Almost self-complementary circulant graphs

Abstract

An almost self-complementary graph is a graph isomorphic to its complement minus a 1-factor. An almost self-complementary circulant graph is called cyclically almost self-complementary if it has an isomorphic almost complement with the same regular cyclic subgroup of the automorphism group. In this paper we prove that a cyclically almost self-complementary circulant of order 2n exists if and only if every prime divisor of n is congruent to 1 modulo 4, thus extending the known result on the existence of self-complementary circulants. We also describe the structure of cyclically almost self-complementary circulants and the action of their automorphism groups. Finally, we exhibit a class of almost self-complementary Cayley graphs on a dihedral group that are isomorphic to cyclically almost self-complementary circulants.