New sufficient condition for Hamiltonian graphs.

*(English)*Zbl 1129.05027Let \(\alpha(G)\) and \(N(v)\) be the independence number of \(G\) and the neighborhood of \(v\) in \(G\), respectively. The main result of this paper is the following theorem.

If \(G\) is a \(k\)-connected \((k\geq2)\) graph of order \(n\), and if \(\max\{\deg(v): v\in S\}\geq n/2\) for every independent set \(S\) of order \(k\), such that \(S\) has two distinct vertices \(x,y\) with \(1\leq| N(x)\cap N(y)| \leq\alpha(G)-1\), then \(G\) is Hamiltonian.

This theorem unifies and extends several well-known sufficient conditions to assure the existence of a Hamiltonian cycle in a simple graph \(G\) of order \(n\geq3\), e.g. Dirac’s, Ore’s, Fan’s and Chen’s conditions. The authors also show that there exist Hamiltonian graphs satisfying the hypothesis of the theorem above but whose Hamiltonicity cannot be assured by any one of the above-mentioned conditions.

If \(G\) is a \(k\)-connected \((k\geq2)\) graph of order \(n\), and if \(\max\{\deg(v): v\in S\}\geq n/2\) for every independent set \(S\) of order \(k\), such that \(S\) has two distinct vertices \(x,y\) with \(1\leq| N(x)\cap N(y)| \leq\alpha(G)-1\), then \(G\) is Hamiltonian.

This theorem unifies and extends several well-known sufficient conditions to assure the existence of a Hamiltonian cycle in a simple graph \(G\) of order \(n\geq3\), e.g. Dirac’s, Ore’s, Fan’s and Chen’s conditions. The authors also show that there exist Hamiltonian graphs satisfying the hypothesis of the theorem above but whose Hamiltonicity cannot be assured by any one of the above-mentioned conditions.

Reviewer: Stanislav Jendrol’ (Košice)

##### MSC:

05C45 | Eulerian and Hamiltonian graphs |

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\textit{K. Zhao} et al., Appl. Math. Lett. 20, No. 1, 116--122 (2007; Zbl 1129.05027)

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##### References:

[1] | Bondy, A.J.; Murty, U.S.R., Graph theory with applications, (1976), American Elsevier New York · Zbl 1226.05083 |

[2] | Chen, G., Hamiltonian graphs involving neighborhood intersections, Discrete math., 112, 253-258, (1993) · Zbl 0782.05055 |

[3] | Chen, G.; Egawa, Y.; Liu, X.; Saito, A, Essential independent set and Hamiltonian cycles, J. graph theory, 21, 243-250, (1996) |

[4] | Dirac, G.A., Some theorems on abstract graphs, Proc. London math. soc., 2, 69-81, (1952) · Zbl 0047.17001 |

[5] | Fan, G., New sufficient conditions for cycles in graphs, J. combin. theory ser. B, 37, 221-227, (1984) · Zbl 0551.05048 |

[6] | Ore, O., Note on Hamiltonian circuits, Amer. math. monthly, 67, 55, (1960) · Zbl 0089.39505 |

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