New bijective links on planar maps via orientations

Eric Fusy (1)

1 ALGORITHMS - Algorithms

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.

Mots clés: bijections; planar maps; bipolar orientations

Domaines: Combinatoire [math.CO]

Eric Fusy. New bijective links on planar maps via orientations. 2009. ⟨hal-00330577v2⟩