On a Ramsey-Tur\'an variant of the Hajnal-Szemer\'edi theorem

TL;DR

This paper extends the Hajnal-Szemerédi theorem by exploring how the absence of large independent sets or large induced K_r-free subgraphs affects the minimum degree conditions needed for a graph to contain a K_r-factor.

Contribution

It introduces new minimum degree conditions under which graphs with restricted independent set sizes contain K_r-factors, generalizing previous results and connecting to Ramsey-Turán theory.

Findings

Graphs with high minimum degree and small large independent sets contain K_r-factors.

Optimal bounds are established for the case when the independence number is small.

Results are extended to graphs with small induced K_r-free subgraphs, including general graphs.

Abstract

A seminal result of Hajnal and Szemer\'{e}di states that if a graph $G$ with $n$ vertices has minimum degree $\delta(G) \ge (r-1)n/r$ for some integer $r \ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples which show optimality of the bound on $\delta(G)$ are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Tur\'an theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results: $\bullet$ For any $r > \ell \ge 2$, if $G$ is a graph satisfying $\delta(G) \ge (r - \ell)n/(r - \ell + 1) +\Omega(n)$ and $\alpha_\ell(G)=o(n)$, that is, a largest $K_\ell$-free induced subgraph has at most $o(n)$ vertices, then $G$ contains a $K_r$-factor. This is optimal for $\ell = r - 1$ and extends a result of Balogh, Molla, and Sharifzadeh who considered the case $r = 3$. $\bullet$ If a graph $G$ satisfies $\delta(G) =\Omega(n)$ and $\alpha_r^*(G) =o(n)$, that is, every induced $K_r$-free $r$-partite subgraph of $G$ has at least one vertex class of size $o(n)$, then it contains a $K_r$-factor. A similar statement is proven for a general graph $H$.