Bounded-degree spanning trees in randomly perturbed graphs
Michael Krivelevich, Matthew Kwan, Benny Sudakov
TL;DR
This work shows that for every fixed dense graph $G$ and every $n$-vertex tree $T$ with bounded maximum degree, a modest random perturbation suffices to embed $T$ into $G\cup R$, where $R\in G(n, c/n)$ and the constant $c=c(\alpha,\Delta)$ depends only on the minimum-density parameter and the tree's degree bound. The authors combine extremal and probabilistic methods, splitting the proof into two cases based on the number of leaves in $T$, and use a multi-phase perturbation together with Szemerédi's regularity lemma and the blow-up lemma to achieve the embedding. A key technical ingredient is a lemma that decomposes a dense graph into super-regular pairs, which supports the embedding in the few-leaves case. The results bridge dense-graph embedding results and random-graph embedding phenomena, providing a robust framework for embedding bounded-degree spanning trees in randomly perturbed graphs and suggesting avenues for extending to broader spanning subgraphs and universality questions.
Abstract
We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizeable body of existing research on randomly perturbed graphs. Specifically, we show that there is $c = c(α,Δ)$ such that if G is an n-vertex graph with minimum degree at least $αn$, and T is an n-vertex tree with maximum degree at most $Δ$ , then if we add cn uniformly random edges to G, the resulting graph will contain T asymptotically almost surely (as $n\to\infty$ ). Our proof uses a lemma concerning the decomposition of a dense graph into super-regular pairs of comparable sizes, which may be of independent interest.