Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Orbits of consistent walk in dart-transitive maps

Micael Toledo, Alejandra Ramos, Primoz Potocnik, Stephen Wilson

Abstract

In a simple graph, a shunt is a symmetry which sends an edge to an incident edge (without fixing their shared vertex). The orbit of this edge under the shunt forms a consistent cycle. The important theorem of Biggs and Conway says that in a dart-transitive graph of valence q, there are exactly q-1 orbits of consistent cycles. These ideas have become a useful tool in the area of graphs symmetries, and generalize easily to consistent walks in graphs which are not simple. These walks are not necessarily cycles, or even circuits. This paper considers these walks and their orbits in the venue of dart-transitive maps and classifies them geometrically.

Orbits of consistent walk in dart-transitive maps

Abstract

In a simple graph, a shunt is a symmetry which sends an edge to an incident edge (without fixing their shared vertex). The orbit of this edge under the shunt forms a consistent cycle. The important theorem of Biggs and Conway says that in a dart-transitive graph of valence q, there are exactly q-1 orbits of consistent cycles. These ideas have become a useful tool in the area of graphs symmetries, and generalize easily to consistent walks in graphs which are not simple. These walks are not necessarily cycles, or even circuits. This paper considers these walks and their orbits in the venue of dart-transitive maps and classifies them geometrically.

Paper Structure

This paper contains 13 sections, 8 theorems, 5 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Maps and their symmetries
  3. Realizations
  4. Automorphisms
  5. Sequences of flags: holes and Petrie paths
  6. Dart-transitive maps
  7. Consistent Walks
  8. Types of $j$-holes: cycles, twinings, beads and bracelets
  9. Example: The map ${\mathcal{M}} = M'_{12,7}$
  10. Example: The map ${\mathcal{N}} = \hbox{\rm D}(M'_{12,7})$
  11. Cunningham's map
  12. Dart-transitive maps having exactly one vertex
  13. Consistent circuits and their orbits

Key Result

Theorem 1.1

If $\Gamma$ is a finite simple $q$-valent graph and $G$ a subgroup of ${\rm Aut}(\Gamma)$ acting transitively on the darts of $\Gamma$, then there are exactly $q-1$ orbits under $G$ of $G$-consistent cyclets.

Figures (14)

  • Figure 1: The right triangle corresponding to the flag $\Phi$.
  • Figure 2: The neighbours of the flag $\Phi$.
  • Figure 3: Different holes and Petrie paths on a $4$-valent map.
  • Figure 4: Two views of a bead.
  • Figure 5: Part of a bracelet.
  • ...and 9 more figures

Theorems & Definitions (16)

  • Theorem 1.1: Bconway
  • Theorem 1.2: MPW3
  • Definition 2.1
  • Remark 3.1
  • Lemma 4.1
  • Remark 4.2
  • Theorem 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 6 more