An Infinite Series of Surfaces with Known 1-Chromatic Number
✍ Scribed by Vladimir P. Korzhik
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 295 KB
- Volume
- 72
- Category
- Article
- ISSN
- 0095-8956
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✦ Synopsis
The 1-chromatic number / 1 (S) of a surface S is the maximum chromatic number of all graphs which can be drawn on the surface so that each edge is crossed by no more than one other edge. It is proved that if 4n+3 is a prime number, n 0, then / 1 (N 8(2n+1) 2) =R(N 8(2n+1) 2) where R(S)=w 1 2 (9+-81&32E(S))x is Ringel's upper bound for / 1 (S), E(S) is the Euler characteristic of S, and N 8(2n+1) 2 is the nonorientable surface of genus 8(2n+1) 2 . By Dirichlet's theorem the arithmetic progression 4n+3, n=1, 2, 3, ..., contains an infinite number of prime integers. As a result the first known infinite series of surfaces with known 1-chromatic number is obtained. 1998 Academic Press / 1 (S) R(S) for S S 0 , ( 1 ) Article No. TB971792 80 0095-8956Â98 25.00