Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges

Abstract

For n ≥ 3 and r = r(n) ≥ 3, let k = k(n) = (k 1 , . . . , k n ) be a sequence of non-negative integers with sum M (k) = n j=1 k j . We assume that M (k) is divisible by r for infinitely many values of n, and restrict our attention to these values. Let X = X(n) be a simple r-uniform hypergraph on the vertex set V = {v 1 , v 2 , . . . , v n } with t edges. We denote by H r (k) the set of all simple r-uniform hypergraphs on the vertex set V with degree sequence k, and let H r (k, X) be the set of all hypergraphs in H r (k) which contain no edge of X. We give an asymptotic enumeration formula for the size of H r (k, X). This formula holds when r 4 k 3 max = o(M (k)), t k 3 max = o(M (k) 2 ) and r t k 4 max = o(M (k) 3 ). Our proof involves the switching method. As a corollary, we obtain an asymptotic formula for the number of hypergraphs in H r (k) which contain every edge of X. We apply this result to find asymptotic expressions for the expected number of perfect matchings and loose Hamilton cycles in a random hypergraph in H r (k) in the regular case.

A threshold result for loose Hamiltonicity in random regular uniform hypergraphs

“…Aldosari and Greenhill [1] applied a switching argument to provide an asymptotic formula for the expected number of loose Hamilton cycles in G(n, r, s) when r, s are slowly-growing. We will use this formula in the proof of Corollary 2.3.…”

“…The first statement follows from substituting ℓ = 1 into Lemma 2.2 and using Stirling's approximation. The proof is completed by comparing this asymptotic expression with the asymptotic expression for EY G given in [1,Corollary 3.2].…”

“…If the probability of an event tends to 1 as n → ∞ along this set, then we say that the event holds asymptotically almost surely (a.a.s.). Since our main focus is on loose Hamilton cycles, we write I (r,s) instead of I (1) (r,s) when ℓ = 1. The case s = 2 (graphs) has been extensively studied.…”

Spanning trees in random regular uniform hypergraphs

“…4. Choose δ (1) , δ (2) ∈ N b , the degree sequence of T ′ 1 and T ′ 2 respectively, such that, for all i,…”

The average number of spanning hypertrees in sparse uniform hypergraphs

“…Suppose that r divides M(k) for infinitely many values of n and take n to infinity along these values. In [1], we found an asymptotic formula for the probability that a random hypergraph from H r (k) contains a given r-uniform hypergraph.…”