[Paper Review] Maximizing the number of edges in three-dimensional colored triangulations whose building blocks are balls

Valentin Bonzom|arXiv (Cornell University)|Feb 18, 2018

Advanced Combinatorial Mathematics57 references11 citations

TL;DR

This paper solves the problem of maximizing edges in 3D colored triangulations where all but at most one building block are 3-balls. It proves that such edge-maximizing triangulations are homeomorphic to the 3-sphere and correspond bijectively to trees in the dual 1-skeleton, with edge counts computed as independent sums over building blocks. The key result is a combinatorial characterization of ball-like blocks via 2-edge-cuts in the dual graph.

ABSTRACT

Colored triangulations offer a generalization of combinatorial maps to higher dimensions. Just like maps are gluings of polygons, colored triangulations are built as gluings of special, higher-dimensional building blocks, such as octahedra, which we call colored building blocks and known in the dual as bubbles. A colored building block is determined by its boundary triangulation, which in the case of polygons is simply characterized by its length. In three dimensions, colored building blocks are labeled by some 2D triangulations and those homeomorphic to the 3-ball are labeled by the subset of planar ones. Similarly to Euler's formula in 2D which provides an upper bound to the number of vertices at fixed number of polygons with given lengths, we look in three dimensions for an upper bound on the number of edges at fixed number of given colored building blocks. In this article we solve this problem when all colored building blocks, except possibly one, are homeomorphic to the 3-ball. To do this, we find a characterization of the way a colored building block homeomorphic to the ball has to be glued to other blocks of arbitrary topology in a colored triangulation which maximizes the number of edges. This local characterization can be extended to the whole triangulation as long as there is at most one colored building block which is not a 3-ball. The triangulations obtained this way are in bijection with trees. The number of edges is given as an independent sum over the building blocks of such a triangulation. In the case of all colored building blocks being homeomorphic to the 3-ball, we show that these triangulations are homeomorphic to the 3-sphere. Those results were only known for the octahedron and for melonic building blocks before. This article is self-contained and can be used as an introduction to colored triangulations and their bubbles from a purely combinatorial point of view.

Motivation & Objective

• To determine the maximum number of edges possible in a 3-dimensional colored triangulation composed of colored building blocks, with all but possibly one homeomorphic to the 3-ball.
• To characterize the gluing structure of a 3-ball building block within such edge-maximizing triangulations.
• To establish a bijection between edge-maximizing triangulations and trees in the dual 1-skeleton.
• To show that when all building blocks are 3-balls, the resulting triangulation is homeomorphic to the 3-sphere.
• To generalize previous results known only for octahedra and melonic blocks to arbitrary planar triangulations as building blocks.

Proposed method

• The paper uses a dual representation of colored triangulations as edge-colored graphs, where 3-bubbles correspond to vertices and 2-edge-cuts to topological excisions of ball-like blocks.
• It applies a recursive argument based on edge-flips and 2-dipole contractions to reduce the graph while preserving topological type and maximizing edge count.
• The method relies on Euler’s formula applied to 3-bubbles (dual to vertices) to derive constraints on bicolored cycles and genera of components.
• It proves that edge-maximizing graphs must have all 3-bubbles planar (genus zero), using the fact that edge flips preserve the number of bicolored cycles under specific conditions.
• The key technical tool is a characterization of topological moves via 2-edge-cuts, showing that a ball-like block can be excised from the triangulation if and only if it is glued via such cuts.
• The construction is extended to show that the entire triangulation is a sphere when all blocks are balls, via a sequence of topological moves from an initial graph to a known spherical graph.

Experimental results

Research questions

• RQ1 What is the maximum number of edges achievable in a 3D colored triangulation when all but one building block are homeomorphic to the 3-ball?
• RQ2 How must a 3-ball building block be glued to other blocks to maximize the number of edges in the triangulation?
• RQ3 Can the edge-maximizing property be characterized combinatorially in terms of the dual 1-skeleton and 2-edge-cuts?
• RQ4 Under what conditions is the entire triangulation homeomorphic to the 3-sphere?
• RQ5 Is there a bijection between edge-maximizing triangulations and combinatorial trees in the dual graph?

Key findings

• The number of edges in an edge-maximizing triangulation is given by an independent sum over the building blocks, with contributions determined by their internal structure and boundary triangulations.
• A building block homeomorphic to the 3-ball must be glued such that it can be excised via a sequence of 2-edge-cuts in the dual 1-skeleton.
• When all building blocks are 3-balls, the resulting triangulation is homeomorphic to the 3-sphere.
• The edge-maximizing triangulations are in bijection with trees, where each node corresponds to a building block and each edge to a 2-edge-cut connection.
• The 3-bubbles (dual to vertices) in such triangulations are all planar, meaning their genera are zero, which is a necessary condition for edge maximality.
• The construction generalizes previous results for octahedra and melonic blocks to arbitrary planar triangulations as building blocks.