Brick assignments and homogeneously almost self-complementary graphs

A graph is called almost self-complementary if it is isomorphic to the graph obtained from its complement by removing a 1-factor. In this paper, we study a special class of vertex-transitive almost self-complementary graphs called homogeneously almost self-complementary. These graphs occur as factors of symmetric index-2 homogeneous factorizations of the “cocktail party graphs” K 2 n − n K 2 . We construct several infinite families of homogeneously almost self-complementary graphs, study their structure, and prove several classification results, including the characterization of all integers n of the form n = p r and n = 2 p with p prime for which there exists a homogeneously almost self-complementary graph on 2n vertices.