Emergent Mind
Cycle Partitions in Dense Regular Digraphs and Oriented Graphs
(2309.11677)
Published Sep 20, 2023
in
math.CO
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 $n0=n0(\alpha)$ 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.
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