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On the number of coloured triangulations of $d$-manifolds. (arXiv:1807.01022v1 [math.CO])
来源于:arXiv
We give superexponential lower and upper bounds on the number of coloured
$d$-dimensional triangulations whose underlying space is a 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 $2^{\Theta(n)} n^{\frac{n}{6}}$ 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. 查看全文>>