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Efficient total coloring of cubic maps of girth 4

Italo J Dejter

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 3-cube 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 $Γ$ 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, a fifth basic operation is found necessary in terms of combinatorial cubic maps $M(Γ)$ of which the graphs $Γ$ 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 coloring.

Efficient total coloring of cubic maps of girth 4

Abstract

Let . A total coloring of a -regular simple graph via colors is an efficient total coloring if each color yields an efficient dominating set, where the 3-cube efficient domination condition applies to the restriction of each color class to the vertex set. Focus was set upon graphs of girth with efficient total colorings of finite simple cubic graphs 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, a fifth basic operation is found necessary in terms of combinatorial cubic maps of which the graphs 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 -belts have mod 4 has an efficient total coloring.

Paper Structure

This paper contains 8 sections, 5 theorems, 8 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Sprays
  3. Extensions
  4. Unfoldings
  5. Exchanges
  6. Amalgams
  7. Toroidally 3-edge connected graphs
  8. Concluding remarks

Key Result

Theorem 6

+1 Let $\Gamma$ be a finite connected simple cubic graph of girth 4. If $\Gamma$ has an ETC with 4 colors, then $|V(\Gamma)|\equiv 0 \mod 4$ and $\Gamma$ has only $\ell$-belts with $\ell\equiv 0 \mod 4$. $\blacktriangleleft$$\blacktriangleleft$

Figures (10)

  • Figure 1: cuadro.
  • Figure 2: The two sequences of exchanges in Example \ref{['tora']}.
  • Figure 3: exchanges not raising and raising genus 1 to 2.
  • Figure 4: Representation for Example \ref{['tor']} of an exchange in the union of two 3-cubes.
  • Figure 5: Two alternate representations as in Figure \ref{['torcuato']} and two related unfoldings.
  • ...and 5 more figures

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • proof
  • Definition 7
  • Definition 8
  • Theorem 9
  • ...and 22 more