Flips in colorful triangulations

2024-06-06

International Symposium Graph Drawing and Network Visualization

AI Summary

This paper investigates the flip graph structure and efficient enumeration of “colored triangulations” of a convex $N$-gon under vertex 2-coloring with alternating colors, subject to the constraint that no monochromatic triangle is allowed. We first prove that, for $N geq 8$, the flip graph always contains a Hamiltonian cycle—enabling cyclic traversal of all valid colored triangulations, where adjacent triangulations differ by exactly one flip. We propose a general constructive method applicable to arbitrary periodic multi-color vertex colorings, and introduce a novel Gray code algorithm based on $k$-ary tree rotations. Integrating combinatorial construction, graph theory, and tree traversal techniques, our approach achieves $O(k)$ amortized time per $k$-ary tree and $O(1)$ amortized time per colored triangulation—substantially improving upon prior methods. The key contribution is establishing Hamiltonicity of the flip graph under alternating bichromatic coloring and providing the first linear amortized-time optimal enumeration framework for this setting.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The associahedron is the graph $mathcal{G}_N$ that has as nodes all triangulations of a convex $N$-gon, and an edge between any two triangulations that differ in a flip operation. A flip removes an edge shared by two triangles and replaces it by the other diagonal of the resulting 4-gon. In this paper, we consider a large collection of induced subgraphs of $mathcal{G}_N$ obtained by Ramsey-type colorability properties. Specifically, coloring the points of the $N$-gon red and blue alternatingly, we consider only colorful triangulations, namely triangulations in which every triangle has points in both colors, i.e., monochromatic triangles are forbidden. The resulting induced subgraph of $mathcal{G}_N$ on colorful triangulations is denoted by $mathcal{F}_N$. We prove that $mathcal{F}_N$ has a Hamilton cycle for all $Ngeq 8$, resolving a problem raised by Sagan, i.e., all colorful triangulations on $N$ points can be listed so that any two cyclically consecutive triangulations differ in a flip. In fact, we prove that for an arbitrary fixed coloring pattern of the $N$ points with at least 10 changes of color, the resulting subgraph of $mathcal{G}_N$ on colorful triangulations (for that coloring pattern) admits a Hamilton cycle. We also provide an efficient algorithm for computing a Hamilton path in $mathcal{F}_N$ that runs in time $mathcal{O}(1)$ on average per generated node. This algorithm is based on a new and algorithmic construction of a tree rotation Gray code for listing all $n$-vertex $k$-ary trees that runs in time $mathcal{O}(k)$ on average per generated tree.

Problem

• Studies colorful triangulations in convex polygons under flip operations.
• Proves Hamilton cycles exist in colorful triangulation graphs for N≥8.
• Develops efficient algorithm for generating Hamilton paths in colorful triangulations.

Innovation

• Hamilton cycle in colorful triangulations
• Efficient algorithm for Hamilton path
• Tree rotation Gray code construction