Triangle packings and 1-factors in oriented graphs

  • Authors:

  • Peter Keevash;Benny Sudakov

  • Affiliations:

  • School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK;Department of Mathematics, UCLA, 520 Portolo Plaza, Los Angeles 90095, United States

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

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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=3mod18 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^tn"i=