Abstract
Let G be a graph and n, k and d be non-negative integers such that \(|V(G)|\geqslant n+2k+d+2\) and \(|V(G)|-n-d \equiv 0 \pmod {2}\). A graph is called an (n, k, d)-graph if deleting any n vertices from G the remaining subgraph of G contains k-matchings and each k-matching in the subgraph can be extended to a defect-d matching. We study the relationships between (n, k, d)-graphs and various closure operations, which are usually considered in the theory of hamiltonian graphs. In particular, we obtain some necessary and sufficient conditions for the existence of (n, k, d)-graphs in terms of these closures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berge, C.: Sur le couplaage maximum d’un graphe. C. R. Acad. Sci. Paris 247, 258–259 (1958)
Bollobás, B., Riordan, O., Ryjác̆ek, Z., Saito, A., Schelp, R.H.: Closure and hamiltonian-connectivity of claw-free graphs. Discrete Math. 195, 67–80 (1999)
Bondy, J.A., Chvátal, V.: A method in graph theory. Discrete Math. 15, 111–136 (1976)
Favaron, O.: On \(k\)-factor-critical graphs. Discuss. Math. Graph Theory 16, 41–51 (1996)
Jin, Z., Yan, H., Yu, Q.: Generalization of matching extensions in graphs (II). Discrete Appl. Math. 155, 1267–1274 (2007)
Liu, G., Yu, Q.: Generalization of matching extensions in graphs. Discrete Math. 231, 311–320 (2001)
Plummer, M.D.: On \(n\)-extendable graphs. Discrete Math. 31, 201–210 (1980)
Plummer, M.D.: Extending matchings in graphs: A survey. Discrete Math. 127, 277–292 (1994)
Plummer, M.D., Saito, A.: Closure and factor-critical graphs. Discrete Math. 215, 171–179 (2000)
Ryjác̆ek, Z.: On a closure concept in claw-free graphs. J. Combin. Theory Ser. B 70, 217–224 (1997)
Sumner, D.P.: Graphs with 1-factors. Proc. Amer. Math. Soc. 42, 8–12 (1974)
Yu, Q.: Characterizations of various matching extensions in graphs. Australas. J. Combin. 7, 55–64 (1993)
Yu, Q., Liu, G.: Graph Factors and Matching Extensions. Springer-Verlag, Berlin (2009)
Acknowledgments
The first author is supported by the National Natural Science Foundation of China Grant No. 11471257 and the Fundamental Research Funds for the Central Universities and the second author is supported by the Discovery Grant (144073) of Natural Sciences and Engineering Research Council of Canada. The authors are indebted to the anonymous referees for their valuable comments and suggestions, in particular, the improved version of Theorem 2.3.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lu, H., Yu, Q. Generalization of Matching Extensions in Graphs (IV): Closures. Graphs and Combinatorics 32, 2009–2018 (2016). https://doi.org/10.1007/s00373-016-1687-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-016-1687-x