Graphs with degree constraints

TL;DR

This paper derives asymptotic formulas for counting graphs with specified degree constraints, generalizing previous results and including new cases like Euler graphs, using analytic combinatorics techniques.

Contribution

It introduces a general framework for enumerating graphs with arbitrary degree constraints, extending known results and deriving new enumeration formulas.

Findings

Asymptotic enumeration formulas for graphs with degree constraints

Extension of results to Euler graphs with even degrees

Application of analytic combinatorics to graph enumeration

Abstract

Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs with a given minimum degree, and establishes new ones, such as the enumeration of Euler graphs, i.e. where all vertices have an even degree. Those results are derived using analytic combinatorics.