Cyclic triangle factors in regular tournaments

TL;DR

This paper proves a longstanding conjecture that large regular tournaments with a number of vertices divisible by three contain a perfect collection of cyclic triangles, advancing understanding of cycle packings in directed graphs.

Contribution

It confirms Cuckler and Yuster's conjecture for large n, showing such tournaments contain a perfect cyclic triangle packing.

Findings

Confirmed the conjecture for sufficiently large n

Established existence of cyclic triangle packings in regular tournaments

Extended results to orientations of graphs with high minimum degree

Abstract

Both Cuckler and Yuster independently conjectured that when $n$ is an odd positive multiple of $3$ every regular tournament on $n$ vertices contains a collection of $n/3$ vertex-disjoint copies of the cyclic triangle. Soon after, Keevash and Sudakov proved that if $G$ is an orientation of a graph on $n$ vertices in which every vertex has both indegree and outdegree at least $(1/2 - o(1))n$, then there exists a collection of vertex-disjoint cyclic triangles that covers all but at most $3$ vertices. In this paper, we resolve the conjecture of Cuckler and Yuster for sufficiently large $n$.