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A model of trees for 5-connected planar triangulations

Éric Fusy

TL;DR

This work constructs a bijection between 5c-triangulations and leg-balanced $5$-regular plane trees, enabling an entirely combinatorial treatment of rooted 5c-triangulations and rooted 5-connected triangulations with root-vertex degree 5. The authors derive explicit algebraic generating functions for these classes via a master bijection built from tree-biorientations and angular maps, and they exploit this correspondence to obtain uniform random generation and succinct encoding procedures. The framework extends the known connections between $k$-connected triangulations and $k$-regular plane trees to the case $k=5$, and it connects to Gao–Wanless–Wormald’s substitution approach through a concrete combinatorial encoding. The paper also discusses algorithmic implications and open questions on bijective extensions to broader classes of irreducible maps.

Abstract

Triangulations of the 5-gon with no separating triangle nor quadrangle, so called 5c-triangulations, are a planar map family closely related to 5-connected planar triangulations. We show that 5c-triangulations are in bijection with 5-regular plane trees satisfying a simple local constraint at inner edges. It yields explicit expressions for the generating functions of rooted 5c-triangulations, and of rooted 5-connected planar triangulations with root-vertex degree 5, these belonging to the same algebraic extension as the generating function of rooted 5-connected planar triangulations computed by Gao, Wanless and Wormald. The bijection also makes it possible to obtain efficient uniform random generation and succinct encoding procedures for 5-connected planar triangulations.

A model of trees for 5-connected planar triangulations

TL;DR

This work constructs a bijection between 5c-triangulations and leg-balanced -regular plane trees, enabling an entirely combinatorial treatment of rooted 5c-triangulations and rooted 5-connected triangulations with root-vertex degree 5. The authors derive explicit algebraic generating functions for these classes via a master bijection built from tree-biorientations and angular maps, and they exploit this correspondence to obtain uniform random generation and succinct encoding procedures. The framework extends the known connections between -connected triangulations and -regular plane trees to the case , and it connects to Gao–Wanless–Wormald’s substitution approach through a concrete combinatorial encoding. The paper also discusses algorithmic implications and open questions on bijective extensions to broader classes of irreducible maps.

Abstract

Triangulations of the 5-gon with no separating triangle nor quadrangle, so called 5c-triangulations, are a planar map family closely related to 5-connected planar triangulations. We show that 5c-triangulations are in bijection with 5-regular plane trees satisfying a simple local constraint at inner edges. It yields explicit expressions for the generating functions of rooted 5c-triangulations, and of rooted 5-connected planar triangulations with root-vertex degree 5, these belonging to the same algebraic extension as the generating function of rooted 5-connected planar triangulations computed by Gao, Wanless and Wormald. The bijection also makes it possible to obtain efficient uniform random generation and succinct encoding procedures for 5-connected planar triangulations.

Paper Structure

This paper contains 15 sections, 12 theorems, 17 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Definitions on maps and orientations
  3. Quadrangular tree-biorientations and bicolored plane trees
  4. The general bijection
  5. Two lemmas giving partial control on black vertex degrees
  6. Application to 5c-triangulations
  7. Tree-biorientations for 5c-triangulations
  8. Specialization of $\Phi$ to 5c-biorientations
  9. Counting results
  10. Generating function of 5c-triangulations
  11. Generating function of 5-connected triangulations with root-vertex degree $5$
  12. Algorithmic applications and concluding remarks
  13. Random generation of 5-connected triangulations
  14. Optimal encoding of 5-connected triangulations
  15. Substitution approach and open questions on bijective extensions

Key Result

Theorem 1.1

There is an explicit bijection $\phi$ between 5c-triangulations with $n\!+\!5$ vertices and leg-balanced 5-regular plane trees with $n$ nodes, for $n\geq 1$.

Figures (7)

  • Figure 1: Left: a 5c-triangulation. Right: the corresponding leg-balanced 5-regular plane tree.
  • Figure 2: Left: a biorientation $O\in\mathcal{O}_4$. Right: the corresponding bimobile $T=\Phi_-(O)$.
  • Figure 3: Top-left: a tree-biorientation $Y\in\mathcal{Q}_3$. Bottom-left: the corresponding bicolored plane tree $T=\Phi(Y)\in\mathcal{T}_3$, which can also be obtained (right column) from the bimobile $\hat{T}=\Phi_-(Y)$ upon deleting the 3 legs at every square vertex, and turning the unique incident inner edge into a leg.
  • Figure 4: The closure mapping $\psi$, from a bicolored plane tree $T\in\mathcal{T}_3$ to a tree-oriented quadrangular dissection $Y\in\mathcal{Q}_3$.
  • Figure 5: (a) The 5c-triangulation $M$ of Figure \ref{['fig:example_main_bijection']}, (b) superimposed with $Q=\iota(M)$; (c) $Q$ is endowed with its minimal $b_5$-rooted regular orientation $X_0$, with the left co-accessibility spanning tree $\tau_0$ colored red; (d) making all edges of $\tau_0$ 2-way and reversing the inner edge incident to $b_5$ yields the 5c-biorientation $Y_0$ of $Q$; (e) the tree $T=\Phi(Y_0)$; (f) the leg-balanced 5-regular plane tree $\tau=\phi(M)$ is obtained as the reduction of $T$.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 4.1
  • ...and 18 more