Hardness of computing clique number and chromatic number for Cayley graphs

Computing the clique number and chromatic number of a general graph are well-known NP-Hard problems. Codenotti et al., (1998) showed that computing clique number and chromatic number are both NP-Hard problems for the class of circulant graphs. We show that computing clique number is NP-Hard for the class of Cayley graphs for the groups Gn, where G is any fixed finite group (e.g., cubelike graphs). We also show that computing chromatic number cannot be done in polynomial time (under the assumption NP≠ZPP) for the same class of graphs. Our presentation uses free Cayley graphs. The proof combines free Cayley graphs with quotient graphs and Goppa codes.