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arXiv · 1603.08198

Tiling directed graphs with tournaments

Andrzej Czygrinow·Louis DeBiasio·Theodore Molla·Andrew Treglown
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Abstract

The Hajnal--Szemer\'edi theorem states that for any integer $r \ge 1$ and any multiple $n$ of $r$, if $G$ is a graph on $n$ vertices and $\delta(G) \ge (1 - 1/r)n$, then $G$ can be partitioned into $n/r$ vertex-disjoint copies of the complete graph on $r$ vertices. We prove a very general analogue of this result for directed graphs: for any integer $r \ge 4$ and any sufficiently large multiple $n$ of $r$, if $G$ is a directed graph on $n$ vertices and every vertex is incident to at least $2(1 - 1/r)n - 1$ directed edges, then $G$ can be partitioned into $n/r$ vertex-disjoint subgraphs of size $r$ each of which contain every tournament on $r$ vertices. A related Tur\'an-type result is also proven.

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Andrzej Czygrinow, Louis DeBiasio, Theodore Molla, Andrew Treglown. 2016-03-27. Tiling directed graphs with tournaments. https://arxiv.org/abs/1603.08198

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