Unified bijections for planar hypermaps with general cycle-length constraints

Bernardi, Olivier; Fusy, Éric

Annales de l’Institut Henri Poincaré D, Tome 7 (2020) no. 1, pp. 75-164.

We present a general bijective approach to planar hypermaps with two main results. First we obtain unified bijections for classes of maps or hypermaps defined by face-degree constraints and girth constraints. To any such class we associate bijectively a class of plane trees characterized by local constraints. This unifies and greatly generalizes several bijections for maps and hypermaps. Second, we present yet another level of generalization of the bijective approach by considering classes of maps with non-uniform girth constraints. More precisely, we consider well-charged maps, which are maps with an assignment of charges (real numbers) to vertices and faces, with the constraints that the length of any cycle of the map is at least equal to the sum of the charges of the vertices and faces enclosed by the cycle. We obtain a bijection between charged hypermaps and a class of plane trees characterized by local constraints.

Classification : 05-XX, 00-XX
Mots-clés : Planar hypermaps, constellations, bijections, mobiles, girth, generating functions, Ising model