Hamilton decompositions of regular tournaments

TL;DR

This paper proves that large regular tournaments can be nearly fully decomposed into edge-disjoint Hamilton cycles, providing an approximate solution to Kelly's 1968 conjecture and extending to almost regular tournaments.

Contribution

It establishes that sufficiently large regular tournaments contain nearly half the number of edges in Hamilton cycles, advancing the understanding of Hamilton decompositions in directed graphs.

Findings

Large regular tournaments contain at least (1/2 - η)n edge-disjoint Hamilton cycles.

The result extends to almost regular tournaments.

Provides an approximate solution to Kelly's conjecture from 1968.

Abstract

We show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each \eta>0 every regular tournament G of sufficiently large order n contains at least (1/2-\eta)n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968. Our result also extends to almost regular tournaments.