Tilings in randomly perturbed dense graphs

TL;DR

This paper determines the minimum number of random edges needed to ensure a dense graph with added randomness contains a perfect tiling of a fixed graph H, using advanced combinatorial tools.

Contribution

It establishes precise thresholds for random edge addition in dense graphs to guarantee perfect H-tilings with high probability, extending previous results.

Findings

Identifies the number of random edges needed for perfect H-tilings

Uses Szemerédi's Regularity lemma and Komlós's result in the proof

Provides probabilistic guarantees for tilings in perturbed dense graphs

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 Szemer\'edi's Regularity lemma as well as a special case of a result of Koml\'os concerning almost perfect $H$-tilings in dense graphs.