Triangle packings and 1-factors in oriented graphs

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 ± 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 ∑ i = 1 t n i ⩽ 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.