TY - JOUR

T1 - Perfect fractional matchings in random hypergraphs

AU - Krivelevich, Michael

PY - 1996/10

Y1 - 1996/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0030492774&partnerID=8YFLogxK

U2 - 10.1002/(sici)1098-2418(199610)9:3<317::aid-rsa4>3.0.co;2-%23

DO - 10.1002/(sici)1098-2418(199610)9:3<317::aid-rsa4>3.0.co;2-%23

M3 - מאמר

AN - SCOPUS:0030492774

VL - 9

SP - 317

EP - 334

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -