On the quantum chromatic number of a graph
Abstract
We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 2, nor if it is 3 in a restricted quantum model; on the other hand, we exhibit a graph on 18 vertices and 44 edges with chromatic number 5 and quantum chromatic number 4.
 Publication:

arXiv eprints
 Pub Date:
 August 2006
 arXiv:
 arXiv:quantph/0608016
 Bibcode:
 2006quant.ph..8016C
 Keywords:

 Quantum Physics;
 Mathematics  Combinatorics
 EPrint:
 7 pages, 1 eps figure