H-factors in graphs with small independence number

Abstract

Let H be an h-vertex graph. The vertex arboricity $ar\left(H\right)$ of H is the least integer r such that $V\left(H\right)$ can be partitioned into r parts and each part induces a forest in H. We show that for sufficiently large $n\in hN$, every n-vertex graph G with $\delta \left(G\right)\ge \max \{(1-\frac{2}{f\left(H\right)}+o(1\left)\right)n,(\frac{1}{2}+o(1\left)\right)n\}$ and $\alpha \left(G\right)=o\left(n\right)$ contains an H-factor, where $f\left(H\right)=2ar\left(H\right)$ or $2ar\left(H\right)-1$. The result can be viewed an analogue of the Alon–Yuster theorem [1] in Ramsey–Turán theory, which generalizes the results of Balogh–Molla–Sharifzadeh [2] and Knierim–Su [21] on clique factors. In particular the degree conditions are asymptotically sharp for infinitely many graphs H which are not cliques.