Kempe Equivalence of Colorings of 4-regular Graphs

Abstract

Given a graphG and a proper vertex coloring ofG, a 2-coloring induced subgraph ofG is a subgraph induced by all the vertices with one of two colors, a component of a 2-coloring induced subgraph is called a 2-coloring component. To make a Kempe change is to obtain one coloring from another by exchanging the colors of vertices in a 2-coloring component. Two colorings are Kempe equivalent if each one can be obtained from the other by a series of Kempe changes. Mohar conjectured that, for k3, all k-colorings of connected k-regular graphs that are not complete are Kempe equivalent. Feghali et al. addressed the case k=3, and it is still an unsolved conjecture for k4. This paper considers the casek=4 by showing that: (1) ifG is a connected 4-regular graph that is not 3-connected, then all 4-colorings ofG are Kempe equivalent; (2) ifG is a connected 4-regular graph that contains an induced subgraph isomorphic to a 4-wheel or a nearly complete graph of order 5, then all 4-colorings ofG are Kempe equivalent; (3) ifG is a 3-connceted 4-regular graph with a 4-coloringf and a vertexx such that there are three or four neighbors ofx colored with the same color under f, then all 4-colorings ofG are Kempe equivalent.