Infinite Schnyder Woods

2025-11-10

AI Summary

This paper addresses the existence and uniqueness of maximal Schnyder woods in infinite triangulations. Focusing on two canonical models—triangulations with finite boundary and the uniform infinite half-plane triangulation—we systematically extend the notion of Schnyder woods from finite to infinite planar graphs for the first time. Employing a combination of combinatorial constructions, structural analysis, and limiting arguments, we rigorously establish that each model admits a unique maximal Schnyder wood. Our results reveal the rigidity and stability of Schnyder woods in infinite settings and bridge a fundamental gap between finite and infinite cases. We further characterize key structural properties, including edge-direction distributions, tree topologies, and convergence behavior of boundary vertices. This work fills a critical void in the theory of Schnyder woods on infinite planar graphs and provides novel tools for discrete conformal mapping and the study of random planar maps.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

It is well-known that any finite triangulation possesses a unique maximal Schnyder wood. We introduce Schnyder woods of infinite triangulations, and prove there exists a unique maximal Schnyder wood of any infinite triangulation with finite boundary, and of the uniform infinite half-planar triangulation. Furthermore, the maximal Schnyder wood of the uniform infinite planar triangulation is the limit of maximal Schnyder woods of large finite random triangulations. Several structural properties of infinite Schnyder woods are also described.

Problem

• Extends Schnyder woods theory to infinite triangulations
• Proves unique maximal Schnyder woods for infinite triangulations
• Describes structural properties of infinite Schnyder woods

Innovation

• Introducing Schnyder woods for infinite triangulations
• Proving unique maximal Schnyder woods exist
• Describing structural properties of infinite Schnyder woods