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arXiv · 1807.01022

On the number of coloured triangulations of $d$-manifolds

Guillaume Chapuy·Guillem Perarnau
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Abstract

We give superexponential lower and upper bounds on the number of coloured $d$-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and $d\geq 3$ is fixed. In the special case of dimension $3$, the lower and upper bounds match up to exponential factors, and we show that there are $2^{O(n)} n^{\frac{n}{6}}$ coloured triangulations of $3$-manifolds with $n$ tetrahedra. Our results also imply that random coloured triangulations of $3$-manifolds have a sublinear number of vertices. Our upper bounds apply in particular to coloured $d$-spheres for which they seem to be the best known bounds in any dimension $d\geq 3$, even though it is often conjectured that exponential bounds hold in this case. We also ask a related question on regular edge-coloured graphs having the property that each $3$-coloured component is planar, which is of independent interest.

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BibTeXRIS

Guillaume Chapuy, Guillem Perarnau. 2020-02-13. On the number of coloured triangulations of $d$-manifolds. https://doi.org/10.1007/s00454-020-00189-w

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