On the Tur\'anability and tileability of oriented graphs

TL;DR

This paper investigates the properties of oriented graphs related to Turánability and tileability, disproving a previous conjecture, and providing new examples and distinctions between these concepts.

Contribution

It disproves a conjecture characterizing Turánable graphs, shows that Turánable graphs may not be tileable, and introduces a new tileable oriented graph.

Findings

Disproved a conjecture on Turánability of oriented graphs

Established that Turánable graphs are not necessarily tileable

Provided a new example of a tileable oriented graph

Abstract

An oriented graph $H$ is Tur\'anable (resp. tileable) if there exist $n_0 \in \mathbb{N}$ such that every semi-regular near-tournament on $n \ge n_0$ vertices contains a copy of $H$ (resp. a perfect $H$-tiling). We disprove a conjectured characterization of Tur\'anable oriented graphs by DeBiasio, Han, Lo, Molla, Piga, and Treglown, show that there are Tur\'anable oriented graphs which are not tileable, and provide a new example of tileable oriented graph.