Abstract
Chvátal and Erdős proved a well-known result that the graph \(G\) with connectivity \(\kappa (G)\) not less than its independence number \(\alpha (G)\) [\(\alpha (G)+1\), \(\alpha (G)-1\), respectively] is Hamiltonian (traceable, Hamiltonian-connected, respectively). In this paper, we strengthen the Chvátal–Erdős theorem to the following: Let \(G\) be a simple 2-connected graph of order large enough such that \(\alpha (G)\le \kappa (G)+1\) [\(\alpha (G)\le \kappa (G)+2\), \(\alpha (G)\le \kappa (G),\) respectively] and such that the number of maximum independent sets of cardinality \(\kappa (G)+1\) [\(\kappa (G)+2\), \(\kappa (G)\), respectively] is at most \(n-2\kappa (G)\) [\(n-2\kappa (G)-1\), \(n-2\kappa (G)+1\), respectively]. Then \(G\) is either Hamiltonian (traceable, Hamiltonian-connected, respectively) or a subgraph of \(K_{k}+((kK_1)\cup K_{n-2k})\) [\(K_{k}+((k+1)K_1\cup K_{n-2k-1})\), \(K_{k}+((k-1)K_1\cup K_{n-2k+1})\), respectively].
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Acknowledgments
The authors are indebted to the anonymous referee for suggesting [7] to make a better bound on the order in Theorem 7 than the original one and the other constructive comments. This research is supported by Nature Science Funds of China and by Specialized Research Fund for the Doctoral Program of Higher Education (No. 20131101110048). The second author (Yinkui Li) is supported by the Project 2014xjz03.
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Chen, G., Li, Y., Ma, H. et al. An Extension of the Chvátal–Erdős Theorem: Counting the Number of Maximum Independent Sets. Graphs and Combinatorics 31, 885–896 (2015). https://doi.org/10.1007/s00373-014-1416-2
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DOI: https://doi.org/10.1007/s00373-014-1416-2