It was shown (Kronk and Mitchen, 1973) that the set of vertices, edges and faces of any normal map on the sphere can be colored with seven colors. In this paper we solve a somewhat different problem: the set of edges and faces of any plane graph with A ~< 3 can be colored by six colors.
β¦ LIBER β¦
On Improving the Edge-Face Coloring Theorem
β Scribed by Daniel P. Sanders; Yue Zhao
- Publisher
- Springer Japan
- Year
- 2001
- Tongue
- English
- Weight
- 135 KB
- Volume
- 17
- Category
- Article
- ISSN
- 0911-0119
No coin nor oath required. For personal study only.
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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