Zhao, Kewen; Lai, Hong-Jian; Shao, Yehong New sufficient condition for Hamiltonian graphs. (English) Zbl 1129.05027 Appl. Math. Lett. 20, No. 1, 116-122 (2007). Let \(\alpha(G)\) 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. Reviewer: Stanislav Jendrol’ (Košice) Cited in 2 Documents MSC: 05C45 Eulerian and Hamiltonian graphs Keywords:Hamiltonian graphs; Dirac condition; Ore condition; Fan condition; Chen condition PDFBibTeX XMLCite \textit{K. Zhao} et al., Appl. Math. Lett. 20, No. 1, 116--122 (2007; Zbl 1129.05027) Full Text: DOI References: [1] Bondy, A. J.; Murty, U. S.R., Graph Theory with Applications (1976), American Elsevier: 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.