Embedding clique-factors in graphs with low $\ell$-independence number

TL;DR

This paper investigates conditions under which graphs with certain minimum degree and independence number constraints contain a perfect clique factor, providing counterexamples and establishing asymptotic bounds for these properties.

Contribution

It provides a negative answer to a conjecture for specific parameters and constructs graphs demonstrating the optimality of degree conditions for clique factors.

Findings

Counterexamples for $oldsymbol{oldsymbol{ extit{ ext{clique-factor}}}}$ existence when $oldsymbol{oldsymbol{ extit{ ext{independence number}}}}$ is small.

Asymptotic bounds on minimum degree for guaranteeing $K_r$-factors under $oldsymbol{oldsymbol{ extit{ ext{independence number}}}}$ constraints.

Identification of the threshold $oldsymbol{oldsymbol{ extit{ ext{cover thresholds}}}}$ related to the problem.

Abstract

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $\delta(G) \ge \max \{ \frac{1}{2},\frac{r - \ell}{r} \}n + o(n)$ and $\alpha_{\ell}(G) = o(n) $ contains a $K_{r}$-factor? We give a negative answer for the case when $\ell\ge \frac{3r}{4}$ by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers $r,\ell$ with $r > \ell \ge \frac{3}{4}r$ and $\mu >0$, there exist $\alpha > 0$ and $N$ such that for every $n\in r\mathbb{N}$ with $n>N$, every $n$-vertex graph $G$ with $\delta(G) \ge \left( \frac{1}{2-\varrho_{\ell}(r-1)} + \mu \right)n $ and $\alpha_{\ell}(G) \le \alpha n$ contains a $K_{r}$-factor. Here $\varrho_{\ell}(r-1)$ is the Ramsey--Tur\'an density for $K_{r-1}$ under the $\ell$-independence number condition.