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Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

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Abstract

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n − 2. These conditions are best possible. A related conjecture also is proposed.

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References

  1. Broersma H., Tuinstra H.: Independence trees and Hamilton cycles. J. Graph Theory 29, 227–237 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Flandrin E., Kaiser T., Kužel R., Li H., Ryjáček Z.: Neighborhood unions and extremal spanning trees. Discret. Math. 308, 2343–2350 (2008)

    Article  MATH  Google Scholar 

  3. Fujisawa J., Matsumura H., Yamashita T.: Degree bounded spanning trees. Graphs Combin. 26, 695–720 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gargano L., Hammar M.: There are spanning spiders in dense graphs (and we know how to find them). Lect. Notes Comput. Sci. 2719, 802–816 (2003)

    Article  MathSciNet  Google Scholar 

  5. Gargano L., Hammar M., Hell P., Stacho L., Vaccaro U.: Spanning spiders and light-splitting switches. Discret. Math. 285, 83–95 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gargano L., Hell P., Stacho L., Vaccaro U.: Spanning trees with bounded number of branch vertices. Lect. Notes Comput. Sci. 2380, 355–365 (2002)

    Article  MathSciNet  Google Scholar 

  7. Kano M., Kyaw A., Matsuda H., Ozeki K., Saito A., Yamashita T.: Spanning trees with small number of leaves in a claw-free graph. Ars. Combin. 103, 137–154 (2012)

    MATH  MathSciNet  Google Scholar 

  8. Matthews M.M., Sumner D.P.: Longest paths and cycles in K 1,3-free graphs. J. Graph Theory 9, 269–277 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  9. Matsuda H., Matsumura H.: On a k-tree containing specified leaves in a graph. Graphs and Combin 22, 371–381 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Neumann-Lara V., Rivera-Campo E.: Spanning trees with bounded degrees. Combinatorica 11, 55–61 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tsugaki M., Yamashita T.: Spanning trees with few leaves. Graphs Combin. 23, 585–598 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Win S.: Existenz von Gerüsten mit vorgeschriebenem Maximalgrad in Graphen (German). Abh. Math. Sem. Univ. Hamburg 43, 263–267 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  13. Win S.: On a conjecture of Las Vergnas concerning certain spanning trees in graphs. Result. Math. 2, 215–224 (1979)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama, 337-8570, Japan

    Haruhide Matsuda

  2. National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Kenta Ozeki

  3. JST, ERATO, Kawarabayashi Large Graph Project, Tokyo, Japan

    Kenta Ozeki

  4. Department of Mathematics, Kinki University, 3-4-1 Kowakae, Higashiosaka, Osaka, 577-8502, Japan

    Tomoki Yamashita

Authors
  1. Haruhide Matsuda
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  2. Kenta Ozeki
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  3. Tomoki Yamashita
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Correspondence to Kenta Ozeki.

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Matsuda, H., Ozeki, K. & Yamashita, T. Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph. Graphs and Combinatorics 30, 429–437 (2014). https://doi.org/10.1007/s00373-012-1277-5

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  • DOI: https://doi.org/10.1007/s00373-012-1277-5

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