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An Extension of the Hajnal–Szemerédi Theorem to Directed Graphs

Published online by Cambridge University Press:  28 October 2014

ANDRZEJ CZYGRINOW ,
LOUIS DeBIASIO ,
H. A. KIERSTEAD  and
THEODORE MOLLA
ANDRZEJ CZYGRINOW
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA (e-mail: aczygri@asu.edu, kierstead@asu.edu)
LOUIS DeBIASIO
Affiliation:
Miami University, Oxford, OH 45056, USA (e-mail: debiasld@miamioh.edu)
H. A. KIERSTEAD
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA (e-mail: aczygri@asu.edu, kierstead@asu.edu)
THEODORE MOLLA
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: molla@illinois.edu)
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Abstract

Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G)⩾ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph $\vv G$ with | $\vv G$ | = ks and δ( $\vv G$ ) ⩾ 2k(s − 1) − 1 contains k disjoint transitive tournaments on s vertices, where δ( $\vv G$ )= minv∈V ( $\vv G$ )d −(v)+d +(v). Our result implies the Hajnal–Szemerédi theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.

Keywords

Primary 05C20Secondary 05C35

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 5 , September 2015 , pp. 754 - 773
Copyright
Copyright © Cambridge University Press 2014 

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