Transitive Triangle Tilings in Oriented Graphs

TL;DR

This paper establishes a minimum degree condition for the existence of perfect transitive triangle tilings in large oriented graphs, extending classical theorems to directed graph settings.

Contribution

It proves an oriented graph analogue of Corrádi and Hajnal's theorem, identifying the exact degree threshold for perfect transitive triangle tilings.

Findings

Existence of perfect transitive triangle tilings under specified degree conditions

Degree threshold of 7n/18 is both sufficient and tight

Extension of classical undirected graph results to directed graphs

Abstract

In this paper, we prove an analogue of Corr\'adi and Hajnal's classical theorem. There exists $n_0$ such that for every $n \in 3\mathbb{Z}$ when $n \ge n_0$ the following holds. If $G$ is an oriented graph on $n$ vertices and every vertex has both indegree and outdegree at least $7n/18$, then $G$ contains a perfect transitive triangle tiling, which is a collection of vertex-disjoint transitive triangles covering every vertex of $G$. This result is best possible, as, for every $n \in 3\mathbb{Z}$, there exists an oriented graph $G$ on $n$ vertices without a perfect transitive triangle tiling in which every vertex has both indegree and outdegree at least $\lceil 7n/18\rceil - 1.$