et al., 2012).

The nonlocal graph coloring game is therefore best understood as a distributed constraint-satisfaction game, equivalent to a homomorphism problem into $k$36, and embedded in the theory of synchronous correlations, operator algebras, and quantum graph parameters (Harris, 2023). The classical sequential games are related only by subject matter and terminology. This distinction is important because several recent results on universality, approximate preservation, non-signalling collapse, and quantum-graph colorings concern the nonlocal formulation specifically, not the adversarial alternating-move games (Botteron et al., 2024).