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A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99–107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n − 1 or G is isomorphic to a complete...
For a graph G, let σ2(G) denote the minimum degree sum of two nonadjacent vertices (when G is complete, we let σ2(G)=∞). In this paper, we show the following two results: (i) Let G be a graph of order n≥4k+3 with σ2(G)≥n and let F be a matching of size k in G such that G−F is 2-connected. Then G−F is hamiltonian or G≅K2+(K2∪Kn−4) or G≅K2¯+(K2∪Kn−4); (ii) Let G be a graph of order n≥16k+1 with σ2(G)≥n...
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