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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)
Let G be a graph. The core of G, denoted by G ∆ , is the subgraph of G induced by the vertices of degree ∆(G), where ∆(G) denotes the maximum degree of G. A k-edge coloring of G is a function f : E(G) → L such that |L| = k and f (e 1) = f (e 2) for all two adjacent edges e 1 and e 2 of G. The chromatic index of G, denoted by χ (G), is the minimum number k(More)
Let P 1 , ..., P k 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 T i of P i such that the edges of T 1 , ..., T k intersect at most (k − 1)(n − k) + (k)(k−1) 2 , where n is the number of points in P 1 ∪ ... ∪ P k. If the intersection of the convex hulls of P 1 , ..., P k(More)
Let X be a set of multicolored points in the plane such that no three points are collinear and each color appears on at most ⌈|X|/2⌉ points. We show the existence of a non-crossing properly colored geometric perfect matching on X (if |X| is even), and the existence of a non-crossing properly colored geometric spanning tree with maximum degree at most 3 on(More)