2 research outputs found
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. (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 , where 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 ) for the same class of graphs.
Our presentation uses free Cayley graphs. The proof combines free Cayley graphs
with quotient graphs and Goppa codes.Comment: 27 page
On free products of graphs
We define a free product of connected simple graphs that is equivalent to
several existing definitions when the graphs are vertex-transitive but differs
otherwise. The new definition is designed for the automorphism group of the
free product to be as large as possible, and we give sufficient criteria for it
to be non-discrete. Finally, we transfer Tits' classification of automorphisms
of trees and simplicity criterion to free products of graphs.Comment: 18 page