Kyungpook Mathematical Journal

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

DOI QR Code

The Basis Number of the Lexicographic Product of Different Ladders with Paths and Cycles

  • Received : 2006.12.11
  • Published : 2008.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In [8] M. Y. Alzoubi and M. M. Jaradat studied the basis number of the composition of paths and cycles with Ladders, Circular ladders and M$\"{o}$bius ladders. Namely, they proved that the basis number of these graphs is 4 except possibly for some cases in each of them. Since the lexicographic product is noncommutative, in this paper we investigate the basis number of the lexicographic product of the different kinds of ladders with paths and cycles. In fact, we prove that the basis number of almost all of these graphs is 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, The basis number of complete multipartite graphs, Ars. combin., 28(1989), 41-49.
  3. A. A. Ali and G. T. Marougi, The basis number of the strong product of graphs, Mutah Lil-Buhoth wa Al-Dirasat, 7(1)(1992), 211-222.
  4. A. A. Ali and G. T. Marougi, The basis number of the Lexicographic product of graphs, Ars. combin., 36(1993), 271-282.
  5. 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
  6. M. Y. Alzoubi, On the basis number of graphs, M. Sc thesis Yarmouk university, (1992).
  7. M. Y. Alzoubi and M. M. M. Jaradat, The basis number of the composition of theta graphs with stars and wheels, Acta Math. Hungar., 103(4)(2004), 201-209.
  8. M. Y. Alzoubi and M. M. M. Jaradat, On the Basis Number of the Composition of Different Ladders with Some Graphs, I. J. M. M. S.,2005(12)(2005), 1861-1868.
  9. J. A. Banks and E. F. Schemeichel, The basis number of n-Cube, J. Combin. Theory ser. B, 33(2)(1982), 95-100. https://doi.org/10.1016/0095-8956(82)90061-2
  10. Bondy, J. A. and Murty, U. S. R.. Graph Theory with Applications, America Elsevier Publishing Co. Inc., New York, 1976.
  11. M. Q. Hailat and M. Y. Alzoubi, The basis number of the composition of graph, Istnbul Univ. Fen Fak. Mat. Der., 53(1994), 43-60.
  12. M. M. Jaradat, On the basis number of the direct product of graphs, Aust. J.Cobin., 27(2003), 293-306.
  13. M. M. M. Jaradat, M. Y. Alzoubi and E. A. Rawashdeh, The basis number of the Lexicographic product of different ladders, SUT Journal of Mathematics, 40(2)(2004), 91-101.
  14. M. M. Jaradat, The basis number of the direct product of a theta graph and a path, Ars. Combin., 75(2005), 105-111.
  15. M. M. M. Jaradat, An upper bound of the basis number of the strong product of graphs, Discussion Mathematicea Graph Theory, 25(3)(2005), 391-406. https://doi.org/10.7151/dmgt.1291
  16. Jonathan Gross and Jay Yellen, Graph Theory and Its Applications, CRC Press New York, 1999.
  17. S. F. MacLane, A combinatorial condition for planar graphs, Fundamenta Math., 28(1937), 22-32.
  18. E. F. Schemeichel, The basis number of graphs, J. Combin. Theory ser. B, 30(2)(1981), 123-129. https://doi.org/10.1016/0095-8956(81)90057-5