Periodic efficient total girth colorings in cubic maps of girth 4 correspond to 3-permutation face colorings

Abstract

Efficient total girth colorings (or ETGCs) of maps of connected simple cubic graphs of girth 4 are reanalyzed in genus-realizing orientable surfaces, where the ETGC condition applies to the restriction of each color class to the vertex set. A necessary condition for such ETGCs to exist is that the maps considered to admit them have only face-cycle lengths divisible by 4. When such ETGCs exist, for which seven constructive tools are provided (four old and three new), the faces of the involved maps are shown to be colorable by the six 3-permutations whenever the face color cycles are length 4 periodic.Moreover, for the vertex coloring of each ETGC, there are two mutually orthogonal subjacent edge colorings leading to two corresponding mutually orthogonal ETGCs distinguishable as the directed ETGC and the reversed ETGC. Furthermore, there are two injections into the set of 3-permutation face colorings: one from the set of periodic directed ETGCs and the other one from the set of periodic reversed ETGCs. Throughout the work, questions and conjectures are posed, among which one asserting that all ETGCs are obtained solely by means of the seven mentioned tools.