Bipartite embedding of trees in the plane
Discrete Applied Mathematics
Lower bounds on the number of crossing-free subgraphs of KN
Computational Geometry: Theory and Applications
Long alternating paths in bicolored point sets
GD'04 Proceedings of the 12th international conference on Graph Drawing
Hamiltonian orthogeodesic alternating paths
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
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We find arbitrarily large finite sets S of points in general position in the plane with the following property. If the points of S are equitably 2-colored (i.e., the sizes of the two color classes differ by at most one), then there is a polygonal line consisting of straight-line segments with endpoints in S , which is Hamiltonian, non-crossing, and alternating (i.e., each point of S is visited exactly once, every two non-consecutive segments are disjoint, and every segment connects points of different colors). We show that the above property holds for so-called double-chains with each of the two chains containing at least one fifth of all the points. Our proof is constructive and can be turned into a linear-time algorithm. On the other hand, we show that the above property does not hold for double-chains in which one of the chains contains at most ≈ 1/29 of all the points.