# On fractional K-factors of random graphs

Simi Haber, Michael Krivelevich

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

## 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 ΣK.v∈K∈Kf(K) ≤ 1 for every v ∈ V and ΣK∈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 G̃τ0 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.

Original language English 441-463 23 Random Structures and Algorithms 30 4 https://doi.org/10.1002/rsa.20144 Published - Jul 2007 Yes

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