Let G be a k-connected graph, and T be a subset of V（G）. If G-T is not connected,then T is said to be a cut-set of G. A k-cut-set T of G is a cut-set of G with │T│=k. Let T bea k-cut-set of a k-connected graph G. If G - T can be partitioned into subgraphs G1 and G2such that │G1│≥ 2, │G2│ 〉 2, then we call T a nontrivial k-cut-set of G. Suppose that G is a（k-1）-connected graph without nontrivial （k - 1）-cut-set. Then we call G a quasi k-connectedgraph. In this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph withoutK4-, then every vertex of G is incident with an edge whose contraction yields a quasi k-connectedgraph, and so there are at least │V（G）│/2 edges of G such that the contraction of every member ofthem results in a quasi k-connected graph.
A graph G has the hourglass property if every induced hourglass S（a tree with a degree sequence 22224） contains two non-adjacent vertices which have a common neighbor in G-V（S）.For an integer k≥4,a graph G has the single k-cycle property if every edge of G,which does not lie in a triangle,lies in a cycle C of order at most k such that C has at least「｜V（C） /2」 edges which do not lie in a triangle,and they are not adjacent.In this paper,we show that every hourglass-free claw-free graph G of δ（G） ≥3 with the single 7-cycle property is Hamiltonian and is best possible;we also show that every claw-free graph G of δ（G） ≥3 with the hourglass property and with single 6-cycle property is Hamiltonian.
