The average number of spanning trees in sparse graphs with given degrees

We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence d=(d1,…,dn), provided that the number of edges is at least n+12dmax4, where dmax is the maximum degree. A key part of our argument involves establishing a concentration result for a certain family of functions over random trees with given degrees, using Prüfer codes

Proof of a conjecture on induced subgraphs of Ramsey graphs

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erd\H{o}s, Faudree and S\'{o}s conjectured that in any C-Ramsey graph there are $\Omega\left(n^{5/2}\right)$ induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samotij, in this paper we prove this conjecture

The average number of spanning trees in sparse graphs with given degrees

We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence d = (d₁, . . . , dn), provided that the number of edges is at least n + 1/2d⁴max, where dmax is the maximum degree. A key part of our argument involves establishing a concentration result for a certain family of functions over random trees with given degrees, using Prüfer codes

Dirac-type theorems in random hypergraphs

For positive integers $d<k$ and $n$ divisible by $k$, let $m_{d}(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_{1}(2,n)=\lceil n/2\rceil$. However, in general, our understanding of the values of $m_{d}(k,n)$ is still very limited, and it is an active topic of research to determine or approximate these values. In this paper we prove a "transference" theorem for Dirac-type results relative to random hypergraphs. Specifically, for any $d0$ and any "not too small" $p$, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and edge probability $p$ typically has the property that every spanning subgraph of $G$ with minimum degree at least $(1+\varepsilon)m_{d}(k,n)p$ has a perfect matching. One interesting aspect of our proof is a "non-constructive" application of the absorbing method, which allows us to prove a bound in terms of $m_{d}(k,n)$ without actually knowing its value