Minors in random regular graphs

Details Minors in random regular graphs Minors in random regular graphs

2008-03-20

arxiv.org/abs/0803.3001v1

Mathematics

Combinatorics

18 pages

Scientific paper

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).

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