Efficient total colorings in cubic maps of girth 4 have equivalent face colorings via the six 3-permutations

Abstract

The faces of maps of finite connected simple cubic graphs of girth 4 in genus-realizing orientable surfaces of face boundary cycle lengths divisible by 4 are shown to be colorable by the six 3-permutations. The resulting face colorings induce corresponding efficient total colorings (or ETCs) with four colors, where the ETC condition applies to the restriction of each color class to the vertex sets, with 2-choosability available for the edge sets that can be refined into one-to-one correspondences between the said face colorings and each one of two possible ETCs in each case.