Maximizing the number of edges in three-dimensional colored triangulations whose building blocks are balls
Résumé
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.