Abstract
If a graph has a unique perfect matching, we call it a UPM-graph. In this paper we study UPM-graphs. It was shown by Kotzig that a connected UPM-graph has a cut edge belonging to its unique perfect matching. We strengthen this result to a further structural characterization. Using the stronger result, we present a characterization of claw-free UPM-graphs, and prove that for any fixed positive integer \(n\), the number of edges of saturated UPM-graphs on \(2n\) vertices form an arithmetic progression from \((2n+2)\lfloor \log _2(n+1)\rfloor -2^{2+\lfloor \log _2(n+1)\rfloor }+n+4\) to \(n^2\) with common difference 2. For a fixed positive integer \(n\), we determine the number of labelled UPM-trees on \(2n\) vertices. For a bipartite UPM-graph which has maximum number of edges, we determine the number of spanning UPM-trees of it.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bal, D., Dudek, A., Yilma, Z.B.: On the maximum number of edges in a hypergraph with a unique perfect matching. Discrete Math. 311, 2577–2580 (2011)
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)
Dudek, A., Schmitt, J.R.: On the size and structure of graphs with a constant number of 1-factors. Discrete Math. 312, 1807–1811 (2012)
Hartke, S.G., Stolee, D., West, D.B., Yancey, M.: Extremal graphs with a given number of perfect matchings. J. Graph Theory 73, 449–468 (2013)
Hendry, G.R.T.: Maximum graphs with a unique \(k\)-factor. J. Combin. Theory B 37, 53–63 (1984)
Hoffmann, A., Sidorowicz, E., Volkmann, L.: Extremal bipartite graphs with a unique \(k\)-factor. Discuss. Math. Graph Theory 26, 181–192 (2006)
Hoffmann, A., Volkmann, L.: On unique \(k\)-factors and unique [1, \(k\)]-factors in graphs. Discrete Math. 278, 127–138 (2004)
Hoffmann, A., Volkmann, L.: Structural remarks on bipartite graphs with unique \(f\)-factors. Graphs Combin. 21, 421–425 (2005)
Jackson, B., Whitty, R.W.: A note concerning graphs with unique \(k\)-factors. J. Graph Theory 13, 577–580 (1989)
Johann, P.: On the structure of graphs with a unique \(k\)-factor. J. Graph Theory 35, 227–243 (2000)
Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 4th edn. Springer, Berlin (2008)
Lovász, L.: On the structure of factorizable graphs. Acta Math. Acad. Sci. Hungar. 23, 179–195 (1972)
Lovász, L., Plummer, M.D.: Matching Theory. Elsevier Science Publishers, B. V., North Holland (1986)
Volkmann, L.: The maximum size of graphs with a unique \(k\)-factor. Combinatorica 24, 531–540 (2004)
Acknowledgments
The authors would like to thank the anonymous referees for their helpful comments on improving the representation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (Grant no. 11101383, 11271338, and 11201432).
Rights and permissions
About this article
Cite this article
Wang, X., Shang, W. & Yuan, J. On Graphs with a Unique Perfect Matching. Graphs and Combinatorics 31, 1765–1777 (2015). https://doi.org/10.1007/s00373-014-1463-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-014-1463-8