On spanning trees and cycles of multicolored point sets with few intersections
نویسندگان
چکیده
Let P1, ..., Pk be a collection of disjoint point sets in < 2 in general position. We prove that for each 1 ≤ i ≤ k we can find a plane spanning tree Ti of Pi such that the edges of T1, ..., Tk intersect at most (k − 1)(n− k) + (k)(k−1) 2 , where n is the number of points in P1 ∪ ...∪Pk. If the intersection of the convex hulls of P1, ..., Pk is non empty, we can find k spanning cycles such that their edges intersect at most (k − 1)n times, this bound is tight. We also prove that if P and Q are disjoint point sets in general position, then the minimum weight spanning trees of P and Q intersect at most 8n times, where |P ∪ Q| = n (the weight of an edge is its length).
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