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Total coloring of regular graphs of girth = degree + 1

Italo J. Dejter

TL;DR

The paper investigates efficient total colorings (ETGCs) of finite connected $k$-regular graphs with girth $k+1$, linking total colorings to efficient domination and introducing concepts such as VEGC and EGC. It develops a constructive framework for cubic graphs ($k=3$, girth $4$), showing ETGCs with $4$ colors can be built from the 3-cube $Q_3$ via four operations (periodic extensions, accordion unfoldings, cycle exchanges, ETCings), and extending these ideas to planar, toroidal, and higher-genus cases. The work also establishes algorithmic approaches to ETGCs and explores how genus can be raised via handles while preserving ETGC structure, as well as partitions of edges into 3-paths and 3-stars. Finally, it provides counterexamples in 4-regular graphs of girth $5$ showing that total colorings need not be efficient, highlighting the limits of ETGC existence. Together, these results map the landscape of ETGC existence across families of regular graphs and surfaces and offer constructive methods for generating ETGCs where possible.

Abstract

Let $2\le k\in\mathbb{Z}$. A total coloring of a $k$-regular simple graph via $k+1$ colors is an {\it 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. In this work, focus is set upon graphs of girth $k+1$. Efficient total colorings of finite connected simple cubic graphs of girth 4 are constructed starting at the 3-cube. It is conjectured that all of them are obtained by means of four basic operations. In contrast, the Robertson 19-vertex $(4,5)$-cage, the alternate union $Pet^k$ of a (Hamilton) $10k$-cycle with $k$ pentagon and $k$-pentagram $5$-cycles, for $k>1$ not divisible by 5, and its double cover $Dod^k$, contain TCs that are nonefficient. Applications to partitions into 3-paths and 3-stars are given.

Total coloring of regular graphs of girth = degree + 1

TL;DR

The paper investigates efficient total colorings (ETGCs) of finite connected -regular graphs with girth , linking total colorings to efficient domination and introducing concepts such as VEGC and EGC. It develops a constructive framework for cubic graphs (, girth ), showing ETGCs with colors can be built from the 3-cube via four operations (periodic extensions, accordion unfoldings, cycle exchanges, ETCings), and extending these ideas to planar, toroidal, and higher-genus cases. The work also establishes algorithmic approaches to ETGCs and explores how genus can be raised via handles while preserving ETGC structure, as well as partitions of edges into 3-paths and 3-stars. Finally, it provides counterexamples in 4-regular graphs of girth showing that total colorings need not be efficient, highlighting the limits of ETGC existence. Together, these results map the landscape of ETGC existence across families of regular graphs and surfaces and offer constructive methods for generating ETGCs where possible.

Abstract

Let . A total coloring of a -regular simple graph via colors is an {\it 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. In this work, focus is set upon graphs of girth . Efficient total colorings of finite connected simple cubic graphs of girth 4 are constructed starting at the 3-cube. It is conjectured that all of them are obtained by means of four basic operations. In contrast, the Robertson 19-vertex -cage, the alternate union of a (Hamilton) -cycle with pentagon and -pentagram -cycles, for not divisible by 5, and its double cover , contain TCs that are nonefficient. Applications to partitions into 3-paths and 3-stars are given.
Paper Structure (8 sections, 15 theorems, 23 equations, 2 figures)

This paper contains 8 sections, 15 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Regular graphs of girth = degree + 1
  3. Planar cases
  4. Toroidal cases
  5. Algorithmic aspects of ETGCs
  6. Raising the genus via handles
  7. Edge-partitions into 3-paths and into 3-stars
  8. 4-regular graphs of girth 5 with TCs but not ETCs

Key Result

Theorem 9

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

Figures (2)

  • Figure 1: TCs of 20-, 40- and 19-vertex 4-regular graphs of girth 5.
  • Figure 2: Stressing in red those vertices and edges in color 0 in $Pet^k$, for $k=2,3,4,6$.

Theorems & Definitions (54)

  • Definition 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Theorem 9
  • proof
  • Definition 10
  • ...and 44 more