Minors in random regular graphs

TL;DR

This paper proves that random r-regular graphs almost surely contain large complete minors proportional to the square root of the number of vertices, confirming a conjecture and establishing bounds on minor sizes.

Contribution

It establishes the existence of large complete minors in random regular graphs, confirming a conjecture and providing bounds that are tight up to a constant factor.

Findings

Large complete minors exist in random r-regular graphs with size proportional to n^{1/2}.

The bounds on minor sizes are optimal up to a constant factor.

The results also determine the order of magnitude of largest minors in G(n,p) during phase transition.

Abstract

We show that there is a constant c>0 so that for any fixed r which is at least 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c n^{1/2} vertices as a minor. This confirms a conjecture of Markstrom. Since any minor of an r-regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph G(n,p) during the phase transition (i.e. when pn is close to 1).