Programming Style: Examples and Counterexamples

The combinatorial structure of a d -dimensional simple convex polytope--as given, for example, by the set of the ( d 1)-regular subgraphs of the facets--can be reconstructed from its abstract graph. However, no polynomial/efficient algorithm is known ...

Denote by ${\mathrm{\Delta }}_M$ the M-dimensional simplex. A map $f:{\mathrm{\Delta }}_M\to {\mathbb{R}}^d$ is an almostr-embedding if $f({\sigma }_1)\cap \dots \cap f({\sigma }_r)=\mathrm{\varnothing }$ whenever ${\sigma }_1,\dots ,{\sigma }_r$ are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that ifris not a prime power and$d\ge 2r+1$, ...${\mathrm{}}_{\mathrm{}\mathrm{}}{\mathbb{}}^{}$$\mathrm{}{\displaystyle \frac{\mathrm{}}{\mathrm{}}}\mathrm{}$${\mathrm{}}_{}{\mathbb{}}^{}$

<P>Mani and Walkup (Mani, P., D. W. Walkup. 1980. A 3-sphere counterexample to the W_v-path conjecture. Math. Oper. Res.5 595–598.) described two triangulated 3-spheres which give rise to “small” counterexamples to the W_v-path and Hirsch conjectures for ...