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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.
We give a simple proof for Kaneko’s theorem which gives a su2cient and necessary condition for the existence of vertex disjoint paths in a graph, each of length at least two, that altogether cover all vertices of the original graph. Moreover we generalize this theorem and give a formula for the maximum number of vertices that can be covered by such a path(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)