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Cubic vertex-transitive graphs of girth six

Primož Potočnik, Janoš Vidali

TL;DR

This work determines the complete classification of cubic vertex-transitive graphs with girth $6$, showing that, apart from the Desargues graph on $20$ vertices, each such graph arises from one of three construction mechanisms: (i) the skeleton of a toroidal map of type $\{6,3\}$, (ii) the truncation of an arc-transitive triangulation of a hyperbolic surface, or (iii) the truncation of a $6$-regular graph with an arc-transitive dihedral scheme. The authors introduce and exploit the signature $(a,b,c)$ to quantify girth-cycle distribution at each vertex, enabling a refined, case-by-case classification. They also provide a framework for larger girth, including a census of signatures observed in graphs up to $1280$ vertices, and discuss the growth of combinatorial possibilities as girth increases. Overall, the paper extends prior girth-5 classifications to girth-6 and sets up a foundation for systematic higher-girth investigations through dihedral-truncation and map-theoretic constructions.

Abstract

In this paper, a complete classification of finite simple cubic vertex-transitive graphs of girth $6$ is obtained. It is proved that every such graph, with the exception of the Desargues graph on $20$ vertices, is either a skeleton of a hexagonal tiling of the torus, the skeleton of the truncation of an arc-transitive triangulation of a closed hyperbolic surface, or the truncation of a $6$-regular graph with respect to an arc-transitive dihedral scheme. Cubic vertex-transitive graphs of girth larger than $6$ are also discussed.

Cubic vertex-transitive graphs of girth six

TL;DR

This work determines the complete classification of cubic vertex-transitive graphs with girth , showing that, apart from the Desargues graph on vertices, each such graph arises from one of three construction mechanisms: (i) the skeleton of a toroidal map of type , (ii) the truncation of an arc-transitive triangulation of a hyperbolic surface, or (iii) the truncation of a -regular graph with an arc-transitive dihedral scheme. The authors introduce and exploit the signature to quantify girth-cycle distribution at each vertex, enabling a refined, case-by-case classification. They also provide a framework for larger girth, including a census of signatures observed in graphs up to vertices, and discuss the growth of combinatorial possibilities as girth increases. Overall, the paper extends prior girth-5 classifications to girth-6 and sets up a foundation for systematic higher-girth investigations through dihedral-truncation and map-theoretic constructions.

Abstract

In this paper, a complete classification of finite simple cubic vertex-transitive graphs of girth is obtained. It is proved that every such graph, with the exception of the Desargues graph on vertices, is either a skeleton of a hexagonal tiling of the torus, the skeleton of the truncation of an arc-transitive triangulation of a closed hyperbolic surface, or the truncation of a -regular graph with respect to an arc-transitive dihedral scheme. Cubic vertex-transitive graphs of girth larger than are also discussed.

Paper Structure

This paper contains 9 sections, 20 theorems, 1 equation, 20 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Graphs
  4. Dihedral schemes
  5. Maps
  6. Three special families of toroidal graphs
  7. Auxiliary results
  8. Proof of Theorem \ref{['thm:g6']}
  9. Larger girths

Key Result

Theorem 1

Let $\Gamma$ be a connected cubic graph. Then $\Gamma$ is vertex-transitive and has girth $6$ if and only if $\Gamma$ is one of the following:

Figures (20)

  • Figure 1: (a) The octahedral graph, a $4$-regular graph. (b) The truncation of the octahedral graph with respect to the dihedral scheme obtained by considering the drawing (a) as a map (i.e., an octahedron). (c) The truncation of the octahedral graph with respect to a different dihedral scheme. Note that in both truncations, vertices of the graph in (a) have been replaced by $4$-cycles.
  • Figure 2: A section of $\Sigma_n \cong \operatorname{SDW}(n, 3)$.
  • Figure 3: The graph $\Psi_{10}$ embedded on a torus. The double edges show one of the $6$-cycles which do not correspond to a face of the embedding.
  • Figure 4: Constructing a graph of girth $6$. The general setting is shown in (a). (b) and (c) show the cases when the arc $u_0 u_1$ lies on seven $6$-cycles, and six $6$-cycles with $T(u_0 u_1) = \{\{2, 2\}, \{2, 0\}\}$, respectively. A contradiction arises in both cases.
  • Figure 5: Constructing a graph of girth $6$ with $c = 6$. (a) The case $T(u_0 u_1) = \{\{2, 2\}, \{1, 1\}\}$, which leads to a contradiction. (b) The case $T(u_0 u_1) = \{\{2, 1\}, \{2, 1\}\}$. (c) The subcase of (b) with $a = b = 5$, which also leads to a contradiction.
  • ...and 15 more figures

Theorems & Definitions (31)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • ...and 21 more