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.
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