A triangle process on regular graphs

Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. In "Triangle-creation processes on cubic graphs", we introduced a new type of switch, called the triangle switch, which creates or deletes a triangle at each step. The aim was to generate graphs with many more triangles than typical in a (uniform) random regular graph. Several Markov chains based on random triangle switches were proposed, and studied on cubic graphs. In particular, the chains were shown to be irreducible. However, irreducibility on $d$-regular graphs, for any $d\geq3$, is far from a straightforward generalisation of the cubic result, and this question was left open in that paper. Here we show that the triangle-switch chain is, in fact, irreducible on $d$-regular graphs with $n$ vertices, for all~$n$ and~$d\geq 3$.