On Hamiltonian cycles and Hamiltonian paths
Information Processing Letters
Graph Theory With Applications
Graph Theory With Applications
An improved degree based condition for Hamiltonian cycles
Information Processing Letters
A sufficient condition for pancyclic graphs
Information Processing Letters
Note: A comprehensive analysis of degree based condition for Hamiltonian cycles
Theoretical Computer Science
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Rahman and Kaykobad proved the following theorem on Hamiltonian paths in graphs. Let G be a connected graph with n vertices. If d(u)+d(v)+δ(u, v) ≥ n+1 for each pair of distinct non-adjacent vertices u and v in G, where δ(u, v) is the length of a shortest path between u and v in G, then G has a Hamiltonian path. It is shown that except for two families of graphs a graph is Hamiltonian if it satisfies the condition in Rahman and Kaykobad's theorem. The result obtained in this note is also an answer for a question posed by Rahman and Kaykobad.