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A sufficient condition for Hamiltonian graphs. (English) Zbl 0838.05078
The following generalization of Ore’s sufficient condition for Hamiltonian graphs is proved. For a pair of non-adjacent vertices $$u$$ and $$v$$ of a simple graph $$G$$ let $$\omega(u, v)$$ be the number of components of the subgraph induced by the neighbours of $$u$$ that contain no neighbour of $$v$$. Let $$\pi(u, v)= \max\{\omega(u, v), \omega(v, u)\}$$. If in a graph $$G$$ of order $$n$$ any pair $$u$$, $$v$$ of non-adjacent vertices satisfies $$\text{deg}(u)+ \text{deg}(v)+ \pi(u, v)\geq n$$, then $$G$$ is Hamiltonian.
MSC:
 05C45 Eulerian and Hamiltonian graphs
Keywords:
Hamiltonian graphs; neighbour
Full Text:
References:
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