Self-complementary two-graphs and almost self-complementary double covers

A graph X is called almost self-complementary with respect to a perfect matching I if it is isomorphic to the graph obtained from its complement X-c by removing the edges of I. A two-graph on a vertex set Omega is a collection T of 3-subsets of Omega such that each 4-subset of Omega contains an even number of elements of I. III this paper we investigate the relationship between self-complementary two-graphs and double covers over complete graphs that are almost self-cornplementary with respect to a set of fibres. In particular, we classify all doubly transitive self-complementary two-graphs, and thus all almost self-complementary graphs Within automorphism group acting 2-transitively on the corresponding perfect matching. (C) 2006 Elsevier Ltd. All rights reserved.