| Abstrakt: |
Asratian and Khachatrian proved that a connected graphGof order at least 3 is hamiltonian ifd(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)|for any pathuwvwithuv ∉ E(G), whereN(x)is the neighborhood of a vertexx. We prove that a graphGwith this condition, which is not complete bipartite, has the following properties:a)For each pair of verticesx, ywith distanced(x, y)≥ 3 and for each integern, d(x, y) ≤ n ≤ |V(G)|− 1, there is anx − ypath of lengthn.(b)For each edgeewhich does not lie on a triangle and for eachn, 4 ≤ n ≤ |V(G)|, there is a cycle of lengthncontaininge.(c)Each vertex ofGlies on a cycle of every length from 4 to |V(G)|. This implies thatGis vertex pancyclic if and only if each vertex ofGlies on a triangle. |
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