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 to be NP-Hard problems. Codenotti et al. (Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. Linear Algebra Appl., 285(1-3): 123–142, 1998) showed that computing the clique number and chromatic number are still NP-Hard problems for the class of circulant graphs. We show that these problems are NP-Hard for the class of Cayley graphs for the groups Gn, where G is any fixed finite group. Our method combines free Cayley graphs with quotient graphs and Goppa codes. In his celebrated 1972 paper [7], Karp established the NP-Completeness of 21 combinatorial problems. Amongst those problems are the CLIQUE problem and the CHROMATIC NUMBER problem. CLIQUE takes a graph X and an integer k and decides whether X contains a clique of size k as a subgraph. CHROMATIC NUMBER takes a graphX and an integer k and decides whether there is a proper colouring of X using at most k colours. The clique number of a graph X is the size of the largest clique contained in X , and is denoted by ω(X). Since deciding whether a general graph X contains a clique of size k is NP-Complete, the problem of computing the clique number of X is NP-Hard. The chromatic number of a graph X is the smallest integer k such that X has a proper k-colouring, and is denoted by χ(X). Again, since deciding whether a general graph X can be coloured properly using at most k colours is NP-Complete, computing the chromatic number of X is NP-Hard. Some of the graph theoretic problems in Karp’s list become easier when restricted to a subclass of graphs. For instance, deciding whether a graph X has a subset of vertices with size k that covers all of the edges of X is NP-Complete. However, if X is bipartite one can use the Hungarian Algorithm (for instance) to find a minimum vertex cover of X in polynomial time. There are also subclasses of graphs for which computing clique number and chromatic number are easy problems. For example, acyclic graphs have easily computable clique numbers, and complete graphs have easily computable chromatic numbers. In 1998, Codenotti, Gerace, and Vigna [4] proved that computing clique number and chromatic number are NP-Hard when restricted to the class of