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TILING DIRECTED GRAPHS WITH TOURNAMENTS
Published online by Cambridge University Press: 14 February 2018
Abstract
The Hajnal–Szemerédi theorem states that for any positive integer $r$ and any multiple
$n$ of
$r$, if
$G$ is a graph on
$n$ vertices and
$\unicode[STIX]{x1D6FF}(G)\geqslant (1-1/r)n$, 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 positive integer
$r$ with
$r\neq 3$ 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 (the case
$r=3$ is different and was handled previously). In fact, this result is a consequence of a tiling result for standard multigraphs (that is multigraphs where there are at most two edges between any pair of vertices). A related Turán-type result is also proven.
- Type
- Research Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s) 2018
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