ETGC existence for toroidally 3-edge-connected cubic graphs with all ℓ-belts divisible by 4

Establish that every toroidally 3-edge-connected simple cubic graph of girth 4 whose facial boundary cycles (ℓ-belts) all have lengths divisible by 4 admits an efficient total girth coloring (ETGC), namely a total coloring with four colors in which each color class is an efficient dominating set and every 4-cycle uses each of the four colors exactly once on its vertices and exactly once on its edges.

Background

The paper studies efficient total colorings (ETCs) of finite simple cubic graphs of girth 4 and introduces the stronger notion of efficient total girth colorings (ETGCs), which are ETCs that also color each girth-4 cycle with all four colors exactly once on vertices and edges. A necessary condition proved earlier (Theorem 9 of the cited work) is that any cubic graph with an ETC must have |V(Γ)| ≡ 0 (mod 4) and only ℓ-belts whose lengths are multiples of 4.

However, Example 7 (Klein-type construction) exhibits a toroidal simple cubic graph of girth 4 whose ℓ-belts all have length ≡ 0 (mod 4) yet that does not admit an ETC. This motivates strengthening the structural hypothesis to a new notion, ‘toroidally 3-edge connected’ (Definition 9), which requires that at least one of the planar unfoldings (xcutout or ycutout) of a toroidal bicutout be 3-edge-connected. The conjecture posits that, under this added condition, the necessary belt-length condition becomes sufficient for the existence of an ETGC.