About the number of oriented Hamiltonian paths and cycles in tournaments

TL;DR

This paper proves that in any tournament and its complement, the number of oriented Hamiltonian paths and cycles of any given type are equal, extending previous results for antidirected paths.

Contribution

It generalizes Rosenfeld's result by showing the equality of counts for Hamiltonian paths and cycles in both a tournament and its complement.

Findings

Tournament and its complement contain equal numbers of oriented Hamiltonian paths.

Tournament and its complement contain equal numbers of oriented Hamiltonian cycles.

Generalizes previous results for antidirected paths.

Abstract

We prove that a tournament and its complement contain the same number of oriented Hamiltonian paths (resp. cycles) of any given type, as a generalization of Rosenfeld's result proved for antidirected paths.