Transitive tournaments and self-complementary graphs

A simple proof is given for a result of Sali and Simonyi on selfcomplementary graphs. ß 2001 John Wiley & Sons, Inc. J Graph Theory 38: 111±112, 2001 Keywords: self-complementary graphs; transitive tournaments Motivated by Sperner capacities of digraphs, A. Sali and G. Simonyi [1] discovered an interesting property of self-complementary graphs. Here a proof of their result is given which is conceptually simpler than the original one. Theorem. (Sali and Simonyi [1]) For any self-complementary graph G on n vertices, the edges of the transitive tournament on n vertices can be partitioned into two isomorphic digraphs whose underlying graphs are isomorphic to G. Proof. Assume that is a complementing permutation on the vertex set ‰nŠ ˆ f1; 2; . . . ; ng of G, i.e., for every 1 x < y n , xy 2 E …G† implies …x† …y† = 2E …G†. The theorem is proved by de®ning a linear order on ‰nŠ such that preserves , i.e., for every xy 2 E…G† such that x < y, it follows that …x† < …y†. It is enough to de®ne on cycles of since if preserves a linear order on each cycle then preserves the sum of the linear orders. Let C ˆ f1; 2; . . . ; kg be a nontrivial cycle of , i.e., …x† ˆ x‡ 1 ( mod k). We may assume that 12 2 E…G†.