Sufficient conditions for perfect mixed tilings

TL;DR

This paper introduces a new framework for establishing sufficient conditions for perfect mixed tilings, extending existing results and resolving a longstanding conjecture in graph theory.

Contribution

It develops a versatile method for perfect mixed tilings, generalizing previous minimum degree conditions and addressing degree sequences and dense graphs.

Findings

Extended minimum degree conditions for perfect $F$-tilings

Unified approach for bounded degree graph embeddings

Asymptotic resolution of Komlós' conjecture

Abstract

We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs $H$ with components of sublinear order. As a corollary, we recover and extend the work of K\"uhn and Osthus regarding sufficient minimum degree conditions for perfect $F$-tilings (for an arbitrary fixed graph $F$) by replacing the $F$-tiling with the aforementioned graphs $H$. Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Koml\'os in a strong sense.