Perfect matchings in random uniform hypergraphs
نویسنده
چکیده
In the random k-uniform hypergraph Hk(n, p) on a vertex set V of size n, each subset of size k of V independently belongs to it with probability p. Motivated by a theorem of Erdős and Rényi [6] regarding when a random graph G(n, p) = H2(n, p) has a perfect matching, Schmidt and Shamir [14] essentially conjectured the following. Conjecture Let k|n for fixed k ≥ 3, and the expected degree d(n, p) = p ( n−1 k−1 ) . Then Pr[Hk(n, p) has a perfect matching] → 0 if d(n, p) − ln n → −∞ e−e −c if d(n, p) − ln n → c 1 if d(n, p) − ln n → ∞ (Erdős and Rényi proved this for G(n, p)). Assuming d(n, p)/n1/2 → ∞, they were able to prove that Hk(n, p) contains a perfect matching with probability 1 − o(1). Frieze and Janson [7] showed that a weaker condition d(n, p)/n1/3 → ∞ was enough. In this paper, we further improve the bound to d(n, p) n1/(5+2/(k−1)) → ∞. A bound for a similar problem about a perfect triangle packing of G(n, p) is also obtained. A perfect triangle packing of a graph is a collection of vertex disjoint triangles whose union is the entire vertex set. Improving a bound p ≥ cn−2/3+1/15 of Krivelevich [10], it is shown that if 3|n and p n−2/3+1/18, then Pr[G(n, p) has a perfect traingle packing] = 1 − o(1).
منابع مشابه
Perfect Matchings in Random r-regular, s-uniform Hypergraphs
The copyright law of the United States (title 17, U.S. Code) governs the making of photocopies or other reproductions of copyrighted material. Any copying of this document without permission of its author may be prohibited by law. Perfect matchings in random i—regular, 5—uniform hypergraphs.
متن کاملExact Minimum Degree Thresholds for Perfect Matchings in Uniform Hypergraphs Iii
We determine the exact minimum l-degree threshold for perfect matchings in k-uniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than 1 2 ( n k−l ) . This extends our previous results [18, 19] that determine the minimum l-degree thresholds for perfect matchings in k-uniform hypergraphs for all l ≥ k/2 and provides two new (exact) threshold...
متن کاملPerfect Fractional Matchings in $k$-Out Hypergraphs
Extending the notion of (random) k-out graphs, we consider when the k-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each r there is a k = k(r) such that the k-out r-uniform hypergraph on n vertices has a perfect fractional matching with high probability (i.e., with probability tending to 1 as n → ∞) and prove an analogous result for r-uniform r-...
متن کاملA Note on Perfect Matchings in Uniform Hypergraphs
We determine the exact minimum `-degree threshold for perfect matchings in kuniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than 12 ( n k−` ) . This extends our previous results that determine the minimum `-degree thresholds for perfect matchings in k-uniform hypergraphs for all ` > k/2 and provides two new (exact) thresholds: (k, `) =...
متن کاملPerfect f-matchings and f-factors in hypergraphs - A combinatorial approach
We prove characterizations of the existence of perfect f -matchings in uniform mengerian and perfect hypergraphs. Moreover, we investigate the f -factor problem in balanced hypergraphs. For uniform balanced hypergraphs we prove two existence theorems with purely combinatorial arguments, whereas for non-uniform balanced hypergraphs we show that the f -factor problem is N P-hard.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Random Struct. Algorithms
دوره 23 شماره
صفحات -
تاریخ انتشار 2003