A degree sequence Koml\'{o}s theorem

TL;DR

This paper strengthens Komlós's theorem by providing a degree sequence condition that allows many vertices to have lower degrees while still guaranteeing an $H$-tiling covering a specified proportion of the graph.

Contribution

It introduces a degree sequence version of Komlós's theorem, broadening the conditions under which $H$-tilings exist in graphs.

Findings

Degree sequence condition is nearly optimal for certain graphs $H$

Allows for many vertices with degrees below the original threshold

Ensures $H$-tilings covering a fixed proportion of vertices

Abstract

An important result of Koml\'os [Tiling Tur\'an theorems, Combinatorica, 2000] yields the asymptotically exact minimum degree threshold that ensures a graph $G$ contains an $H$-tiling covering an $x$th proportion of the vertices of $G$ (for any fixed $x \in (0,1)$ and graph $H$). We give a degree sequence strengthening of this result which allows for a large proportion of the vertices in the host graph $G$ to have degree substantially smaller than that required by Koml\'os' theorem. We also demonstrate that for certain graphs $H$, the degree sequence condition is essentially best possible in more than one sense.