Perfect fractional matchings in random hypergraphs

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Abstract

Given an r-uniform hypergraph H = (V, E) on \V\=n vertices, a real-valued function f : -E→R+ is called a perfect fractional matching if ∑ v∈e f(e) ≤ 1 for all v ∈ V and ∑ e∈E f(e) = n/r. Considering a random r-uniform hypergraph process of n vertices, we show that with probability tending to 1 as n→∞, at the very moment t0 when the last isolated vertex disappears, the hypergraph H,to has a perfect fractional matching. This result is clearly best possible. As a consequence, we derive that if p(n) = (In n + w(n))/ (n-1/r-1), where w(n) is any function tending to infinity with n, then with probability tending to 1 a random r-uniform hypergraph on n vertices with edge probability p has a perfect fractional matching. Similar results hold also for random r-partite hypergraphs.

Original languageEnglish
Pages (from-to)317-334
Number of pages18
JournalRandom Structures and Algorithms
Volume9
Issue number3
DOIs
StatePublished - Oct 1996

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