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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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