Book contents
- Frontmatter
- Contents
- Preface
- Farewell to Paul Erdős
- Toast to Paul Erdős
- List of Contributors
- Paul Erdős: Some Unsolved Problems
- Menger's Theorem for a Countable Source Set
- On Extremal Set Partitions in Cartesian Product Spaces
- Matchings in Lattice Graphs and Hamming Graphs
- Reconstructing a Graph from its Neighborhood Lists
- Threshold Functions for H-factors
- A Rate for the Erdős–Turán Law
- Deterministic Graph Games and a Probabilistic Intuition
- On Oriented Embedding of the Binary Tree into the Hypercube
- Potential Theory on Distance-Regular Graphs
- On the Length of the Longest Increasing Subsequence in a Random Permutation
- On Richardson's Model on the Hypercube
- Random Permutations: Some Group-Theoretic Aspects
- Ramsey Problems with Bounded Degree Spread
- Hamilton Cycles in Random Regular Digraphs
- On Triangle Contact Graphs
- A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies
- Lattice Points of Cut Cones
- The Growth of Infinite Graphs: Boundedness and Finite Spreading
- Amalgamated Factorizations of Complete Graphs
- Ramsey Size Linear Graphs
- Turán–Ramsey Theorems and Kp-Independence Numbers
- Nearly Equal Distances in the Plane
- Clique Partitions of Chordal Graphs
- On Intersecting Chains in Boolean Algebras
- On the Maximum Number of Triangles in Wheel-Free Graphs
- Blocking Sets in SQS(2v)
- (1,2)-Factorizations of General Eulerian Nearly Regular Graphs
- Oriented Hamilton Cycles in Oriented Graphs
- Minimization Problems for Infinite n-Connected Graphs
- On Universal Threshold Graphs
- Image Partition Regularity of Matrices
- Extremal Graph Problems for Graphs with a Color-Critical Vertex
- A Note on ω1 → ω1 Functions
- Topological Cliques in Graphs
- Local-Global Phenomena in Graphs
- On Random Generation of the Symmetric Group
- On Vertex-Edge-Critically n-Connected Graphs
- On a Conjecture of Erdős and Čudakov
- A Random Recolouring Method for Graphs and Hypergraphs
- Obstructions for the Disk and the Cylinder Embedding Extension Problems
- A Ramsey-Type Theorem in the Plane
- The Enumeration of Self-Avoiding Walks and Domains on a Lattice
- An Extension of Foster's Network Theorem
- Randomised Approximation in the Tutte Plane
- On Crossing Numbers, and some Unsolved Problems
On Triangle Contact Graphs
Published online by Cambridge University Press: 06 December 2010
- Frontmatter
- Contents
- Preface
- Farewell to Paul Erdős
- Toast to Paul Erdős
- List of Contributors
- Paul Erdős: Some Unsolved Problems
- Menger's Theorem for a Countable Source Set
- On Extremal Set Partitions in Cartesian Product Spaces
- Matchings in Lattice Graphs and Hamming Graphs
- Reconstructing a Graph from its Neighborhood Lists
- Threshold Functions for H-factors
- A Rate for the Erdős–Turán Law
- Deterministic Graph Games and a Probabilistic Intuition
- On Oriented Embedding of the Binary Tree into the Hypercube
- Potential Theory on Distance-Regular Graphs
- On the Length of the Longest Increasing Subsequence in a Random Permutation
- On Richardson's Model on the Hypercube
- Random Permutations: Some Group-Theoretic Aspects
- Ramsey Problems with Bounded Degree Spread
- Hamilton Cycles in Random Regular Digraphs
- On Triangle Contact Graphs
- A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies
- Lattice Points of Cut Cones
- The Growth of Infinite Graphs: Boundedness and Finite Spreading
- Amalgamated Factorizations of Complete Graphs
- Ramsey Size Linear Graphs
- Turán–Ramsey Theorems and Kp-Independence Numbers
- Nearly Equal Distances in the Plane
- Clique Partitions of Chordal Graphs
- On Intersecting Chains in Boolean Algebras
- On the Maximum Number of Triangles in Wheel-Free Graphs
- Blocking Sets in SQS(2v)
- (1,2)-Factorizations of General Eulerian Nearly Regular Graphs
- Oriented Hamilton Cycles in Oriented Graphs
- Minimization Problems for Infinite n-Connected Graphs
- On Universal Threshold Graphs
- Image Partition Regularity of Matrices
- Extremal Graph Problems for Graphs with a Color-Critical Vertex
- A Note on ω1 → ω1 Functions
- Topological Cliques in Graphs
- Local-Global Phenomena in Graphs
- On Random Generation of the Symmetric Group
- On Vertex-Edge-Critically n-Connected Graphs
- On a Conjecture of Erdős and Čudakov
- A Random Recolouring Method for Graphs and Hypergraphs
- Obstructions for the Disk and the Cylinder Embedding Extension Problems
- A Ramsey-Type Theorem in the Plane
- The Enumeration of Self-Avoiding Walks and Domains on a Lattice
- An Extension of Foster's Network Theorem
- Randomised Approximation in the Tutte Plane
- On Crossing Numbers, and some Unsolved Problems
Summary
It is proved that any plane graph may be represented by a triangle contact system, that is a collection of triangular disks which are disjoint except at contact points, each contact point being a node of exactly one triangle. Representations using contacts of T- or Y-shaped objects follow. Moreover, there is a one-to-one mapping between all the triangular contact representations of a maximal plane graph and all its partitions into three Schnyder trees.
Introduction: on graph drawing
An old problem of geometry consists of representing a simple plane graph G by means of a collection of disks in one-to-one correspondence with the vertices of G. These disks may only intersect pairwise in at most one point, the corresponding contacts representing the edges of G. The case of disks with no prescribed shape is solved by merely drawing for each vertex v a closed curve around v and cutting the edges half way. The difficulty arises when the disks have to be of a specified shape. The famous case of circular disks, solved by the Andreev–Thurston circle packing theorem [1], involves questions of numerical analysis: the coordinates of the centers and radii are not rational, and are computed by means of convergent series. This problem is still up to date, and considered in many research works. In the present paper we will consider triangular disks.
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- Combinatorics, Geometry and ProbabilityA Tribute to Paul Erdös, pp. 165 - 178Publisher: Cambridge University PressPrint publication year: 1997