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Threshold Functions for H-factors

Published online by Cambridge University Press:  12 September 2008

Noga Alon  and
Raphael Yuster
Noga Alon
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
Raphael Yuster
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
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Abstract

Let H be a graph on h vertices, and G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n/h vertex disjoint copies of H. The fractional arboricity of H is , where the maximum is taken over all subgraphs (V′, E′) of H with |V′| > 1. Let δ(H) denote the minimum degree of a vertex of H. It is shown that if δ(H) < a(H), then n−1/a(H) is a sharp threshold function for the property that the random graph G(n, p) contains an H-factor. That is, there are two positive constants c and C so that for p(n) = cn−1/a(H) almost surely G(n, p(n)) does not have an H-factor, whereas for p(n) = Cn−1/a(H), almost surely G(n, p(n)) contains an H-factor (provided h divides n). A special case of this answers a problem of Erdős.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 2 , June 1993 , pp. 137 - 144
Copyright
Copyright © Cambridge University Press 1993

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