0 IntroductionA cycle (path) in a graph G is called a hamiltonian cycle (path) in G if it con-tains all the vertices of G.A graph is hamiltonian (traceable) if it has a hamiltoniancycle (path).The neighborhood of a vertex v,denoted N(v),is the set of all verticesadjacent to v.We define the distance between two vertices u and v,denoted dist(u,v),as the minmum of the lengths of all u-v paths.Let NC2=min|N(u)∪N(v)|,DC2=min{max{d(u),d(v)}},where the minimum is taken over all pairs of vertices u,vthat are at distance two in the graph.Let δrepresent the minimum degree of G.Referto [5] for other terminology.
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