科研之友
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摘要
Let G be a graph, and 6 (G) and a (G) be the minimum degree and the independence number of G, respectively. For a vertex v is an element of V(G), d(v) and N(v) represent the degree of v and the neighborhood of v in G, respectively. A number of sufficient conditions for a connected simple graph G of order n to be Hamiltonian have been proved. Among them are the well known Dirac condition (1952) (delta(G) >= n/2) and Ore condition (1960) (for any pair of independent vertices u and v, d(u) + d(v) >= n). In 1984 Fan generalized these two conditions and proved that if G is a 2-connected graph of order n and max{d(x), d(y)} n/2 for each pair of nonadjacent vertices x, y with distance 2 in G, then G is Hamiltonian. In 1993, Chen proved that if G is a 2-connected graph of order n, and if max{d(x), d(y)} : n/2 for each pair of nonadjacent vertices x, y with 1 <= vertical bar N(x) boolean AND N(y)vertical bar <= alpha(G) - 1, then G is Hamiltonian. In 1996, Chen, Egawa, Liu and Saito further showed that if G is a k-connected graph of order n, and if max{d(v) : v is an element of S} >= n/2 for every independent set S of G with vertical bar S vertical bar = k which has two distinct vertices x, y is an element of S such that the distance between x and y is 2, then G is Hamiltonian. In this paper, we generalize all the above conditions and prove that if G is a k-connected graph of order n, and if max{d(v) : v is an element of S} >= n/2 for every independent set S of G with vertical bar S vertical bar = k which has two distinct vertices x, y is an element of S satisfying 1 <= vertical bar N(x) boolean AND N (y)vertical bar <= alpha (G) - 1, then G is Hamiltonian.
