Clique-factors in graphs with sublinear $\ell$-independence number
Abstract
Given a graph $G$ and an integer $\ell\ge 2$, we denote by $\alpha_{\ell}(G)$ the maximum size of a $K_{\ell}$-free subset of vertices in $V(G)$. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$-vertex graphs $G$ with $\alpha_{\ell}(G) = o(n)$, which can be seen as a Ramsey--Turán variant of the celebrated Hajnal--Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$-factors in $n$-vertex graphs $G$ with $\alpha_\ell(G)=n^{1-o(1)}$ for all $r\ge \ell\ge 2$.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.02169
- arXiv:
- arXiv:2203.02169
- Bibcode:
- 2022arXiv220302169H
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- arXiv admin note: text overlap with arXiv:2111.10512