Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments

TL;DR

This paper proves Kelly's conjecture for large regular tournaments by demonstrating that large regular digraphs with robust expansion can be decomposed into edge-disjoint Hamilton cycles, confirming several longstanding conjectures.

Contribution

It introduces a new method for decomposing graphs based on robust expansion, proving Kelly's conjecture for large tournaments and related results.

Findings

Proves Kelly's conjecture for large n.

Confirms a conjecture of Erdős on Hamilton cycle packing.

Provides new bounds for Hamilton cycle packings in undirected graphs.

Abstract

A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdos on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich.