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

DOI:
https://doi.org/10.1090/S0002-9939-1979-0539644-4

MathSciNet review:
539644

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Abstract | References | Similar Articles | Additional Information

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 , the inequality

*P*of intersection of the lines, is the number of lines of the arrangement that contain

*P*. This strengthens recent results of Simmons and Grünbaum.

**[1]**G. L. Alexanderson and John E. Wetzel,*Dissections of a plane oval*, Amer. Math. Monthly**84**(1977), 442-449. MR**0513837 (58:23976)****[2]**L. Fejes Tóth,*A combinatorial problem concerning oriented lines in the plane*, Amer. Math. Monthly**82**(1975), 387-389. MR**1537693****[3]**Branko Grünbaum,*Two-coloring the faces of arrangements*, Period. Math. Hungar. (to appear). MR**590382 (82h:51029)****[4]**Ilona Palásti,*The ratio of black and white polygons of a map generated by general straight lines*, Period. Math. Hungar.**7**(1976), 91-94. MR**0438236 (55:11154)****[5]**G. J. Simmons,*A linear partitioning of the plane which extends the known solutions of four combinatorial problems*, MAA Southwestern Section annual meeting, March 24-25, 1972, Albuquerque, New Mexico (title only), Amer. Math. Monthly**79**(1972), 824.**[6]**-,*A maximal*2-*arrangement of sixteen lines in the projective plane*, Period. Math. Hungar.**4**(1973), 21-23. MR**0333941 (48:12260)****[7]**John E. Wetzel,*Dissections of a simply-connected plane domain*, Amer. Math. Monthly**85**(1978), 660-661. MR**508229 (80d:54042)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0539644-4

Keywords:
Two-colorings,
two-arrangements,
arrangements of lines,
arrangements of pseudolines

Article copyright:
© Copyright 1979
American Mathematical Society