Skip to main content

On Graphs with a Unique Perfect Matching

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, access via your institution.

Fig. 1

References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hendry, G.R.T.: Maximum graphs with a unique \(k\)-factor. J. Combin. Theory B 37, 53–63 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hoffmann, A., Sidorowicz, E., Volkmann, L.: Extremal bipartite graphs with a unique \(k\)-factor. Discuss. Math. Graph Theory 26, 181–192 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hoffmann, A., Volkmann, L.: On unique \(k\)-factors and unique [1, \(k\)]-factors in graphs. Discrete Math. 278, 127–138 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hoffmann, A., Volkmann, L.: Structural remarks on bipartite graphs with unique \(f\)-factors. Graphs Combin. 21, 421–425 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jackson, B., Whitty, R.W.: A note concerning graphs with unique \(k\)-factors. J. Graph Theory 13, 577–580 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Johann, P.: On the structure of graphs with a unique \(k\)-factor. J. Graph Theory 35, 227–243 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 4th edn. Springer, Berlin (2008)

    Google Scholar 

  12. Lovász, L.: On the structure of factorizable graphs. Acta Math. Acad. Sci. Hungar. 23, 179–195 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lovász, L., Plummer, M.D.: Matching Theory. Elsevier Science Publishers, B. V., North Holland (1986)

    MATH  Google Scholar 

  14. Volkmann, L.: The maximum size of graphs with a unique \(k\)-factor. Combinatorica 24, 531–540 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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

Authors

Corresponding author

Correspondence to Xiumei Wang.

Additional information

Supported by NSFC (Grant no. 11101383, 11271338, and 11201432).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-014-1463-8

Keywords

  • Perfect matching
  • Trees
  • Claw-free graphs
  • Saturated graphs