New bijective links on planar maps via orientations

Eric Fusy (1)

1 ALGORITHMS - Algorithms

This article describes new bijective links on planar maps, which are of incremental complexity and present original features. The first two bijections $\Phi_{1,2}$ are correspondences on oriented planar maps. They can be considered as variations on the classical edge-poset construction for bipolar orientations on graphs, suitably adapted so as to operate only on the embeddings in a simple local way. In turn, $\Phi_{1,2}$ yield two new bijections $F_{1,2}$ between families of (rooted) maps. (i) By identifying maps with specific constrained orientations, $\Phi_2\circ\Phi_1$ specialises to a bijection $F_1$ between 2-connected maps and irreducible triangulations; (ii) $F_1$ gives rise to a bijection $F_2$ between loopless maps and triangulations, observing that these decompose respectively into 2-connected maps and into irreducible triangulations in a parallel way.

Mots clés: bijections; planar maps; bipolar orientations

Domaines: Combinatoire [math.CO]

Eric Fusy. New bijective links on planar maps via orientations. 2008. ⟨hal-00330577v1⟩