Equilibrium graphs and rational trees
European Journal of Combinatorics
Threshold functions for Extension Statements
Journal of Combinatorial Theory Series A
Journal of Combinatorial Theory Series A
Matching and covering the vertices of a random graph by copies of a given graph
Discrete Mathematics
Bounding Ramsey numbers through large deviation inequalities
Random Structures & Algorithms
Hamilton cycles in random subgraphs of pseudo-random graphs
Discrete Mathematics
Triangle Factors in Random Graphs
Combinatorics, Probability and Computing
Corrigendum: On fractional K-factors of random graphs
Random Structures & Algorithms
Research paper: Combinatorial and computational aspects of graph packing and graph decomposition
Computer Science Review
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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 $\widetilde{G}$ 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 ${\widetilde{G}}_{\tau_0}$ 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