Sharp threshold for the appearance of certain spanning trees in random graphs

Authors:
Dan Hefetz;Michael Krivelevich;Tibor Szabó
Affiliations:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK;School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978 Israel;Institute of Mathematics, Free University Berlin, 14195 Berlin, Germany
Venue:
Random Structures & Algorithms
Year:
2012

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Abstract

We prove that a given tree T on n vertices with bounded maximum degree is contained asymptotically almost surely in the binomial random graph \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}G(n,\frac{(1+\varepsilon) \log n}{n})\end{align*}\end{document} **image** provided that T belongs to one of the following two classes: (1) T has linearly many leaves; (2) T has a path of linear length all of whose vertices have degree two in T. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.