Cycle Partitions in Dense Regular Digraphs and Oriented Graphs

TL;DR

This paper proves a long-standing conjecture by Jackson for large graphs, showing that dense regular digraphs and oriented graphs can be decomposed into a small number of vertex-disjoint cycles, extending understanding of Hamiltonian properties.

Contribution

It establishes that dense regular digraphs and oriented graphs can be covered by few cycles, confirming Jackson's conjecture for large graphs and generalizing cycle partition results.

Findings

Dense regular digraphs can be covered by at most n/(d+1) cycles.

Oriented graphs can be covered by at most n/(2d+1) cycles.

The conjecture holds for sufficiently large n.

Abstract

A conjecture of Jackson from 1981 states that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large $n$. In fact we prove a more general result that for all $\alpha>0$, there exists $n_0=n_0(\alpha)$ such that every $d$-regular digraph on $n\geq n_0$ vertices with $d \geq \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles, and moreover that if $G$ is an oriented graph, then at most $n/(2d+1)$ cycles suffice.