Hardness of computing clique number and chromatic number for Cayley graphs

Abstract

Computing the clique number and chromatic number of a general graph are well-known 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][124][125][126][127][128][129][130][131][132][133][134][135][136][137][138][139][140][141][142] 1998) showed that computing clique number and chromatic number are still 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 G n , 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 P = NP) for the same class of graphs. Our presentation uses free Cayley graphs. The proof combines free Cayley graphs with quotient graphs and Goppa codes.In his celebrated 1972 paper [10], 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 graph X 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 ω(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 χ(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.

Probe Machine Based Computing Model for Maximum Clique Problem

“…The maximum clique problem is a classic combinatorial optimization problem in graph theory, and is one of the first problems shown to be NP‐complete [ 22 , 23 ] . The concept of clique may trace its history to the research of social science.…”

“…A variety of algorithms have been proposed so far. These algorithms can be briefly categorized into two classes: exact [ 20–29 ] and heuristic algorithms [ 30–35 ] . The drawback of exact algorithms is that the time complexity of the algorithms increases exponentially with the size of the problem.…”

The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

“…In general, finding the clique number of a Cayley graph is NP-hard [GR17]. It is not possible to compute the exact clique number of X = P P (q, 2d, I) for q > 2000 using the current computational power.…”

Strong cliques in vertex‐transitive graphs

“…It is easy to see that for every graph we have . For some results regarding chromatic number of vertex‐transitive graphs (and some of the subclasses of vertex‐transitive graphs), see, for example, [7,10,18]. For a graph , we denote by the clique cover number of , that is, is the minimal number of cliques that cover the vertex set of .…”