Long alternating paths in bicolored point sets

Authors:
Jan Kynčl;János Pach;Géza Tóth
Affiliations:
Department of Applied Mathematics and, Institute for Theoretical Computer Science, Charles University, Prague, Czech Republic;City College, CUNY and Courant Institute of Mathematical Sciences, New York University, New York, NY;Rényi Institute, Hungarian Academy of Sciences, Budapest, Hungary
Venue:
GD'04 Proceedings of the 12th international conference on Graph Drawing
Year:
2004

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Abstract

Given n red and n blue points in convex position in the plane, we show that there exists a noncrossing alternating path of length $n+c\sqrt{n\over \log n}$. We disprove a conjecture of Erdős by constructing an example without any such path of length greater than ${4\over 3}n+c'\sqrt{n}$.