Making multigraphs simple by a sequence of double edge swaps

TL;DR

This paper proves that any multigraph with a given degree sequence can be converted into a simple graph through a finite sequence of double edge swaps, advancing understanding in graph transformations and sampling.

Contribution

It provides a new theoretical result showing the reachability of simple graphs from multigraphs via double edge swaps involving loops or multiple edges.

Findings

Any loopy multigraph can be transformed into a simple graph.

The result answers a question posed by Janson.

It has implications for Markov chain Monte Carlo methods in graph sampling.

Abstract

We show that any loopy multigraph with a graphical degree sequence can be transformed into a simple graph by a finite sequence of double edge swaps with each swap involving at least one loop or multiple edge. Our result answers a question of Janson motivated by random graph theory, and it adds to the rich literature on reachability of double edge swaps with applications in Markov chain Monte Carlo sampling from the uniform distribution of graphs with prescribed degrees.