Tiling directed graphs with tournaments

TL;DR

This paper extends the Hajnal--Szemerédi theorem to directed graphs, showing that under certain degree conditions, such graphs can be partitioned into subgraphs containing all tournaments of a fixed size.

Contribution

It provides a generalization of a classical graph partitioning theorem to directed graphs with conditions ensuring the existence of tournament-containing subgraphs.

Findings

Directed graph analogue of Hajnal--Szemerédi theorem proven.

Partitioning into subgraphs containing all tournaments on r vertices established.

A Turán-type result related to these partitions also demonstrated.

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.