Abstract: 1. Let P be the union of two disjoint sets, R and B, having n and m points in the plane, respectively, where n and m are even. Let K(R) and K(B) be the complete Euclidean graphs over R and B, respectively. An edge pq of (K(R) U K(B)) is a red edge if p,q in R and blue if p,q in B. An l-monchromatic perfect matching, M, is a perfect matching of P such that (i) M \subset (K(R) U K(B)), (ii) the set of red edges and blue edges in M induce a perfect plane matching of R and B, respectively, (iii) Bottleneck(M) <= l. The l-monochromatic perfect matching crossing number of P, denoted by cr(P, l), is the minimum number of crossings in an l-monochromatic perfect matching of P. We give asymptotically tight bounds on cr(P, l) for different values of l. 2. Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. We show that computing a non-crossing bichromatic spanning tree T, such that the length of the longest edge in T is minimized is NP-hard for points in general position.