The canonical coloring graph of trees and cycles

Abstract

For a graph G and an ordering of the vertices π, the set of canonical k-colorings of G under π is the set of non-isomorphic proper k-colorings of G that are lexicographically least under π. The canonical coloring graph Can^π_k(G) is the graph with vertex set the canonical colorings of G and two vertices are adjacent if the colorings differ in exactly one place. This is a natural variation of the color graph C_k(G) where all colorings are considered. We show that every graph has a canonical coloring graph which is disconnected; that trees have canonical coloring graphs that are Hamiltonian; and cycles have canonical coloring graphs that are connected.