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- W2981812780 abstract "A combinatorial map is the embedding of a graph on a surface (orientable or not), considered up to deformation. We describe a bijective method, called opening, that allows to reduce a map into a smaller map on the same surface, with only one face, along with some additional decorations called blossoms. This construction generalizes the opening described in the case of planar maps in [Sch97].Several papers from the 90's used advanced calculation methods to obtain properties on the generating series of maps on a given surface. In particular, in case the surface is orientable, this series can be written as a rational function of the generating series of some trees. This is valid both in case the maps are enumerated by their number of edges only [BenCan91], by both their number of vertices and faces [BenCanRic93]. A similar weaker result was also obtained in the case of non-orientable surfaces [AG00]. Actually, these rationality properties concerning the generating series of maps imply strong structural properties concerning the maps themselves, and providing a combinatorial interpretation of these properties has been an important motivation in the development of the bijective combinatorics of maps.The opening algorithm that we describe produces a map that can be further successively decomposed into smaller maps along with additional decorations. A deep analysis of the maps obtained this way, and their generating series, then allows to recover in a combinatorial way the rationality results described earlier.A k-triangulation of a finite polygon is a set of diagonals, maximal for the set-inclusion, such that no k+1 of its diagonals are pairwise crossing. A k-star is a set of 2k+1 points and 2k+1 diagonals such that each point is adjacent to its two opposite points. The work of [PilSan07] showed that a k-triangulation can be decomposed into a complex k-stars, and that multitriangulations can be obtained one from another by a succession of local elementary operations called flips.Our purpose is to extend these results to the case of multitriangulations on any surface. In this regard, we first study a class of multitriangulations of a polygon with an infinite number of sides, and extend to this context the main results of [PilSan07]. Using the classical construction of the universal cover of a surface, we then hope to reduce the case of a multitriangulations in any surface to that of a periodic multitriangulation of an infinite polygon. We present some element of such a proof, along with some conjectures that would allow to conclude." @default.
- W2981812780 created "2019-11-01" @default.
- W2981812780 creator A5046826935 @default.
- W2981812780 date "2019-09-24" @default.
- W2981812780 modified "2023-09-23" @default.
- W2981812780 title "Blossoming bijections, multitriangulations : What about other surfaces?" @default.
- W2981812780 hasPublicationYear "2019" @default.
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