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On the length of longest alternating paths for multicoloured point sets in convex position. (English) Zbl 1101.05033
Summary: Let $$P$$ be a set of points in $$\mathbb R^2$$ in general position such that each point is coloured with one of $$k$$ colours. An alternating path of $$P$$ is a simple polygonal whose edges are straight line segments joining pairs of elements of $$P$$ with different colours. In this paper we prove the following: Suppose that each colour class has cardinality $$s$$ and $$P$$ is the set of vertices of a convex polygon. Then $$P$$ always has an alternating path with at least $$(k-1)s$$ elements. Our bound is asymptotically sharp for odd values of $$k$$.

MSC:
 05C15 Coloring of graphs and hypergraphs 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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References:
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