Threshold Functions for H-factors

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.