On fractionalK-factors of random graphs

Abstract

Let K be a graph on r vertices and let G = (V,E) be another graph on ∣V ∣ = n vertices. Denote the set of all copies of K in G by 𝒦. A non-negative real-valued function f : 𝒦→ ℝ+ is called a fractional K-factor if ∑ K:v∈K∈𝒦f(K) ≤ 1 for every v ∈ V and ∑ K∈𝒦f(K) = n/r. For a non-empty graph K let d(K) = e(K)/v(K) and d(1)(K) = e(K)/(v(K) - 1). We say that K is strictly K1-balanced if for every proper subgraph K′ ⊊ K, d(1)(K′) < d(1)(K). We say that K is imbalanced if it has a subgraph K′ such that d(K′) > d(K). Considering a random graph process on n vertices, we show that if K is strictly K1-balanced, then with probability tending to 1 as n →∞, at the first moment τ0 when every vertex is covered by a copy of K, the graph has a fractional K-factor. This result is the best possible. As a consequence, if K is K1-balanced, we derive the threshold probability function for a random graph to have a fractional K-factor. On the other hand, we show that if K is an imbalanced graph, then for asymptotically almost every graph process there is a gap between τ0 and the appearance of a fractional K-factor. We also introduce and apply a criteria for perfect fractional matchings in hypergraphs in terms of expansion properties. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007