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TILING DIRECTED GRAPHS WITH TOURNAMENTS

Published online by Cambridge University Press:  14 February 2018

ANDRZEJ CZYGRINOW ,
LOUIS DEBIASIO ,
THEODORE MOLLA  and
ANDREW TREGLOWN
ANDRZEJ CZYGRINOW
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85281, USA; aczygri@asu.edu
LOUIS DEBIASIO
Affiliation:
Department of Mathematics, Miami University, Oxford, OH 45056, USA; debiasld@miamioh.edu
THEODORE MOLLA
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA; molla@usf.edu
ANDREW TREGLOWN
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK; a.c.treglown@bham.ac.uk

Abstract

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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
Forum of Mathematics, Sigma , Volume 6 , 2018 , e2
Creative Commons
Creative Common License - CCCreative Common License - BY
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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