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arXiv · 2311.00950

A robust version of the multipartite Hajnal--Szemerédi theorem

Jie Han·Jie Hu·Donglei Yang
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Abstract

In this note we show the following strengthening of a multipartite version of the Hajnal--Szemerédi theorem. For an integer $r \ge 3$ and $γ> 0$, there exists a constant $C$ such that if $p\ge Cn^{-2/r}(\log n)^{1/{r \choose 2}}$ and $G$ is a balanced $r$-partite graph with each vertex class of size $n$ and $δ^\ast(G)\ge (1-1/r+γ)n$, then with high probability the random subgraph $G(p)$ of $G$ contains a $K_r$-factor. We also use it to derive corresponding transversal versions.

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Jie Han, Jie Hu, Donglei Yang. 2023-11-02. A robust version of the multipartite Hajnal--Szemerédi theorem. https://arxiv.org/abs/2311.00950

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