Advertisement

Journal of Mathematical Sciences

, Volume 184, Issue 5, pp 564–572 | Cite as

Bounds of the number of leaves of spanning trees

  • A. V. BankevichEmail author
  • D. V. Karpov
Article
  • 46 Downloads

We prove that every connected graph with s vertices of degree not 2 has a spanning tree with at least \( \frac{1}{4}\left( {s - 2} \right) \) + 2 leaves.

Let G be a connected graph of girth g with υ vertices. Let maximal chain of successively adjacent vertices of degree 2 in the graph G does not exceed k ≥ 1. We prove that G has a spanning tree with at least α g,k (υ(G) − k − 2) + 2 leaves, where \( {\alpha_{g,k}} = \frac{{\left[ {\frac{{g + 1}}{2}} \right]}}{{\left[ {\frac{{g + 1}}{2}} \right]\left( {k + 3} \right) + 1}} \) for k < g − 2; \( {\alpha_{g,k}} = \frac{{g - 2}}{{\left( {g - 1} \right)\left( {k + 2} \right)}} \) for k ≥ g − 2.

We present infinite series of examples showing that all these bounds are tight. Bibliography: 12 titles.

Keywords

Span Tree Connected Graph Adjacent Vertex Infinite Series Maximal Chain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. A. Storer, “Constructing full spanning trees for cubic graphs,” Inform. Process. Lett., 13, 8–11 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    J. R. Griggs, D. J. Kleitman, and A. Shastri, “Spanning trees with many leaves in cubic graphs,” J. Graph Theory, 13, 669–695 (1989).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    D. J. Kleitman and D. B. West, “Spanning trees with many leaves,” SIAM J. Discrete Math., 4, 99–106 (1999).MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. R. Griggs and M. Wu, “Spanning trees in graphs of minimum degree 4 or 5,” Discrete Math., 104. 167–183 (1992).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    N. Alon, “Transversal numbers of uniform hypergraphs,” Graphs and Combinatorics, 6, 1–4 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    G. Ding, T. Johnson, and P. Seymour, “Spanning trees with many leaves,” J. Graph Theory, 37, 189–197 (2001).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Y. Caro, D. B. West. and R. Yuster, “Connected domination and spanning trees with many leaves,” SIAM J. Discrete Math., 13, 202–211 (2000).MathSciNetCrossRefGoogle Scholar
  8. 8.
    P. S. Bonsma., “Spanning trees with many leaves in graphs with minimum degree three,” SIAM J. Discrete Math., 22, 920–937 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    P. S. Bonsma and F. Zickfeld, “Spanning trees with many leaves in graphs without diamonds and blossoms,” LATIN 2008, 531–543 (2008).MathSciNetGoogle Scholar
  10. 10.
    N. V. Gravin, “Constructing spanning trees with many leaves,” Zap. Nauchn. Semin. POMI, 381, 31–46 (2010).Google Scholar
  11. 11.
    D. V. Karpov, “Spanning trees with many leaves,” Zap. Naacha. Semin. POMI, 381, 78–87 (2010).Google Scholar
  12. 12.
    F. Harary, Graph Theory, Addison-Wesley (1969).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

Personalised recommendations