Fast Sequential Creation of Random Realizations of Degree Sequences

On Fractional Realizations of Graph Degree Sequences

We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope; we characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of threshold and pseudo-split graphs, and we characterize their re...

Non-crossing Tree Realizations of Ordered Degree Sequences

We investigate the enumeration of non-crossing tree realizations of integer sequences, and we consider a special case in four parameters, that can be seen as a four-dimensional tetrahedron that generalizes Pascal’s triangle and the Catalan numbers. This work is motivated by the study of ambiguities in categorial grammars.

Packings and Realizations of Degree Sequences with Specified Substructures

Graphic Sequences Have Realizations Containing Bisections of Large Degree

A bisection of a graph is a balanced bipartite spanning subgraph. Bollobás and Scott conjectured that every graphG has a bisectionH such that degH(v) ≥ bdegG(v)/2c for all vertices v. We prove a degree sequence version of this conjecture: given a graphic sequence π, we show that π has a realization G containing a bisection H where degH(v) ≥ b(degG(v)− 1)/2c for all vertices v. This bound is ver...

On uniform sampling simple directed graph realizations of degree sequences

Choosing a uniformly sampled simple directed graph realization of a degree sequence has many applications, in particular in social networks where self-loops are commonly not allowed. It has been shown in the past that one can perform a Markov chain arc-switching algorithm to sample a simple directed graph uniformly by performing two types of switches: a 2-switch and a directed 3-cycle reorienta...