A quantum protocol to win the graph colouring game on all Hadamard graphs

This paper deals with graph colouring games, an example of pseudo-telepathy, in which two provers can convince a verifier that a graph $G$ is $c$-colourable where $c$ is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph $G_N$ with $N=c=16$. Their protocol applies only to Hadamard graphs where $N$ is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of $G_{12}$ having 1609 vertices, with $c=12$. Combined with a result of Frankl and Rodl, our result shows that all sufficiently large Hadamard graphs yield pseudo-telepathy games.