Perfect fractional matchings in random hypergraphs

Abstract

Given an r-uniform hypergraph H = (V, E ) on ( V ( = n vertices, a real-valued function f : E+ R f is called a perfect fractional matching if C, , , f(e) 5 1 for all u E V and C e 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 + m , at the very moment to when the last isolated vertex disappears, the hypergraph H,, has a perfect fractional matching. This result is clearly best possible. As a consequence, we derive that if p ( n ) = (Inn + w(n)) / (;I;), 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.