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Completely Independent Spanning Trees on Complete Graphs, Complete Bipartite Graphs and Complete Tripartite Graphs

  • Kung-Jui Pai
  • Shyue-Ming Tang
  • Jou-Ming Chang
  • Jinn-Shyong Yang
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 20)

Abstract

Let T 1, T 2,…, T k be spanning trees in a graph G. If for any two vertices x, y of G, the paths from x to y in T 1, T 2,…, T k are vertex-disjoint except end vertices x and y, then T 1, T 2,…, T k are called completely independent spanning trees in G. In 2001, Hasunuma gave a conjecture that there are k completely independent spanning trees in any 2k-connected graph. Péterfalvi disproved the conjecture in 2012. In this paper, we shall prove that there are \(\lfloor\frac{n}{2}\rfloor\) completely independent spanning trees in a complete graph with \(n ( \geqslant 4)\) vertices. Then, we prove that there are \(\lfloor\frac{n}{2}\rfloor\) completely independent spanning trees in a complete bipartite graph K m,n where \(m\geqslant n\geqslant 4\). Next, we also prove that there are \(\lfloor\frac{n_1+n_2}{2}\rfloor\) completely independent spanning trees in a complete tripartite graph \(K_{n_3,n_2,n_1}\) where \(n_3\geqslant n_2\geqslant n_1\) and \(n_1+n_2\geqslant 4\). As a result, the Hasunuma’s conjecture holds for complete graphs and complete m-partite graphs where m ∈ {2,3}.

Keywords

completely independent spanning trees edge-disjoint spanning trees complete graphs complete bipartite graphs complete tripartite graphs 

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References

  1. 1.
    Hasunuma, T.: Completely independent spanning trees in the underlying graph of a line digraph. Discrete Math. 234, 149–157 (2001)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Hasunuma, T.: Completely Independent Spanning Trees in Maximal Planar Graphs. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 235–245. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Hasunuma, T., Morisaka, C.: Completely independent spanning trees in torus networks. Networks 60, 59–69 (2012)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Péterfalvi, F.: Two counterexamples on completely independent spanning trees. Discrete Math. 312, 808–810 (2012)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kung-Jui Pai
    • 1
  • Shyue-Ming Tang
    • 2
  • Jou-Ming Chang
    • 3
  • Jinn-Shyong Yang
    • 3
  1. 1.Department of Industrial Engineering and ManagementMing Chi University of TechnologyNew Taipei CityTaiwan, ROC
  2. 2.Fu Hsing Kang SchoolNational Defense UniversityTaipeiTaiwan, ROC
  3. 3.Institute of Information and Decision SciencesNational Taipei College of BusinessTaipeiTaiwan, ROC

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