Rooted bicubic planar maps via Dyck paths

TL;DR

This paper establishes a combinatorial bijection between rooted bicubic planar maps and Dyck paths, providing a constructive framework for understanding their decomposition into 3-connected components.

Contribution

It introduces an explicit bijection linking bicubic planar maps to Dyck paths with colored ascents, enabling constructive assembly from 3-connected building blocks.

Findings

Bijection between rooted bicubic planar maps and Dyck paths with colored ascents.

Primitive maps can generate larger maps through two specific insertion operations.

Prism graphs can produce maps with up to 6n rootings for all n ≥ 11.

Abstract

We provide a combinatorial proof of Tutte's decomposition of rooted bicubic planar maps into 3-connected components. Motivated by the framework of Bell transformations, we establish an explicit bijection between rooted bicubic planar maps on $2n$ vertices and Dyck paths of semilength $3n$ with ascents of length divisible by 3, where each $3j$-ascent is colored using one of $g_j$ colors corresponding to the rooted 3-connected bicubic maps on $2j$ vertices. Our bijection gives a constructive method for assembling all rooted bicubic planar maps from their 3-connected building blocks. We give a simple proof for the fact that every 3-connected bicubic planar map on $2n$ vertices with $n \geq 4$ can be obtained from a smaller primitive map through just two insertion operations that add either 4 or 6 vertices. Finally, we briefly discuss rootings of 3-connected bicubic maps, providing lower bounds on the minimal number of rootings and showing that prism graphs can be used in combination with our insertion operations to generate maps with the maximum of $6n$ distinct rootings for all $n \geq 11$.