A Quantum Protocol to Win the Graph Colouring Game on All Hadamard Graphs

David AVIS

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E89-A    No.5    pp.1378-1381
Publication Date: 2006/05/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e89-a.5.1378
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
graph colouring game,  pseudo-telepathy,  Hadamard graph,  quantum chromatic number,  

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This paper deals with graph colouring games, an example of pseudo-telepathy, in which two players 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 GN 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 G12 having 1609 vertices, with c=12. Moreover combined with a result of Godsil and Newman, our result shows that all Hadamard graphs GN (N ≥ 12) and c=N yield pseudo-telepathy games.