On fractional K-factors of random graphs

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 Σ_K.v∈K∈Kf(K) ≤ 1 for every v ∈ V and Σ_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 G̃ on n vertices, we show that if K is strictly K ₁-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 G̃_τ0 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.