Shallow excluded minors and improved graph decompositions
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Short paths in expander graphs
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
The order of the largest complete minor in a random graph
Random Structures & Algorithms
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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\sqrt{n} $ vertices as a minor. This confirms a conjecture of Markström (Ars Combinatoria 70 (2004) 289–295). 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). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009