Koml\'os's tiling theorem via graphon covers

TL;DR

This paper extends Komlos's tiling theorem to the setting of graphons, providing a new proof and a stability version, thereby broadening the understanding of vertex-disjoint subgraph coverings in large graphs.

Contribution

It introduces a graphon version of Komlos's theorem and a stability result, using advanced machinery from graphon tiling theory.

Findings

Established a graphon analogue of Komlos's tiling theorem.

Proved a stability version of the theorem.

Connected finite graph results with graphon theory.

Abstract

Komlos [Komlos: Tiling Turan Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus essentially extending the Hajnal-Szemeredi theorem which deals with the case when H is a clique. We give a proof of a graphon version of Komlos's theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladky, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komlos's theorem.