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

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 m