A two-coloring inequality for Euclidean two-arrangements

Simmons, Gustavus J.; Wetzel, John E.

doi:10.1090/S0002-9939-1979-0539644-4

A two-coloring inequality for Euclidean two-arrangements
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by Gustavus J. Simmons and John E. Wetzel

Proc. Amer. Math. Soc. 77 (1979), 124-127

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

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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 \[ r \leqslant 2g - 2 - \sum \limits _P {(\lambda (P) - 2)} \] 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.

References

G. L. Alexanderson and John E. Wetzel, Dissections of a plane oval, Amer. Math. Monthly 84 (1977), no. 6, 442–449. MR 513837, DOI 10.2307/2321901
L. Fejes Toth, Research Problems: A Combinatorial Problem Concerning Oriented Lines in the Plane, Amer. Math. Monthly 82 (1975), no. 4, 387–389. MR 1537693, DOI 10.2307/2318414
B. Grünbaum, Two-coloring the faces of arrangements, Period. Math. Hungar. 11 (1980), no. 3, 181–185. MR 590382, DOI 10.1007/BF02026614
Ilona Palásti, The ratio of black and white polygons of a map generated by general straight lines, Period. Math. Hungar. 7 (1976), no. 2, 91–94. MR 438236, DOI 10.1007/BF02082683

A linear partitioning of the plane which extends the known solutions of four combinatorial problems

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G. J. Simmons, A maximal $2$-arrangement of sixteen lines in the projective plane, Period. Math. Hungar. 4 (1973), 21–23. MR 333941, DOI 10.1007/BF02018032
John E. Wetzel, Dissections of a simply-connected plane domain, Amer. Math. Monthly 85 (1978), no. 8, 660–661. MR 508229