Yoshimi Egawa

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Let G be a graph and n ≥ 2 an integer. We prove that the following are equivalent: (i) there is a partition (V1, . . . , Vm) of V (G) such that each Vi induces one of stars K1,1, . . . ,K1,n, and (ii) for every subset S of V (G), G\S has at most n|S| components with the property that each of their blocks is an odd order complete graph.
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(More)