Abstract
Corner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs that can be skeleton of a corner polyhedron: as planar maps, they are the duals of some particular bipartite triangulations, which we call hereafter corner triangulations.In this paper we count corner polyhedral graphs by determining the generating function of the corner triangulations with respect to the number of vertices: we obtain an explicit rational expression for it in terms of the Catalan gen- erating function. We first show that this result can be derived using Tutte's classical compositional approach. Then, in order to explain the occurrence of the Catalan series we give a direct algebraic decomposition of corner triangu- lations: in particular we exhibit a family of almond triangulations that admit a recursive decomposition structurally equivalent to the decomposition of binary trees. Finally we sketch a direct bijection between binary trees and almond triangulations. Our combinatorial analysis yields a simpler alternative to the algorithm of Eppstein and Mumford for endowing a corner polyhedral graph with the cycle cover structure needed to realize it as a polyhedral graph.
Highlights
In the recent paper Eppstein and Mumford (2014), the authors introduced and characterized a new family of 3D polyhedra: a polyhedron P in R3 is a corner polyhedron if v0 = (0, 0, 0) is a vertex of P, and: the boundary of P has the topology of the 2-sphere, each edge of P is parallel to one of the coordinate axis, exactly 3 edges of P meet at each vertex, all vertices of P but s are visible from infinity in the direction (1, 1, 1)
The skeleton of P can be viewed as embedded on the boundary sphere of P, and up to such homeomorphisms it is a cubic planar map P, which we canonically consider as rooted on the edge (v0, v1)
In this paper we prove the following theorem: Theorem 2 The generating series Ec(z) of corner triangulations with respect to their number of black faces is algebraic of degree 2 over Q(t)
Summary
An Eulerian triangulation is a planar map with black and white triangular faces, such that adjacent faces have different colors It is rooted if one of its edges is marked. Let a corner quasitriangulation be a rooted planar map without bridges nor multiple edges, whose non-root faces are black or white triangles such that adjacent triangles have different colors, and all canonically oriented clockwise triangles are the boundaries of the white faces. In particular we prove: Theorem 3 The generating series of almond triangulations with respect to the number of black faces is the unique formal power series solution of the equation. We return to Theorem 3 to give a simple direct bijective correspondence between binary trees and almond triangulations in the style of the earlier bijective proofs given by the last two authors for many families of planar maps. This is an extended abstract, full detail will appear elsewhere
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