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arXiv · 1708.09243

Tilings in randomly perturbed dense graphs

József Balogh·Andrew Treglown·Adam Zsolt Wagner
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Abstract

A perfect $H$-tiling in a graph $G$ is a collection of vertex-disjoint copies of a graph $H$ in $G$ that together cover all the vertices in $G$. In this paper we investigate perfect $H$-tilings in a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds $m$ random edges to it. Specifically, for any fixed graph $H$, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect $H$-tiling with high probability. Our proof utilises Szemerédi's Regularity lemma as well as a special case of a result of Komlós concerning almost perfect $H$-tilings in dense graphs.

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József Balogh, Andrew Treglown, Adam Zsolt Wagner. 2018-05-11. Tilings in randomly perturbed dense graphs. https://arxiv.org/abs/1708.09243

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