An extension of the Hajnal-Szemeredi theorem to directed graphs

TL;DR

This paper extends the Hajnal-Szemeredi theorem from undirected graphs to directed graphs, establishing optimal degree conditions for partitioning into transitive tournaments, and proposes conjectures for further generalizations.

Contribution

It introduces a directed graph analogue of the Hajnal-Szemeredi theorem with optimal degree bounds and explores broader conjectures for multigraphs and other tournaments.

Findings

Directed analogue of Hajnal-Szemeredi theorem proven

Optimal degree bound for directed graphs established

Conjectures supported by asymptotic results

Abstract

Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph D with |D|=ks and minimum (total) degree at least 2k(s-1)-1 contains k vertex disjoint transitive tournaments on s vertices. Our result implies the Hajnal-Szemeredi Theorem, and the 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.