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

A graph XX is called almost self-complementary with respect to a perfect matching II if it is isomorphic to the graph obtained from its complement Xc by removing the edges of II. A two-graph on a vertex set ΩΩ is a collection TT of 3-subsets of ΩΩ such that each 4-subset of ΩΩ contains an even number of elements of TT. In this paper we investigate the relationship between self-complementary two-graphs and double covers over complete graphs that are almost self-complementary 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 with an automorphism group acting 2-transitively on the corresponding perfect matching.