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Tight factorizations of girth-$g$-regular graphs

Italo J. Dejter

TL;DR

This work studies tight factorizations of girth-$g$-regular graphs (egc graphs), where a $g$-girth-coloring assigns $g$ colors to edges so that every girth-$g$ cycle is colored bijectively. It develops a range of constructions (triangle replacement, toroidal cutouts, partial line graphs, barrels) and analyzes both obstructions and positive realizations across girths $3$ and $4$ (and higher), yielding multiple egc examples (e.g., $K_4$, $Q_4$ in two ways, Armanios–Wells) and identifying notable non-egc cases (e.g., bipartite complement of the Heawood graph). The paper connects these combinatorial structures to Hamiltonicity, Hamilton decompositions, and opens geometric avenues by embedding these graphs into 3D constructions, including compounds of PL Möbius strips and polylinks of hollow triangles, thereby linking graph colorings to concrete geometric realizations and symmetry groups. Overall, it provides both concrete classifications and robust methods for generating and verifying egc colorings while highlighting obstructions and rich geometric interpretations. $K_4$, $Q_4$, and barrel/mutant-barrel constructions emerge as central motifs, illustrating how tight factorizations drive both combinatorial and geometric insights with potential applications in optimization and spatial design.

Abstract

Girth-regular graphs with equal girth, regular degree and chromatic index are studied for the determination of 1-factorizations with each 1-factor intersecting every girth cycle. Applications to hamiltonian decomposability and to 3-dimensional geometry are given. Applications are suggested for priority assignment and optimization problems.

Tight factorizations of girth-$g$-regular graphs

TL;DR

This work studies tight factorizations of girth--regular graphs (egc graphs), where a -girth-coloring assigns colors to edges so that every girth- cycle is colored bijectively. It develops a range of constructions (triangle replacement, toroidal cutouts, partial line graphs, barrels) and analyzes both obstructions and positive realizations across girths and (and higher), yielding multiple egc examples (e.g., , in two ways, Armanios–Wells) and identifying notable non-egc cases (e.g., bipartite complement of the Heawood graph). The paper connects these combinatorial structures to Hamiltonicity, Hamilton decompositions, and opens geometric avenues by embedding these graphs into 3D constructions, including compounds of PL Möbius strips and polylinks of hollow triangles, thereby linking graph colorings to concrete geometric realizations and symmetry groups. Overall, it provides both concrete classifications and robust methods for generating and verifying egc colorings while highlighting obstructions and rich geometric interpretations. , , and barrel/mutant-barrel constructions emerge as central motifs, illustrating how tight factorizations drive both combinatorial and geometric insights with potential applications in optimization and spatial design.

Abstract

Girth-regular graphs with equal girth, regular degree and chromatic index are studied for the determination of 1-factorizations with each 1-factor intersecting every girth cycle. Applications to hamiltonian decomposability and to 3-dimensional geometry are given. Applications are suggested for priority assignment and optimization problems.

Paper Structure

This paper contains 16 sections, 20 theorems, 48 equations, 20 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Egc girth-3-regular graphs
  3. Egc girth-4-regular graphs
  4. The 4-cube as a twice-egc girth-4-regular graph
  5. Toroidal representation tables
  6. The bipartite complement of the Heawood graph
  7. Prisms of types 4443, 3221 and 3221
  8. Egc 1111-graphs
  9. Egc girth-g-regular graphs, g larger than 4
  10. Hamilton cycles and hamiltonian decomposability
  11. Applications to 3-dimensional geometry
  12. Compound of four PL Möbius strips
  13. Use of double egc girth-4-regular 4-cube
  14. Petrie polygons
  15. Automorphism group of the compound
  16. ...and 1 more sections

Key Result

Theorem 1

PV There is only one $(e_1)(e_2)(e_3)$-graph $\Gamma$ with $(e_1)(e_2)(e_3)=222=2^3$ namely $\Gamma=K_4$. Moreover, $\Gamma=K_4$ is egc. All other proper $(e_1)(e_2)(e_3)$-graphs are $1^20$-graphs, but not necessarily egc.

Figures (20)

  • Figure 1: Producing egc $(e_1)(e_2)(e_3)$-graphs $\Gamma$ that are triangle-replaced graphs from graphs $\Gamma'$. In particular, pairs $(\Gamma)(\Gamma')$ represented here are: (a)(d), (b)(e), (c)(f), (g)(h), (h)(m), (i)(n), (j)(o), (k)(p), (l)(q) and (s)(t). In additon, item (r) is an example of a generalized snark other than the Petersen graph. Items (u) and (v) are related to the dodecahedral graph and to the Coxeter graph, respectively.
  • Figure 2: Cutouts of $(e_1)(e_2)(e_3)(e_4)$-graphs: (a) is the Folkman graph $\mathbb{D}5$; (b) is for a graph embeddable into the Klein bottle; (c) exemplifies Theorem \ref{['re']}2(e). (d) is for the 4-cube in Theorem \ref{['re']}1(c); (e) and (f) exemplify Theorem \ref{['re']}3(b); (g) exemplifies Theorem \ref{['re']}3(c); (h) and (i) exemplify Theorem \ref{['re']}2(b); (j) exemplifies Theorem \ref{['re']}3(a);
  • Figure 3: On top of the figure there are represented three 1-factorizations of $K_{4,4}$. Below them, corresponding toroidal cutouts of $Q_4$ are drawn, with 0-, 1- and 2-color 4-cycles.
  • Figure 4: The biipartite complement $H$ of the Heawood-graph, with vertex set $V(H)\!=\!\{ij; i\in\{+,-\}$$j\in\mathbb{Z}_7\}$ is depicted on the upper left, (a), of the figure; its edges $\{+j,-j\}$$\{+j,-(j+2)\}$$\{+j,-(j+3)\}$ and $\{+j,-(j+4)\}$ for $j\in\mathbb{Z}_7$ are denoted $jj$$j(j+2)$$j(j+3)$ and $j(j+4)$ respectively, where addition is taken$\pmod{7}$. Its associated graph ${\rm GA}(H)$ is depicted on the right, (b), of the figure. This forces the coloring of the subgraph of ${\rm GA}(H)$ in the lower left, (c), of the figure.
  • Figure 5: Cases of prisms of: truncated octahedron $32^21$ in (a)--(g), $K_{3,3}$$4^33$ in (h), $ST_4$$31^3$ in (i) and a 16-vertex graph $31^3$ in (j), where the first two cases are shown not to be egc by arguments presented in the text, and the last two cases are explicitly shown to be egc.
  • ...and 15 more figures

Theorems & Definitions (63)

  • Theorem 1
  • proof
  • Remark 2
  • Theorem 4
  • proof
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Lemma 9
  • ...and 53 more