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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
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References:
[1] ASRATYAN A. S., KHACHATRYAN N. K.: Two theorems on hamiltonian graphs. Mat. Zametki 35 (1984), 55-61. · Zbl 0552.05038
[2] FAUDREE R. J., GOULD R. J., JACOBSON R. S., SCHELP R. H.: Neighborhood unions and hamiltonian properties in graphs. J. Combin. Theory Ser. B 46 (1989), 1-20. · Zbl 0677.05056
[3] FRAISSE P.: A new sufficient condition for hamiltonian graphs. J. Graph Theory 10 (1986), 405-409. · Zbl 0606.05043
[4] GOULD R. J.: Updating the hamiltonian problem - a survey. J. Graph Theory 15 (1991), 121-157. · Zbl 0746.05039
[5] ORE O.: Note on hamiltonian circuits. Amer. Math. Monthly 67 (1960), 5. · Zbl 0089.39505
[6] TIAN F.: A note on the paper ”A new sufficient condition for hamiltonian graphs”. Preprint, 1989.
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