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Given n red and n blue points in the plane and a planar straight line matching between the red and the blue points, the matching can be extended into a bipartite planar straight line spanning tree. That is, any red-blue planar matching can be completed into a crossing-free red-blue spanning tree. Such a tree can be constructed in O(n log n) time.

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)

Consider a planar straight line graph (PSLG), G, with k connected components, k 2. We show that if no component is a singleton, we can always find a vertex in one component that sees an entire edge in another component. This implies that when the vertices of G are colored, so that adjacent vertices have different colors, then (1) we can augment G with k − 1… (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)

Nowadays the term monochromatic and heterochromatic (or rainbow , multicolored) subgraphs of an edge colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochro-matic subgraphs, such… (More)

A {1, 3, · · · , 2n − 1}-factor of a graph G is defined to be a spanning subgraph of G, each degree of whose vertices is one of {1, 3, · · · , 2n − 1}, where n is a positive integer. In this paper, we give a sufficient condition for a graph to have a {1, 3, · · · , 2n − 1}-factor.