Abstract
We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different faces), we build a new plane map with a distinguished vertex and two distinguished half-edges directed toward the vertex. The faces of the new map have the same degree as those of the original map, except at the locations of the distinguished corners, where each receives an extra degree: this is the location of the distinguished half-edges. This bijection provides a sampling algorithm for uniform maps with prescribed face degrees and allows to recover Tutte's famous counting formula for bipartite and quasibipartite plane maps.
 In addition, we explain how to decompose the previous bijection into two more elementary ones, which each transfer a degree from one face of the map to another face. In particular, these transfer bijections are simpler to manipulate than the previous one and this point of view simplifies the proofs.
Highlights
This paper is the extended version of [Bet19] and a sequel to [Bet14], in which we presented two bijections on plane quadrangulations with a boundary
We show how to generalize these bijections to bipartite and, in some cases, quasibipartite plane maps
As the sum of the face degrees equals twice the number of edges, the number of faces with an odd degree must be even, so that quasibipartite maps are the simplest maps to consider after bipartite maps
Summary
This paper is the extended version of [Bet19] and a sequel to [Bet14], in which we presented two bijections on plane quadrangulations with a boundary. We show how to generalize these bijections to bipartite and, in some cases, quasibipartite plane maps. By a slight modification, we may transform a distinguished edge into an extra degree-2 face and use twice the bijections interpreting the following identities in order to transfer both corners of the extra face to the desired faces. In a very recent work, Louf [Lou19] introduced a new family of bijections accounting for formulas on plane maps arising from the so-called KP hierarchy His bijections strongly rely on the mechanism of sliding along a path but, in his case, the path is somehow local ( arbitrary long) as it is canonically defined from only one vertex using a depth-first search exploration of the map. The definitions of these sets depend on the section they appear in; they are always clearly defined (with helping pictographs) at the beginning of the section in question
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