Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching

Abstract

We give the solution to the following question of C. D. Godsil[2]: Among the bipartite graphsG with a unique perfect matching and such that a bipartite graph obtains when the edges of the matching are contracted, characterize those having the property thatG +G, whereG + is the bipartite multigraph whose adjacency matrix,B +, is diagonally similar to the inverse of the adjacency matrix ofG put in lower-triangular form. The characterization is thatG must be obtainable from a bipartite graph by adding, to each vertex, a neighbor of degree one. Our approach relies on the association of a directed graph to each pair (G, M) of a bipartite graphG and a perfect matchingM ofG.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    M. Aigner, Combinatorial Theory,Springer-Verlag New York, (1979).

    Google Scholar 

  2. [2]

    C. D. Godsil, Inverses of trees,Combinatorica 5 (1985), 33–39.

    Google Scholar 

  3. [3]

    W.Mayeda, Graph Theory,Wiley-Interscience, (1972).

  4. [4]

    H.Ryser, Combinatorial Mathematics,MAA Cants Mathematical Monographs No.14 (1963).

  5. [5]

    R. Simon, Trees with 1-factors: degree distribution,Congressum Numerantium,45 (1984), 147–159.

    Google Scholar 

  6. [6]

    R.Simion, Trees with 1-factors and oriented trees,manuscript.

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Simion, R., Cao, D.S. Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching. Combinatorica 9, 85–89 (1989). https://doi.org/10.1007/BF02122687

Download citation

AMS subject classification (1980)

  • 05 C 05
  • 06 A10