Bounded-Degree Spanning Trees in Randomly Perturbed Graphs

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, ∆) 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 ∆, then if we add cn uniformly random edges to G, the resulting graph will contain T asymptotically almost surely (as n → ∞). 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.

Embedding spanning bounded degree subgraphs in randomly perturbed graphs

“…Krivelevich, Kwan and Sudakov [16] studied the corresponding problem for the containment of bounded degree trees and showed that p = ω(1/n) is sufficient in this case. For p = ω(1/n) it is already possible to find any almost spanning bounded degree tree in G(n, p) [4].…”

Embedding Spanning Bounded Degree Graphs in Randomly Perturbed Graphs

“…Krivelevich et al [28] studied the corresponding problem for the containment of spanning trees of maximum degree in G α ∪ G(n, p). For p = c(ε, )/n, it is already possible to find any almost spanning bounded degree tree on (1 − ε)n vertices in G(n, p) [4].…”

How many randomly colored edges make a randomly colored dense graph rainbow Hamiltonian or rainbow connected?

“…Here, it is worth mentioning that in the Erdős-Rényi random graph, G n m ( , ), the threshold for Hamiltonicity is n nn (log + log log ) . G H m , has since been studied in a number of other contexts (see, eg, [1,3,9,10,13] (1)) ] color a typical graph of minimum degree δn, then w.h.p. the resultant graph will contain a rainbow-Hamilton cycle.…”