Proper Z4 x Z2-colorings: structural characterization with application to some snarks

Abstract

A proper abelian coloring of a cubic graph G by a finite abelian group A is any proper edge-coloring of G by the non-zero elements of A such that the sum of the colors of the three edges incident to any vertex v of G equals zero. It is known that cyclic groups of order smaller than 10 do not color all bridgeless cubic graphs, and that all abelian groups of order at least 12 do. This leaves the question open for the four so called exceptional groups Z4 x Z2, Z3 x Z3, Z10 and Z11 for snarks. It is conjectured in literature that every cubic graph has a proper abelian coloring by each exceptional group and it is further known that the existence of a proper Z4 x Z2-coloring of G implies the existence of a proper coloring of G by all the remaining exceptional groups. In this paper, we give a characterization of a proper Z4 x Z2-coloring in terms of the existence of a matching M in a 2-factor F of G with particular properties. Moreover, in order to modify an arbitrary matching M so that it meets the requirements of the characterization, we first introduce an incidence structure of the cycles of F in relation to the cycles of G - M. Further, we provide a sufficient condition under which M can be modified into a desired matching in terms of particular properties of the introduced incidence structure. We conclude the paper by applying the results to some oddness two snarks, in particular to permutation snarks. We believe that the approach of this paper with some additional refinements extends to larger classes of snarks, if not to all in general.