K_1,3-covering red and blue points in the plane
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We say that a finite set of red and blue points in the plane in general position can be K_1,3-covered if the set can be partitioned into subsets of size 4, with 3 points of one color and 1 point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set R of r red points and a set B of b blue points in the plane in general position, how many points of R∪ B can be K_1,3-covered? and we prove the following results: (1) If r=3g+h and b=3h+g, for some non-negative integers g and h, then there are point sets R∪ B, like {1,3}-equitable sets (i.e., r=3b or b=3r) and linearly separable sets, that can be K_1,3-covered. (2) If r=3g+h, b=3h+g and the points in R∪ B are in convex position, then at least r+b-4 points can be K_1,3-covered, and this bound is tight. (3) There are arbitrarily large point sets R∪ B in general position, with |R|=|B|+1, such that at most r+b-5 points can be K_1,3-covered. (4) If b< r< 3b, then at least 8/9(r+b-8) points of R∪ B can be K_1,3-covered. For r>3b, there are too many red points and at least r-3b of them will remain uncovered in any K_1,3-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.
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