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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A two-coloring inequality for Euclidean two-arrangements


Authors: Gustavus J. Simmons and John E. Wetzel
Journal: Proc. Amer. Math. Soc. 77 (1979), 124-127
MSC: Primary 51M20; Secondary 05C15
MathSciNet review: 539644
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Abstract: We prove that for any properly two-colored arrangement of lines in the Euclidean plane having, say, r red and g green regions with $ r \geqslant g$, the inequality

$\displaystyle r \leqslant 2g - 2 - \sum\limits_P {(\lambda (P) - 2)} $

holds, where for each point P of intersection of the lines, $ \lambda (P)$ is the number of lines of the arrangement that contain P. This strengthens recent results of Simmons and Grünbaum.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0539644-4
PII: S 0002-9939(1979)0539644-4
Keywords: Two-colorings, two-arrangements, arrangements of lines, arrangements of pseudolines
Article copyright: © Copyright 1979 American Mathematical Society