Kneser colorings of polyhedra
It is well known that a 1-dimensional simplicial complex \(K^ 1\) (a graph) imbedded in a 2-dimensional manifold \(M^ 2\) has chromatic number bounded above by a constant c which depends only on the topology of \(M^ 2\). The author has generalized this result to higher dimensional simplicial complexes \(K^ n\). [See, for example Ill. J. Math. 25, 464- 469 (1981; Zbl 0488.05034)]. In the present paper the author reverses that viewpoint, by showing that if \(K^ n\) can be suitably colored by not too many colors, then it piecewise linearly imbeds in a given \({\mathbb{R}}^ m\). Here is a sample result. Let \(G(X^ n)\) denote the graph whose vertices \(X^ n_ i\) are closures of the non-singular edge-less components of the underlying polyhedron \(X^ n\) of \(K^ n\), with \(X^ n_ i\) adjacent to \(X^ n_ j\) if and only if \(X^ n_ i\) is disjoint from \(X^ n_ j\). If \(G(X^ n)\) is bichromatic and \(n\neq 2\), then \(K^ n\) piecewise linearly imbeds in \({\mathbb{R}}^{2n}\).
embedding polyhedra into Euclidean spaces
scientific article; zbMATH DE number 4147110