Abstract
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 bipartite graph. In this paper, we study the existence of cycles of various lengths in a Hamiltonian graph G given the existence of a pair of vertices that have a high degree sum but are not adjacent on any Hamiltonian cycle in G.
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References
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Hakimi S.L., Schmeichel E.F.: A cycle structure theorem for Hamiltonian graphs. J. Combin. Theory Ser. B 45, 99–107 (1988)
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Ferrara, M., Jacobson, M.S. & Harris, A. Cycle Lengths in Hamiltonian Graphs with a Pair of Vertices Having Large Degree Sum. Graphs and Combinatorics 26, 215–223 (2010). https://doi.org/10.1007/s00373-010-0915-z
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DOI: https://doi.org/10.1007/s00373-010-0915-z