## Abstract Let us call a finite subset __X__ of a Euclidean __m__βspace E^m^ __Ramsey__ if for any positive integer __r__ there is an integer __n__ = __n__(__X;r__) such that in any partition of E^n^ into __r__ classes __C__~1~,β¦, __C~r~__, some __C~i~__ contains a set __X__' which is the image of
A seven-color theorem on the sphere
β Scribed by Hudson V. Kronk; John Mitchem
- Publisher
- Elsevier Science
- Year
- 1973
- Tongue
- English
- Weight
- 731 KB
- Volume
- 5
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
β¦ Synopsis
Ringel has shown that the set of vertices and regions of any norm:4 map on the sphere can be admissibly colored by six colors. In this paper, it is shown that the set of vertices, edges and regions of any normd map on the sphere tit? be admissibly colored with seven colors.
π SIMILAR VOLUMES
Theorem 1. For any integers r, d>1 there exists an integer T=T(d, r), such that given sets A 1 , ..., A d+1 /R d in general position, consisting of T points each, one can find disjoint (d+1)-point sets S 1 , ..., S r such that each S i contains exactly one point of each A j , j=1, 2, ..., d+1, and t
## Abstract A major event in 1976 was the announcement that the Four Color Conjecture (4CC) had at long last become the Four Color Theorem (4CT). The proof by W. Haken, K. Appel, and J. Koch is published in the __Illinois Journal of Mathematics__, and their twoβpart article outlines the nature and
Lawrence [2, Theorem 3] and Borodin and Kostochka [1, Lemma 2' 1 both give the same theorem about vertex colorings of graphs (Corollary 1 below). But Lawrence's proof, although powerful, is a little long, and Borodin and Kostoehka state the result without a proof.
## Abstract In 1965 Ringel raised a 6 color problem for graphs that can be stated in at least three different forms. In particular, is it possible to color the vertices and faces of every plane graph with 6 colors so that any two adjacent or incident elements are colored differently? This 6 color p
Any graph imbedded on the Klein Bottle has a chromatic number of at most 6. Here we show that if the graph has no triangles, then its chromatic number is at most 4. The results for arbitrary girth are also included, 2s well as graphs which show all but one of these bounds to be best possible. These