On directed versions of the Hajnal--Szemer\'edi theorem

TL;DR

This paper extends the Hajnal--Szemerédi theorem to directed graphs, establishing minimum degree conditions for perfect packings of tournaments, including cyclic and transitive cases, using advanced combinatorial tools.

Contribution

It proves a directed analogue of the Hajnal--Szemerédi theorem for large digraphs with minimum degree conditions ensuring perfect tournament packings.

Findings

Proves the conjecture for cyclic triangles.

Asymptotically confirms the conjecture for all transitive tournaments.

Utilizes hypergraph matchings and the Directed Graph Removal lemma.

Abstract

We say that a (di)graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemer\'edi theorem characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. In this paper we prove the following analogue for directed graphs: Suppose that $T$ is a tournament on $r$ vertices and $G$ is a digraph of sufficiently large order $n$ where $r$ divides $n$. If $G$ has minimum in- and outdegree at least $ (1-1/r)n$ then $G$ contains a perfect $T$-packing. In the case when $T$ is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla (for large digraphs). Furthermore, in the case when $T$ is transitive we conjecture that it suffices for every vertex in $G$ to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all $r \geq 3$. Our approach makes use of a result of Keevash and Mycroft concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal lemma.