Triangle packings and 1-factors in oriented graphs
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 1. Under the same hypotheses, we also show that one can embed any prescribed almost 1-factor, i.e. for any sequence n1,...,nt with n1+...+nt < n-O(1) we can find a vertex-disjoint collection of directed cycles with lengths n1,...,nt. In addition, under quite general conditions on the ni we can remove the O(1) additive error and find a prescribed 1-factor.