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.