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On the number of coloured triangulations of -manifolds
We give superexponential lower and upper bounds on the number of coloured
-dimensional triangulations whose underlying space is an oriented manifold,
when the number of simplices goes to infinity and is fixed. In the
special case of dimension , the lower and upper bounds match up to
exponential factors, and we show that there are
coloured triangulations of -manifolds with tetrahedra. Our results also
imply that random coloured triangulations of -manifolds have a sublinear
number of vertices. Our upper bounds apply in particular to coloured
-spheres for which they seem to be the best known bounds in any dimension
, 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 -coloured component is planar, which is of independent
interest.Comment: 15 pages. New version, proof of the lower bound correcte
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