Cycle Partitions in Dense Regular Digraphs and Oriented Graphs

Abstract

A conjecture of Jackson from 1981 states that every $d$-regular oriented graph on $n$ vertices with $n\le 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large $n$. In fact we prove a more general result that for all $\hat{I}\pm >0$, there exists $n_0=n_0(\hat{I}\pm )$ such that every $d$-regular digraph on $n\ge n_0$ vertices with $d\ge \hat{I}\pm n$ 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.

33 pages, 1 figure