Graphs and Combinatorics

, Volume 21, Issue 4, pp 421–425 | Cite as

Structural Remarks on Bipartite Graphs with Unique f-Factors

Article

Abstract

In this note we will derive some structural results for a bipartite graph G with a unique f-factor. Two necessary conditions will be that G is saturated, meaning that the addition of any edge leads to a second f-factor, and that f A , f B ≥1. Here f A and f B are defined as the minimum of f over the vertices in the two partite sets A and B of G, respectively. Our main result states that G has at least f A + f B vertices for which d G (v) = f(v) holds.

Keywords

Unique f-factor Bipartite graphs Extremal graphs 

Mathematics Subject Classificiation

Primary 05C70 Secondary 05C35 

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References

  1. 1.
    Chartrand, G., Lesniak, L.: Graphs and Digraphs 3rd edition. Chapman and Hall, London 1996Google Scholar
  2. 2.
    Hoffmann, A., Sidorowicz, E., Volkmann, L.: Extremal bipartite graphs with a unique k-factor, submittedGoogle Scholar
  3. 3.
    Jackson, B., Whitty, R.W.: A note concerning graphs with unique f-factors. J. Graph Theory 9 577–580 (1989)Google Scholar
  4. 4.
    Johann, P., On the structure of graphs with a unique k-factor. J. Graph Theory 35 227–243 (2000)Google Scholar
  5. 5.
    Volkmann, L.: The maximum size of graphs with a unique k-factor. Combinatorica 24 531–540 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Watson Wyatt Deutschland GmbHMunichGermany
  2. 2.Lehrstuhl II für MathematikRWTH-AachenAachenGermany

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