The average number of spanning hypertrees in sparse uniform hypergraphs

Abstract

An r-uniform hypergraph H consists of a set of vertices V and a set of edges whose elements are r-subsets of V . We define a hypertree to be a connected hypergraph which contains no cycles. A hypertree spans a hypergraph H if it is a subhypergraph of H which contains all vertices of H. Greenhill, Isaev, Kwan and McKay (2017) gave an asymptotic formula for the average number of spanning trees in graphs with given, sparse degree sequence. We prove an analogous result for r-uniform hypergraphs with given degree sequence k = (k 1 , . . . , k n ). Our formula holds whenwhere k is the average degree and k max is the maximum degree.

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,…”

“…Greenhill, Kwan and Wind [18] determined asymptotic distribution of spanning trees for random cubic graphs, that is, when (r, s) = (3,2). The aim of this paper is to generalise the results of [18] to random r-regular s-uniform hypergraphs for any fixed r, s 2 with (r, s) = (2, 2).…”

“…(In fact a more general result is proved in [2], which covers irregular degree sequences and allows s and the maximum degree to grow slowly with n. ) In the graph case, the asymptotic formula for EY G was known up to a constant factor by the results of McKay [23] (who also considered irregular, slowly-growing degrees), and then this constant factor was calculated precisely in [18,Theorem 1.1].…”