An Ore-type Alon-Yuster Theorem

TL;DR

This paper proves a conjecture relating degree conditions in large graphs to the existence of near-perfect $H$-tilings, extending Ore-type theorems to tiling problems.

Contribution

It confirms a conjecture by establishing degree sum conditions that guarantee $H$-tilings covering almost all vertices in large graphs.

Findings

Degree sum condition ensures $H$-tilings covering all but a bounded number of vertices.

The result applies to any fixed graph $H$, with the bound depending only on $H$.

The theorem generalizes Ore-type conditions to graph tiling problems.

Abstract

A graph $G$ admits an $H$-tiling if it contains a collection of vertex-disjoint copies of $H$. In this paper, we confirm a conjecture proposed by K\"{u}hn, Osthus, and Treglown by showing that for any given graph $H$, there exists a constant $C(H)$ such that the following holds. If $G$ is a sufficiently large $n$-vertex graph satisfying $d(x) + d(y) \geq 2\left(1 - 1/\chi_{\text{cr}}(H)\right)n$ for all nonadjacent vertices $x, y \in V(G)$, then $G$ contains an $H$-tiling covering all but at most $C(H)$ vertices. Here $\chi_{\text{cr}}(H)$ denotes the critical chromatic number of $H$.