New bijective links on planar maps via orientations

TL;DR

This paper introduces new bijections between classes of planar maps and triangulations, providing combinatorial insights and alternative proofs for known enumerative equivalences.

Contribution

It establishes novel bijections linking bipolar orientations, irreducible triangulations, and loopless maps, enriching the combinatorial understanding of planar maps.

Findings

Bijection between bipolar orientations and transversal structures on triangulations

Specialization to bijection between non-separable maps and irreducible triangulations

Alternative proof of equinumerosity between loopless maps and triangulations

Abstract

This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific "transversal structures" on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yields in turn a bijection between rooted loopless maps and rooted triangulations, based on the observation that loopless maps and triangulations are decomposed in a similar way into components that are respectively non-separable maps and irreducible triangulations. This gives another bijective proof (after Wormald's construction published in 1980) of the fact that rooted loopless maps with $n$ edges are equinumerous to rooted triangulations with $n$ inner vertices.