Long paths with endpoints in given vertex-subsets of graphs Original Research Article

Let image be a connected graph of order n, t a real number with image and image with image. In this paper, we study the problem of some long paths to maintain their one or two different endpoints in M. We obtain the following two results: (1) for any vertex image, there exists a vertex image and a path P with the two endpoints v and u to satisfy image, image, image; (2) there exists either a cycle C to cover all vertices of M or a path P with two different endpoints image and image in M to satisfy image, where image.