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In this paper we consider graphs which have no k vertex-disjoint cycles. For given integers k, let f (k,) be the maximum order of a graph G with independence number (G) , which has no k vertex-disjoint cycles. We prove that f (k,) = 3k + 2 − 3 if 1 5 or 1 k 2, and f (k,) 3k + 2 − 3 in general. We also prove the following results: (1) there exists a constant(More)
Let k ≥ 2 be an integer. We show that if G is a (k + 1)-connected graph and each pair of nonadjacent vertices in G has degree sum at least |G| + 1, then for each subset S of V (G) with |S| = k, G has a spanning tree such that S is the set of endvertices. This result generalizes Ore's theorem which guarantees the existence of a Hamilton path connecting any(More)