Isomorphism Classes of Vertex-Transitive Tournaments

TL;DR

This paper derives a formula for counting isomorphism classes of vertex-transitive tournaments of prime order, introducing Cayley tournaments to establish a one-to-one correspondence.

Contribution

It provides a new formula for the number of such classes and introduces Cayley tournaments as a key concept for classification.

Findings

Established a one-to-one correspondence between vertex-transitive and Cayley tournaments of prime order.

Derived a formula for counting isomorphism classes of vertex-transitive tournaments.

Introduced Cayley tournaments as a subclass of Cayley digraphs.

Abstract

Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.