Abstract
The \(\{K_2,C_n\}\)-factor of a graph is a spanning subgraph whose each component is either \(K_2\) or \(C_n\). In this paper, a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the \(\{K_2,C_{2i+1}| i\geqslant 2 \}\)-factor for any graph is obtained, which answers a problem due to Gao and Wang (J Oper Res Soc China, 2021. https://doi.org/10.1007/s40305-021-00357-6).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu, Q.L., Liu, G.Z.: Graph Factors and Matching Extensions. Springer, Verlag (2009)
Chvátal, V.: Tough graphs and Hamiltonian circuits. Discret. Math. 5, 215–228 (1973)
Zhou, S.Z., Wu, J.C., Zhang, T.: The existence of \(P_{\geqslant 3}\)-factor covered graphs. Discuss. Math. Graph Theory 37(4), 1055–1065 (2017)
Liu, G.Z., Zhang, L.J.: Toughness and the existence of fractional \(k\)-factors of graphs. Discret. Math. 308, 1741–1748 (2008)
Gao, W., Chen, Y.J., Wang, Y.Q.: Network vulnerability parameter and results on two surfaces. Int. J. Intell. Syst. 36(8), 4392–4414 (2021)
Yuan, Y., Hao, R.X.: Toughness condition for the existence of all fractional \((a, b, k)\)-critical graphs. Discret. Math. 342, 2308–2314 (2019)
Yang, J.B., Ma, Y.H., Liu, G.Z.: Fractional \((g, f)\)-factors in graphs. Appl. Math. -J. Chin. Univ. Ser. A 16, 385–390 (2001)
Kano, M., Lu, H.L., Yu, Q.L.: Component factors with large components in graphs. Appl. Math. Lett. 23, 385–389 (2010)
Scheinerman, E., Ullman, D.: Fractional Graph Theory: A Rational Approach to the Theory of Graphs. John Wiley, New York (1997)
Zhang, L.J., Liu, G.Z.: Fractional \(k\)-factor of graphs. J. Syst. Sci. Math. Sci. 21, 88–92 (2001)
Gao, W., Wang, W.F.: Remarks on component factors. J. Oper. Res. Soc. China (2021). https://doi.org/10.1007/s40305-021-00357-6
Woodall, D.: The binding number of a graph and its Anderson number. J. Comb. Theory Ser. B 15, 225–255 (1973)
Anderson, I.: Perfect matchings of a graph. J. Comb. Theory Ser. B 10(3), 183–186 (1971)
Katerinis, P., Woodall, D.R.: Binding numbers of graphs and the existence of \(k\)-factors. Q. J. Math. Oxf. II. Ser. 38(2), 221–228 (1987)
Zhou, S.Z., Sun, Z.R.: Binding number conditions for \(P_{\geqslant 2}\)-factor and \(P_{\geqslant 3}\)-factor uniform graphs. Discret. Math. 343(3), 111715 (2020)
Cornuejols, G., Pulleyblank, W.R.: Perfect triangle-free \(2\)-matchings. Math. Program. Stud. 13, 1–7 (1980)
Acknowledgements
The authors thank the anonymous referees for the helpful comments and suggestions.
Author information
Authors and Affiliations
Contributions
The authors are listed in alphabetic order in this paper, and we think that each author’s contribution is equivalent.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
The research was supported by the National Natural Science Foundation of China (No. 62006196).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Guan, XX., Ma, TL. & Shi, C. Tight Toughness, Isolated Toughness and Binding Number Bounds for the \(\{K_2,C_n\}\)-Factors. J. Oper. Res. Soc. China (2023). https://doi.org/10.1007/s40305-023-00485-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40305-023-00485-1