Perturbation of dense graphs

TL;DR

This paper demonstrates that for many spanning subgraph problems in randomly perturbed graphs, the minimum degree condition can be replaced with a density condition, broadening the applicability of existing results.

Contribution

It extends previous work by showing that density conditions suffice for embedding various spanning subgraphs in perturbed graphs, relaxing the minimum degree requirement.

Findings

Density conditions replace minimum degree in embedding problems

Results apply to F-factors, bounded degree graphs, and Hamilton cycles

Strengthens prior theorems in the field

Abstract

In the past two decades, various properties of randomly perturbed/augmented (hyper)graphs have been intensively studied, since the model was introduced by Bohman, Frieze and Martin in 2003. The model usually considers a deterministic graph $G$ with minimum degree condition, perturbed/augmented by a binomial random graph $G(n,p)$ on the same vertex set. In this paper, we show that for many problems of finding spanning subgraphs, one can indeed relax the minimum degree condition to a density condition. This includes the embedding problem for $F$-factors when $F$ is not a forest, graphs with bounded maximum degree, $r$-th power of $k$-uniform tight Hamilton cycles for $r,k\ge 2$, and $k$-uniform Hamilton $\ell$-cycles for $\ell\in[2,k-1]$. These results strengthen the results of Balogh, Treglown, and Wagner, of B\"{o}ttcher, Montgomery, Parczyk, and Person, and of Chang, Han and Thoma.