$H$-factors in graphs with small independence number

TL;DR

This paper establishes degree conditions under which large graphs with small independence number contain an $H$-factor, generalizing classical theorems to a broader class of graphs with minimal independence number.

Contribution

It provides asymptotically sharp degree thresholds for the existence of $H$-factors in graphs with small independence number, extending the Alon–Yuster theorem.

Findings

Degree conditions guarantee $H$-factors in graphs with small independence number.

Results generalize known theorems for clique factors to broader graphs.

Conditions are asymptotically sharp for infinitely many non-clique graphs.

Abstract

Let $H$ be an $h$-vertex graph. The vertex arboricity $ar(H)$ of $H$ is the least integer $r$ such that $V(H)$ can be partitioned into $r$ parts and each part induces a forest in $H$. We show that for sufficiently large $n\in h\mathbb{N}$, every $n$-vertex graph $G$ with $\delta(G)\geq \max\left\{\left(1-\frac{2}{f(H)}+o(1)\right)n, \left(\frac{1}{2}+o(1)\right)n\right\}$ and $\alpha(G)=o(n)$ contains an $H$-factor, where $f(H)=2ar(H)$ or $2ar(H)-1$. The result can be viewed an analogue of the Alon--Yuster theorem \cite{MR1376050} in Ramsey--Tur\'{a}n theory, which generalises the results of Balogh--Molla--Sharifzadeh~\cite{MR3570984} and Knierm--Su~\cite{MR4193066} on clique factors. In particular the degree conditions are asymptotically sharp for infinitely many graphs $H$ which are not cliques.