On Neighbourly Triangulations
Karanbir S. Sarkaria · Transactions of the American Mathematical Society · 1983
A simplicial complex is called $d$-neighbourly if any $d + 1$ vertices determine a $d$-simplex. We give methods for constructing $1$-neighbourly triangulations of $3$- and $4$-manifolds; further we discuss some relationships between $d$-neighbourly triangulations, chromatic numbers and the problem of finding upper and lower bounds on the number of simplices and locating the zeros of the characteristic polynomial of a triangulation. A triangulation of an orientable manifold is called order-orientable if there exists some ordering of the vertices which orients the manifold. We give necessary conditions for their existence; also we construct such triangulations on $3$-dimensional handlebodies and discuss the problem of recognising finite monotone subsets of an affine space by using these ideas.