Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지

The Basis Number of the Cartesian Product of a Path with a Circular Ladder, a Möbius Ladder and a Net

  • Alzoubi, Maref Y. (Department of Mathematics, Yarmouk University) ;
  • Jaradat, Mohammed M.M. (Department of Mathematics, Yarmouk University)
  • Received : 2004.10.13
  • Published : 2007.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

The basis number of a graph G is the least positive integer $k$ such that G has a $k$-fold basis. In this paper, we prove that the basis number of the cartesian product of a path with a circular ladder, a M$\ddot{o}$bius ladder and path with a net is exactly 3. This improves the upper bound of the basis number of these graphs for a general theorem on the cartesian product of graphs obtained by Ali and Marougi, see [2]. Also, by this general result, the cartesian product of a theta graph with a M$\ddot{o}$bius ladder is at most 5. But in section 3 we prove that it is at most 4.

Keywords

References

  1. A. A. Ali, The basis number of the direct product of paths and cycles, Ars Combin., 27(1989), 155-163.
  2. A. A. Ali and G. T. Marougi, The basis number of the cartesian product of some graphs, The J. of the Indian Math. Soc., 58(1992), 123-134.
  3. A. A. Ali and G. T. Marougi, The basis number of the cartesian product of some special graphs, Mu'tah Lil-Buhooth Wa Al-Dirasat, 8(2)(1993), 83-100.
  4. A. S. Alsardary and J. Wojciechowski, The basis number of the powers of the complete graph, Discrete Math., 188(1-3)(1998), 13-25. https://doi.org/10.1016/S0012-365X(97)00271-9
  5. M. Y. Alzoubi and M. M. M. Jaradat, The basis number of the composition of theta graph with stars and wheels, Acta Math. Hungar., 103(3)(2004), 201-209.
  6. J. A. Banks and E. F. Schmeichel, The basis number of n-cube, J. Combin. Theory Ser., B 33(2)(1982), 95-100.
  7. J. A. Bondy and U. S. R. Murty, Graph theory with applications, America Elsevier Publishing Co. Inc., New York, 1976.
  8. R. Diestel, Graph theory, Graduate Texts in Mathematics, 173, Springer-Verlage, New York, 1997.
  9. M. Q. Hailat and M. Y. Alzoubi, The basis number of the composition of graphs, Istanbul Univ. Fen Fak. Mat. Der., 53(1994), 43-60.
  10. F. Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Massachusetts, 1971.
  11. W. Imrich and S. Klavzar, Product graphs: Structure and recognition, Wiley- Interscience Series in Discrete Mathematics and Optimization, 2000.
  12. M. M. M. Jaradat, On the basis number of the direct product of graphs, Australas. J. Combin., 27(2003), 293-306.
  13. M. M. M. Jaradat, The basis number of the direct product of a theta graph and a path, Ars combin., 75(2005), 105-111.
  14. M. M. M. Jaradat and M. Y. Alzoubi, On the basis number of the semi-strong product of bipartite graphs with cycles, Kyungpook Math. J., 45(1)(2005), 45-53.
  15. M. M. M. Jaradat and M. Y. Alzoubi, An upper bound of the basis number of the lexicographic product of graphs, Australas. J. Combin., 32(2005), 305-312.
  16. M. M. M. Jaradat, The basis number of the strong product of paths and cycles with bipartite graphs, Missouri Journal of Mathematical Sciences (accepted).
  17. S. MacLane, A combinatorial condition for planar graphs, Fundameta Math., 28(1937), 22-32.
  18. E.F. Schmeichel, The basis number of a graph, J. Combin. Theory ser., B 30(2)(1981), 123-129.