Quantum chromatic number

This report gather some notable results found in [SS12], giving also details on some foundation results presented in [CMN+07]. The results covered in Section 2 are answers given in [SS12] to questions of the article [CMN+07]. Graph parameters are studied a lot as they are a way to highlight a general structure from a graph. In that sense, the chromatic number is studied a lot as it shows the multipartitions. The chromatic number can be seen as the smallest number needed to color a graph, with respecting the condition that two adjacent vertices must not have the same color. But it can also be studied with the following protocol. We consider an integer c and a graph G. We denote [c] = {1, . . . , n}. A referee gives to Alice and Bob two vertices that are either the same one, or belong to one edge. Alice and Bob answer two colors in [c] that should be the same if they were given the same vertex, and different in the other case. The chromatic number coincide with the smallest c for which Alice and Bob have a probability 1 winning strategy.