Hardness of Computing Clique Number and Chromatic Number For Cayley Graphs

TL;DR

This paper proves that computing clique and chromatic numbers remains NP-Hard for Cayley graphs over groups G^n, extending known hardness results from circulant graphs to a broader class using advanced algebraic techniques.

Contribution

It establishes NP-Hardness of clique and chromatic number computations for Cayley graphs over groups G^n, generalizing previous results and employing free Cayley graphs, quotient graphs, and Goppa codes.

Findings

Clique number computation is NP-Hard for Cayley graphs over G^n.

Chromatic number computation is NP-Hard for the same class.

Uses algebraic techniques like free Cayley graphs and Goppa codes.

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. \emph{Linear Algebra Appl.}, 285(1-3): 123--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 $\text{P}\neq \text{NP}$) for the same class of graphs. Our presentation uses free Cayley graphs. The proof combines free Cayley graphs with quotient graphs and Goppa codes.