Planar triangulations, bridgeless planar maps and Tamari intervals

TL;DR

This paper introduces a direct bijection between planar 3-connected triangulations and bridgeless planar maps using a new combinatorial object called 'sticky trees', unifying various enumerated structures in planar graph theory.

Contribution

It presents the first direct, non-recursive bijection linking these classes of planar maps and triangulations via sticky trees, connecting multiple combinatorial objects.

Findings

Established a bijection between planar 3-connected triangulations and bridgeless planar maps.

Linked sticky trees to Tamari lattice intervals and closed flows on forests.

Reproduced known enumeration results through new bijective methods.

Abstract

We present a direct bijection between planar 3-connected triangulations and bridgeless planar maps, which were first enumerated by Tutte (1962) and Walsh and Lehman (1975) respectively. Previously known bijections by Wormald (1980) and Fusy (2010) are all defined recursively. Our direct bijection passes by a new class of combinatorial objects called "sticky trees". We also present bijections between sticky trees, intervals in the Tamari lattices and closed flows on forests. With our bijections, we recover several known enumerative results about these objects. We thus show that sticky trees can serve as a nexus of bijective links among all these equi-enumerated objects.