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arXiv · 1806.00493

Near-perfect clique-factors in sparse pseudorandom graphs

Jie Han·Yoshiharu Kohayakawa·Yury Person
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Abstract

We prove that, for any $t\ge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$\lambda$ satisfying $\lambda\le c d^{t-1}/n^{t-2}$ contains vertex-disjoint copies of $K_t$ covering all but at most $n^{1-1/(8t^4)}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Sz\'abo [\emph{Triangle factors in sparse pseudo-random graphs}, Combinatorica \textbf{24} (2004), pp.~403--426] that $(n,d,\lambda)$-graphs with $n\in 3\mathbb{N}$ and $\lambda\leq cd^{2}/n$ for a suitably small absolute constant~$c>0$ contain triangle-factors.

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Jie Han, Yoshiharu Kohayakawa, Yury Person. 2018-06-01. Near-perfect clique-factors in sparse pseudorandom graphs. https://arxiv.org/abs/1806.00493

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