A digraph whose degree sequence has a unique vertex labeled realization is called threshold. In this paper we present several characterizations of threshold digraphs and their degree sequences, and show these characterizations to be equivalent. Using this result, we obtain a new, short proof of the Fulkerson-Chen theorem on degree sequences of general digraphs.

Let a i , b i , i = 0, 1, 2, . . . be drawn uniformly and independently from the unit interval, and let t be a fixed real number. Let a site (i, j) ∈ N 2 be open if a i + b j ≤ t, and closed otherwise. We obtain a simple, exact expression for the probability Θ(t) that there is an infinite path (oriented or not) of open sites, containing the origin. Θ(t) is continuous and has continuous first derivative except at the critical point (t = 1), near which it has critical exponent (3 − √ 5)/2.

When generating a random graph, if more structure is desired than is given in the popular Erdős-Renyi model, one method is to generate a degree sequence first then create a graph with this degree sequence. Blitzstein and Diaconis[1] (among others) developed a sequential algorithm to create a random graph from a degree sequence. This algorithm is assured to always terminate in a graph with the desired degree sequence; unfortunately, it is slow. This work focuses on the subroutine of the previous algorithm which determines the candidate edges, improving the runtime of the overall algorithm from O(mn 2) to O(mn).