A triangle process on regular graphs

TL;DR

This paper explores triangle switches, a type of local graph operation, and demonstrates their ability to connect all regular graphs of degree at least 3, enabling Markov chains to generate graphs with many triangles.

Contribution

It introduces and analyzes triangle switches, proving they connect all d-regular graphs for d≥3, facilitating new graph generation methods with high triangle counts.

Findings

Triangle switches connect all d-regular graphs for d≥3.

Markov chains using triangle switches are irreducible on these graphs.

The study extends to 2-regular graphs, analyzing their connectivity.

Abstract

Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. We study a restricted set of switches, called triangle switches. Each triangle switch creates or deletes at least one triangle. Triangle switches can be used to define Markov chains which generate graphs with a given degree sequence and with many more triangles (3-cycles) than is typical in a uniformly random graph with the same degrees. We show that the set of triangle switches connects the set of all $d$-regular graphs on $n$ vertices, for all $d\geq 3$. Hence, any Markov chain which assigns positive probability to all triangle switches is irreducible on these graphs. We also investigate this question for 2-regular graphs.