Universality for bounded degree spanning trees in randomly perturbed graphs

TL;DR

This paper establishes the threshold for embedding all bounded degree spanning trees in dense graphs with added random edges, advancing understanding of graph universality in probabilistic combinatorics.

Contribution

It proves that adding a suitable random graph to a dense graph guarantees the containment of all bounded degree spanning trees with high probability.

Findings

Threshold C depends only on density and maximum degree

All bounded degree spanning trees are contained with high probability

Results extend previous work on graph universality in random perturbations

Abstract

We solve a problem of Krivelevich, Kwan and Sudakov [SIAM Journal on Discrete Mathematics 31 (2017), 155-171] concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph $G_\alpha$ on $n$ vertices with $\delta(G_\alpha)\ge \alpha n$ for $\alpha>0$ and we add to it the binomial random graph $G(n,C/n)$, then with high probability the graph $G_\alpha\cup G(n,C/n)$ contains copies of all spanning trees with maximum degree at most $\Delta$ simultaneously, where $C$ depends only on $\alpha$ and $\Delta$.