On Koml\'os' tiling theorem in random graphs

TL;DR

This paper improves a theorem on packing subgraphs in random graphs, reducing uncovered vertices and providing bounds for specific ranges of edge probability, advancing understanding of graph tiling in probabilistic settings.

Contribution

It enhances Komlós' tiling theorem in random graphs by reducing the number of uncovered vertices and establishing bounds for certain probabilities, answering open questions.

Findings

Achieved a tighter bound on uncovered vertices in $H$-packings.

Provided an upper bound on $H$-packing size for specific $p$ ranges.

Extended the theorem to cover more cases with improved results.

Abstract

Conlon, Gowers, Samotij, and Schacht showed that for a given graph $H$ and a constant $\gamma > 0$, there exists $C > 0$ such that if $p \ge Cn^{-1/m_2(H)}$ then asymptotically almost surely every spanning subgraph $G$ of the random graph $\mathcal{G}(n,p)$ with minimum degree at least $\delta(G) \ge (1 - 1/\chi_{\mathrm{cr}}(H) + \gamma )np$ contains an $H$-packing that covers all but at most $\gamma n$ vertices. Here, $\chi_{\mathrm{cr}}(H)$ denotes the critical chromatic threshold, a parameter introduced by Koml\'os. We show that this theorem can be bootstraped to obtain an $H$-packing covering all but at most $\gamma (C/p)^{m_2(H)}$ vertices, which is strictly smaller when $p > C n^{-1/m_2(H)}$. In the case where $H = K_3$ this answers the question of Balogh, Lee, and Samotij. Furthermore, we give an upper bound on the size of an $H$-packing for certain ranges of $p$.