Triangle packings and 1-factors in oriented graphs

TL;DR

This paper proves that dense oriented graphs contain near-perfect triangle packings and can embed almost any prescribed collection of directed cycles, advancing understanding of cycle packings in directed graphs.

Contribution

It establishes near-complete triangle packings in dense oriented graphs and shows how to embed almost any prescribed 1-factor, answering longstanding questions in the field.

Findings

Dense oriented graphs with high minimum degree contain almost perfect triangle packings.

Any almost 1-factor can be embedded in such graphs under certain degree conditions.

The results are tight, with explicit constructions showing the bounds are best possible.

Abstract

An oriented graph is a directed graph which can be obtained from a simple undirected graph by orienting its edges. In this paper we show that any oriented graph G on n vertices with minimum indegree and outdegree at least (1/2-o(1))n contains a packing of cyclic triangles covering all but at most 3 vertices. This almost answers a question of Cuckler and Yuster and is best possible, since for n = 3 mod 18 there is a tournament with no perfect triangle packing and with all indegrees and outdegrees (n-1)/2 or (n-1)/2 \pm 1. Under the same hypotheses, we also show that one can embed any prescribed almost 1-factor, i.e. for any sequence n_1,...,n_t with n_1+...+n_t < n-O(1) we can find a vertex-disjoint collection of directed cycles with lengths n_1,...,n_t. In addition, under quite general conditions on the n_i we can remove the O(1) additive error and find a prescribed 1-factor.