Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extending graph total colorings to cell complexes

Italo J. Dejter

TL;DR

This work extends graph total colorings to 2-cell complexes by defining efficient total cell colorings (ETCC) and their strict variants (SETCC). It provides explicit, double-periodic constructions on planar lattices that extend to toroidal 2-cell complexes derived from square, triangular, hexagonal, and trihexagonal tilings, linking ETC/EDS concepts to tiling combinatorics and Hadwiger–Nelson style colorings. By exploiting toroidal quotients and duals (e.g., Heawood as a dual), the paper demonstrates concrete colorings with color sets of sizes $5$ and $7$ and shows how edge-removal techniques yield SETCCs for Archimedean tilings. It concludes with a survey of constructed toroidal complexes, highlights minimal examples, and poses open questions about generalizing ETCC/SETCC to broader classes of $r$-cell complexes and other tiling schemes.

Abstract

Let $2\le k\in\mathbb{Z}$. A total coloring of a simple connected regular graph via color set $ \{0,1,\ldots, k\}$ is said to be {\it efficient} 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 2-cell complexes whose 1-skeletons, namely their induced 1-cell complexes, are toroidal graphs. Each such 2-cell complex is said to cover its induced 1-skeleton. An efficient total coloring of one such skeleton induces an efficient total cell coloring of its covering 2-cell complex if it assigns a vertex-and-edge $k$-color set to the border skeleton of each of its 2-cells, with the consequently missing color in $\{0,1,\ldots,k\}$ assigned to the 2-cell itself, so that the two adjacent 2-cells along any 1-cell are assigned different colors. Applications are given for plane tilings, cycle products, toroidal triangulations, honeycombs and star-of-David tilings.

Extending graph total colorings to cell complexes

TL;DR

This work extends graph total colorings to 2-cell complexes by defining efficient total cell colorings (ETCC) and their strict variants (SETCC). It provides explicit, double-periodic constructions on planar lattices that extend to toroidal 2-cell complexes derived from square, triangular, hexagonal, and trihexagonal tilings, linking ETC/EDS concepts to tiling combinatorics and Hadwiger–Nelson style colorings. By exploiting toroidal quotients and duals (e.g., Heawood as a dual), the paper demonstrates concrete colorings with color sets of sizes and and shows how edge-removal techniques yield SETCCs for Archimedean tilings. It concludes with a survey of constructed toroidal complexes, highlights minimal examples, and poses open questions about generalizing ETCC/SETCC to broader classes of -cell complexes and other tiling schemes.

Abstract

Let . A total coloring of a simple connected regular graph via color set is said to be {\it efficient} 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 2-cell complexes whose 1-skeletons, namely their induced 1-cell complexes, are toroidal graphs. Each such 2-cell complex is said to cover its induced 1-skeleton. An efficient total coloring of one such skeleton induces an efficient total cell coloring of its covering 2-cell complex if it assigns a vertex-and-edge -color set to the border skeleton of each of its 2-cells, with the consequently missing color in assigned to the 2-cell itself, so that the two adjacent 2-cells along any 1-cell are assigned different colors. Applications are given for plane tilings, cycle products, toroidal triangulations, honeycombs and star-of-David tilings.
Paper Structure (8 sections, 13 theorems, 10 equations, 9 figures)

This paper contains 8 sections, 13 theorems, 10 equations, 9 figures.

Table of Contents

  1. Total colorings and efficient dominating sets
  2. Combinatorial cell complexes
  3. Euclidean tilings by convex regular polygons
  4. SETCCs based on the square tiling
  5. SETCCs based on the triangular tiling
  6. SETCCs based on the hexagonal tiling
  7. SETCCs based on the trihexagonal tiling
  8. Other SETCCs derived from the triangular tiling

Key Result

Theorem 4

CW A Hausdorff space $\Psi(X)$ is homeomorphic to a CW complex if and only if there exists a partition of $\Psi(X)$ into "open cells" $e_\alpha^r$, each with a corresponding closure ("or closed cell" $\bar{e}_\alpha^r:=closure_{\Psi(X)}(c_\alpha^r)$ that satisfies

Figures (9)

  • Figure 1: SETCCs for the square and C&R $4.8^2$ tilings. Toroidal SETCC via square tiling.
  • Figure 2: SETCC for the triangular and C&R $3^3.4^2$ tilings.
  • Figure 3: ETCC and SETCC mod 7 for square tiling based on the triangular tiling.
  • Figure 4: Two SETCCs for C&R $3^2.4.3.4$ tiling based on the triangular tiling.
  • Figure 5: SETCCs based on the triangular and hexagonal tilings.
  • ...and 4 more figures

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Remark 3
  • Theorem 4
  • Definition 5
  • Definition 6
  • Example 7
  • Theorem 8
  • proof
  • Corollary 9
  • ...and 23 more