The distribution of the number of small cuts in a random planar triangulation

Zhicheng Gao,Gilles Schaeffer

doi:10.46298/dmtcs.2797

Abstract

We enumerate rooted 3-connected (2-connected) planar triangulations with respect to the vertices and 3-cuts (2-cuts). Consequently, we show that the distribution of the number of 3-cuts in a random rooted 3-connected planar triangulation with $n+3$ vertices is asymptotically normal with mean $(10/27)n$ and variance $(320/729)n$, and the distribution of the number of 2-cuts in a random 2-connected planar triangulation with $n+2$ vertices is asymptotically normal with mean $(8/27)n$ and variance $(152/729)n$. We also show that the distribution of the number of 3-connected components in a random 2-connected triangulation with $n+2$ vertices is asymptotically normal with mean $n/3$ and variance $\frac{8}{ 27}n$ .

Highlights

Throughout the paper, a triangulation is a connected graph G embedded in the plane with no edge crossings such that every face has degree 3
In this paper we studied the distribution of the number of 3-cuts (2-cuts) in a random rooted 3-connected (2-connected) triangulation
The approach used in this paper can be used to study the distribution of the number of k-cuts (1 ≤ k ≤ 3) in a random k-connected map

Summary

Introduction

Throughout the paper, a triangulation is a connected graph G embedded in the plane with no edge crossings such that every face has degree 3 (bounded by a triangle). A classical theorem of Whitney [17] says that every 4-connected planar triangulation contains a cycle through all the vertices (such a cycle is called a Hamilton cycle). It is known that the length of a longest cycle (path) in a 3-connected triangulation depends heavily on the number of 3-cuts [12, 14]. Another classical theorem of Whitney [18] states that every 3-connected planar graph has a unique embedding in the plane, and a 2-connected planar graph may have many embeddings in the plane using switchings at the 2-cuts. Minimum length cuts in planar maps were considered in the physics literature

Enumeration of triangulations with respect to small cuts

Proof of Theorem 1

Concluding remarks