Efficient total colorings of cubic maps of girth 4 and related topics

TL;DR

This paper investigates efficient total colorings of cubic graphs with girth 4, introduces new basic operations for constructing such colorings, and conjectures conditions for their existence in toroidally 3-edge-connected graphs.

Contribution

It identifies two additional basic operations needed for efficient total colorings and proposes a conjecture on their existence in certain cubic graphs.

Findings

Two new basic operations are necessary for efficient total colorings.

A conjecture relates efficient total colorings to toroidally 3-edge-connected graphs with specific belt properties.

Application of new operations yields total perfect code partitions in girth 4 cubic graphs.

Abstract

Let $2\le k\in\mathbb{Z}$. A total coloring of a$k$-regular simple graph via $k+1$ colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. Focus was set upon graphs of girth $k+1$ with efficient total colorings of finite simple cubic graphs $\Gamma$ of girth 4 built up from the 3-cube and leading to a conjecture that all of those colorings were obtained by means of four basic operations. In the present work, two more basic operations are found necessary in terms of combinatorial cubic maps $M(\Gamma)$ of which the graphs $\Gamma$ are their 1-skeletons. This takes to conjecturing that any simple cubic graph that is toroidally 3-edge-connected (defined in the work) and whose $\ell$-belts have $\ell\equiv 0$ mod 4 has an efficient total coloringAn application of one of the two new basic operations yields total perfect code partitions in $g$-toroidal cubic graphs of girth 4 via semi-total colorings that reduce to total colorings, for every $0<g\in\mathbb{Z}$.