Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better understanding of the connections between these objects. In this article we prove a characterization of 3-connected planar graphs as those graphs admitting rigid geodesic embeddings, a bijection between Schnyder labelings and rigid geodesic embeddings, a strong version of the Brightwell-Trotter theorem.

Abstract. A d-angulation is a planar map with faces of degree d. We present for each integer d ≥ 3 a bijection between the class of d-angulations of girth d and a class of decorated plane trees. Each of the bijections is obtained by specializing a “master bijection ” which extends an earlier construction of the first author. Bijections already existed for triangulations (d = 3) and for quadrangulations (d = 4). As a matter of fact, our construction unifies a bijection by Fusy, Poulalhon and Schaeffer for triangulations and a bijection by Schaeffer for quadrangulations. For d ≥ 5, both the bijections and the enumerative results are new. We also extend our bijections so as to enumerate p-gonal d-angulations, that is, d-angulations with a simple boundary of length p. We thereby recover bijectively the results of Brown for p-gonal triangulations and quadrangulations and establish new results for d ≥ 5. A key ingredient in our proofs is a class of orientations characterizing d-angulations of girth d. Earlier results by Schnyder and by De Fraisseyx and Ossona de Mendez showed that triangulations of girth 3 and quadrangulations of girth 4 are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a d-angulation has girth d if and only if the graph obtained by duplicating each edge d − 2 times admits an orientation having indegree d at each inner vertex. 1.

boundaries

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α: V → N unifies many different combinatorial structures, including the afore mentioned. We call these orientations α-orientations. The main focus of this paper are bounds for the maximum number of α-orientations that a planar map with n vertices can have, for different instances of α. We give examples of triangulations with 2.37 n Schnyder woods, 3-connected planar maps with 3.209 n Schnyder woods and inner triangulations with 2.91 n bipolar orientations. These lower bounds are accompanied by upper bounds of 3.56 n, 8 n and 3.97 n respectively. We also show that for any planar map M and any α the number of α-orientations is bounded from above by 3.73 n and describe a family of maps which have at least 2.598 n α-orientations.

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of an α-orientation unifies many different combinatorial structures, including the afore mentioned. We ask for the number of αorientations and also for special instances thereof, such as Schnyder woods and bipolar orientations. The main focus of this paper are bounds for the maximum number of such structures that a planar map with n vertices can have. We give examples of triangulations with 2.37 n Schnyder woods, 3-connected planar maps with 3.209 n Schnyder woods and inner triangulations with 2.91 n bipolar orientations. These lower bounds are accompanied by upper bounds of 3.56 n, 8 n and 3.97 n respectively. We also show that for any planar map M and any α the number of α-orientations is bounded from above by 3.73 n and present a family of maps which have at least 2.598 n α-orientations for n big enough. 1

Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how some recent schemes for compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here have linear time complexity.

Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or near-maximal planar graph. We briefly survey the connections and analogies between these three kinds of labelings, and their uses in designing efficient geometric algorithms.

Let G be a directed graph and k a positive integer. We define the k-color graph of G (Dk(G) for short) as the directed graph having all k-colorings of G as node set, and where two k-colorings β and ϕ are joined by a directed edge β → ϕ if ϕ is obtained from β by choosing a vertex v and recoloring v so that its color is different from the colors of all its out-neighbors. We investigate reachability questions in Dk(G). In particular we want to know whether a fixed legal k ′-coloring ψ of G with k ′ ≤ k is reachable in Dk(G) from every possible initial k-coloring β. Interesting instances of this problem arise when G is planar and the orientation is an arbitrary α-orientation for fixed α. Our main result is that reachability can be guaranteed if the orientation has maximal out-degree ≤ k − 1 and an accessible pseudo-sink. 1

Abstract. Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to d-angulations (plane graphs with faces of degree d) for all d ≥ 3. A Schnyder decomposition is a set of d spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly d − 2 of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the d-angulation is d. As in the case of Schnyder woods (d = 3), there are alternative formulationsintermsoforientations (“fractional ” orientations when d ≥ 5)and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed d-angulation of girth d is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on d-regular plane graphs of mincut d rooted at a vertex v ∗ ) are decompositions into d spanning trees rooted atv ∗ such that each edge not incidentto v ∗ isused in opposite directions by two trees. Additionally, for even values of d, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d = 4, these correspond to wellstudied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees). In the case d = 4, the dual of even Schnyder decompositions yields (planar) orthogonal and straight-line drawing algorithms. For a 4-regular plane graph G of mincut 4 with n vertices plus a marked vertex v, the vertices of G\v are placed on a (n−1)×(n−1) grid according to a permutation pattern, and in the orthogonal drawing each of the 2n−2 edges of G\v has exactly one bend. Embedding also the marked vertex v is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to v. We propose a further compaction step for the drawing algorithm and show that the obtained grid-size is strongly concentrated around 25n/32 ×25n/32 for a uniformly random instance with n vertices. 1.

This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two transversal bipolar orientations. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with n vertices, the grid size of the drawing is asymptotically with high probability 11n/27 × 11n/27 up to an additive error of O ( √ n). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size (⌈n/2 ⌉ − 1) × ⌊n/2⌋.