Reflexive coloring complexes for 3-edge-colorings of cubic graphs

Abstract

Given a 3-colorable graph X, the 3-coloring complex B(X) is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of X. Two color classes C,D∈V(B(X)) are joined by an edge if C and D appear together in a 3-coloring of X. The graph B(X) is 3-colorable. Graphs for which B(B(X)) is isomorphic to X are termed reflexive graphs. In this paper, we consider 3-edge-colorings of cubic graphs for which we allow half-edges. Then we consider the 3-coloring complexes of their line graphs. The main result of this paper is a surprising outcome that the line graph of any connected cubic triangle-free outerplanar graph is reflexive. We also exhibit some other interesting classes of reflexive line graphs.