Partitioning transitive tournaments into isomorphic digraphs

TL;DR

This paper explores conditions for decomposing complete graphs into isomorphic subgraphs with orientations that form transitive tournaments, extending previous results on self-complementary graphs.

Contribution

It generalizes earlier theorems on orientations of self-complementary graphs to decompositions into multiple isomorphic subgraphs, providing sufficient and necessary conditions.

Findings

Sufficient conditions for orientations in decompositions of complete graphs.

Decomposition of odd complete graphs into Hamiltonian cycles admits such orientations.

Partial characterization of when these orientations are possible.

Abstract

In an earlier paper the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositions of odd vertex complete graphs to Hamiltonian cycles admit such an orientation. These conditions are further generalized and some necessary conditions are given as well.