New bijective links on planar maps

Fusy, Eric

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008).

Résumé

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.

Détail

DOI : 10.46298/dmtcs.3628