Random graphs with forbidden vertex degrees

TL;DR

This paper analyzes the structure of random graphs conditioned on vertex degrees belonging to a specific subset, revealing asymptotic degree distributions and conditions for giant component emergence, with applications to even and odd degree sets.

Contribution

It introduces a framework for studying conditioned random graphs with degree constraints, deriving asymptotic degree distributions and structural properties under certain hypotheses.

Findings

Empirical degree distribution is asymptotically Poisson.

Conditions for the existence of a giant component are established.

Applications include even and odd degree constrained graphs.

Abstract

We study the random graph G_{n,\lambda/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter \mux given as the root of a certain `characteristic equation' of S that maximises a certain function \psis(\mu). Subject to a hypothesis on S, we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set S including the sets of (respectively) even and odd numbers. The random \emph{even} graph is related to the random-cluster model on the complete graph K_n.