On the girth of forbidden subgraphs of coloring graphs

Abstract

The k-coloring graph of G, denoted Ck(G), is the graph whose vertex set is the proper k-colorings of the vertices of G, and where two k-colorings are adjacent if and only if they differ at exactly one vertex. A graph H is called permissible if there exists a graph G and a natural number k such that H is an induced subgraph of Ck(G), otherwise H is called forbidden. We show that every graph is either permissible or some subdivision of it is permissible. Furthermore, we prove if every edge in a triangle-free permissible graph is subdivided, the resulting graph is permissible, and all further subdivisions are also permissible. Although subdivisions tend to increase the girth of a graph, we conjecture that there exist forbidden subgraphs of arbitrarily large girth. Finally, we define forbidden subgraphs of girth seven and nine, the largest girths found so far.