An Ore-type condition for $H$-tilings in graphs

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 Kühn, 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 $d(x)+d(y)\ge 2\left(1-1/\ddot{I}{‡}_{\text{cr}}(H)\right)n$ for all nonadjacent vertices $x,y\in V(G)$, then $G$ contains an $H$-tiling covering all but at most $C(H)$ vertices. Here $\ddot{I}{‡}_{\text{cr}}(H)$ denotes the critical chromatic number of $H$.