Bounded-degree spanning trees in randomly perturbed graphs

TL;DR

This paper proves that adding a small number of random edges to a dense graph guarantees the embedding of any bounded-degree spanning tree, bridging fixed dense and random graph embedding problems.

Contribution

It introduces a new result showing that modest random perturbations in dense graphs ensure spanning tree embeddings, extending existing theories in graph embedding and perturbation.

Findings

Adding random edges guarantees spanning tree embedding in dense graphs

The method involves decomposing dense graphs into super-regular pairs

The results hold asymptotically as the number of vertices grows

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 $c = c(\alpha,\Delta)$ such that if G is an n-vertex graph with minimum degree at least $\alpha n$, and T is an n-vertex tree with maximum degree at most $\Delta$ , 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.