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On Directed Versions of the Hajnal–Szemerédi Theorem

Published online by Cambridge University Press:  05 February 2015

ANDREW TREGLOWN
ANDREW TREGLOWN*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK (e-mail: [email protected])
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Abstract

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.

In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 6 , November 2015 , pp. 873 - 928
Copyright
Copyright © Cambridge University Press 2015 

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