Minors in random regular graphs
Nikolaos Fountoulakis2009, Random Structures and Algorithms
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Abstract
We show that there is a constant c so that for fixed r ≥ 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c √ n vertices as a minor. This confirms a conjecture of Markström . 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 Gn,p during the phase transition (i.e. when pn → 1).
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