Minors, connectivity, and diameter in randomly perturbed sparse graphs

TL;DR

This paper investigates how adding a small number of random edges to sparse graphs affects their minors, topological minors, connectivity, and diameter, revealing tight bounds and asymptotic behaviors.

Contribution

It establishes tight bounds for minors and topological minors in perturbed sparse graphs and analyzes their connectivity and diameter.

Findings

Adding b3 n random edges yields large minors proportional to n/b1(G)

Random perturbation increases connectivity and reduces diameter in sparse graphs

Results are tight up to logarithmic factors

Abstract

Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every $n$-vertex graph $G$ contains a complete minor of order $\Omega(n/\alpha(G))$. We prove that adding $\xi n$ random edges, where $\xi > 0$ is arbitrarily small yet fixed, to an $n$-vertex graph $G$ satisfying $\alpha(G) \leq \zeta(\xi)n$ asymptotically almost surely results in a graph containing a complete minor of order $\tilde \Omega \left( n/\sqrt{\alpha(G)}\right)$; this result is tight up to the implicit logarithmic terms. For complete topological minors, we prove that there exists a constant $C>0$ such that adding $C n$ random edges to a graph $G$ satisfying $\delta(G) = \omega(1)$, asymptotically almost surely results in a graph containing a complete topological minor of order $\tilde \Omega(\min\{\delta(G),\sqrt{n}\})$; this result is tight up to the implicit logarithmic terms. Finally, extending results of Bohman, Frieze, Krivelevich, and Martin for the dense case, we analyse the asymptotic behaviour of the vertex-connectivity and the diameter of randomly perturbed sparse graphs.