Unified bijections for maps with prescribed degrees and girth

A bijection for plane graphs and its applications

“…These bijections typically associate a tree (decorated in a certain way) to a map with specific constraints (e.g., no loops, no multiple edges, a restriction on the face-degrees). In particular, a different bijection for outer-triangular plane graphs was given in [4], relying on the same canonical orientations as the ones used here, but not going through eulerian triangulations. The bijection in [4] is more precise than the one of Theorem 1.1 in the sense that the corresponding decorated trees, called mobiles, keep track of the face-degree distribution of the outer-triangular plane graphs.…”

“…In particular, a different bijection for outer-triangular plane graphs was given in [4], relying on the same canonical orientations as the ones used here, but not going through eulerian triangulations. The bijection in [4] is more precise than the one of Theorem 1.1 in the sense that the corresponding decorated trees, called mobiles, keep track of the face-degree distribution of the outer-triangular plane graphs. However, the price to pay is that the mobiles in [4] are significantly more complicated than the oriented binary trees appearing in Theorem 1.1.…”

“…The bijection in [4] is more precise than the one of Theorem 1.1 in the sense that the corresponding decorated trees, called mobiles, keep track of the face-degree distribution of the outer-triangular plane graphs. However, the price to pay is that the mobiles in [4] are significantly more complicated than the oriented binary trees appearing in Theorem 1.1. It suggests that, when forgetting the precise face-degrees (and recording just the number of edges and faces), the mobiles in [4] should simplify into oriented binary trees.…”

“…However, the price to pay is that the mobiles in [4] are significantly more complicated than the oriented binary trees appearing in Theorem 1.1. It suggests that, when forgetting the precise face-degrees (and recording just the number of edges and faces), the mobiles in [4] should simplify into oriented binary trees. Such a simplification does not seem easy to define on tree-structures.…”

“…The bijection is illustrated in Figure 1 and described in Section 2. Both steps of the bijection rely crucially on the existence of certain canonical orientations, introduced by Bernardi and Fusy, in [4] for outer-triangular plane graphs and in [5] for eulerian triangulations. In the first step of the bijection, the canonical orientation is used to define some local operations which transform the outertriangular plane graph into an eulerian triangulation; Figure 2: The plot of b(µ), which is the growth rate of the expected number of embeddings of random planar graphs with n vertices and ⌊µn⌋ edges.…”