Classification of ETCs via five constructive operations

Establish that every efficient total coloring of any finite connected simple cubic graph of girth 4 can be obtained solely by applying the five constructive operations defined in the paper—sprays, extensions, unfoldings, exchanges, and amalgams—with sprays providing the base case on the 3-cube Q3.

Background

Building on prior work that proposed four operations (sprays, extensions, unfoldings, exchanges) to generate all observed ETGCs from the 3-cube, this paper introduces a fifth operation, amalgam, shown to be necessary by new constructions. Throughout the paper, the authors develop these operations and demonstrate how they produce ETGCs (and related edge-girth colorings on prisms) across planar and toroidal settings.

The conjecture asserts a complete constructive classification: no efficient total coloring (ETC) of a finite connected simple cubic graph of girth 4 exists outside those obtainable by iterating the five operations. This elevates the framework from a toolbox of constructions to a proposed exhaustive generative scheme.

References

Conjecture ETCs of finite connected simple cubic graphs $\Gamma$ of girth 4 are obtained solely by means of the following five constructive operations: Sprays (Definition~\ref{sabado}), yielding the smallest such $\Gamma$, namely $\Gamma=Q_3$ (as in Corollary~\ref{t1}), Extensions (Definition~\ref{pe}), Unfoldings (Definition~\ref{unfold}), exchanges (Definition~\ref{exchange}) and Amalgams (Definition~\ref{amalgam}).

— Efficient total coloring of cubic maps of girth 4  (2604.02991 - Dejter, 3 Apr 2026) in Conjecture (label ‘con1’), Section “Concluding remarks”