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We consider the following problem. Let n ≥ 2, b ≥ 1 and q ≥ 2 be integers. Let R and B be two disjoint sets of n red points and bn blue points in the plane, respectively, such that no three points of R∪B lie on the same line. Let n = n1 + n2 + · · · + nq be an integer-partition of n such that 1 ≤ ni for every 1 ≤ i ≤ q. Then we want to partition R ∪ B into… (More)

The tree partition number of an r-edge-colored graph G, denoted by t r (G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. We determine t 2 (K(n 1 , n 2 ,. .. , n k)) of the complete k-partite graph K(n 1 , n 2 ,. .. , n k). In… (More)

An r-edge-coloring of a graph is an assignment of r colors to the edges of the graph. An exactly r-edge-coloring of a graph is an r-edge-coloring of the graph that uses all r colors. A matching of an edge-colored graph is called rainbow matching, if no two edges have the same color in the matching. In this paper, we prove that an exactly r-edge-colored… (More)