Abstract
Let P be a set of points in 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 .
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