Cycle Partitions in Dense Regular Digraphs and Oriented Graphs
Allan Lo, Viresh Patel, Mehmet Akif Yıldız
TL;DR
The paper proves Jackson's conjecture for large $n$ by establishing a general cycle-cover bound for dense $d$-regular digraphs and oriented graphs: every such graph on $n$ vertices with $d\ge \alpha n$ can be covered by at most $n/(d+1)$ vertex-disjoint cycles, and if the graph is oriented, by at most $n/(2d+1)$ cycles, with cycles of length at least $d/2$. The approach leverages a structure theorem that partitions the graph into a small number of robust expander blocks, then uses balanced path systems and $Q$-contractions to balance the partition and reduce to cycle covers on contracted graphs; long cycles are obtained via the components of an auxiliary graph $S(\mathcal{P})$. A sequence of technical lemmas—balancingpathsystemcombined, balancedpath, and regdigraph—makes this reduction rigorous, employing flow methods, augmentation arguments, and robust-expansion properties. The results generalize prior work on regular graphs and extend cycle-cover and Hamiltonicity results to dense regular digraphs and oriented graphs, with implications for path covers, edge-disjoint cycles, and PMH-type properties in connected or bipartite contexts.
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 $α>0$, there exists $n_0=n_0(α)$ such that every $d$-regular digraph on $n\geq n_0$ vertices with $d \geq αn $ 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.