A model of trees for 5-connected planar triangulations

TL;DR

This paper introduces a bijection between 5-connected planar triangulations and 5-regular plane trees, enabling explicit enumeration, efficient random generation, and encoding of these complex structures.

Contribution

It establishes a novel bijection that links 5-connected planar triangulations to 5-regular plane trees, providing new enumeration formulas and algorithms.

Findings

Derived explicit generating functions for rooted 5c-triangulations.

Connected 5c-triangulations with 5-regular plane trees via bijection.

Enabled uniform random generation and encoding of 5-connected triangulations.

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.