Triangle-creation processes on cubic graphs

TL;DR

This paper introduces a new triangle switch operation on cubic graphs, analyzes Markov chains performing these switches, and provides bounds on the number of triangles in the long-term behavior.

Contribution

It defines the triangle switch move, proves irreducibility of related Markov chains on 3-regular graphs, and establishes bounds on triangle counts over time.

Findings

Markov chains with triangle switches are irreducible on 3-regular graphs.

Bounds on the number of triangles are obtained in linear time.

Results hold regardless of initial graph.

Abstract

An edge switch is an operation which makes a local change in a graph while maintaining the degree of every vertex. We introduce a switch move, called a triangle switch, which creates or deletes at least one triangle. Specifically, a make move is a triangle switch which chooses a path $zwvxy$ of length 4 and replaces it by a triangle $vxwv$ and an edge $yz$, while a break move performs the reverse operation. We consider various Markov chains which perform random triangle switches, and assume that every possible make or break move has positive probability of being performed. Our first result is that any such Markov chain is irreducible on the set of all 3-regular graphs with vertex set $\{1,2,\ldots, n\}$. For a particular, natural Markov chain of this type, we obtain a non-trivial linear upper and lower bounds on the number of triangles in the long run. These bounds are almost surely obtained in linear time, irrespective of the starting graph.