On the Basis Number of the Semi-Strong Product of Bipartite Graphs with Cycles

  • Jaradat, M.M.M. (Department of Mathematics, Yarmouk University) ;
  • Alzoubi, Maref Y. (Department of Mathematics, Yarmouk University)
  • 투고 : 2003.12.08
  • 발행 : 2005.03.23

초록

A basis of the cycle space C (G) is d-fold if each edge occurs in at most d cycles of C(G). The basis number, b(G), of a graph G is defined to be the least integer d such that G has a d-fold basis for its cycle space. MacLane proved that a graph G is planar if and only if $b(G)\;{\leq}\;2$. Schmeichel showed that for $n\;{\geq}\;5,\;b(K_{n}\;{\bullet}\;P_{2})\;{\leq}\;1\;+\;b(K_n)$. Ali proved that for n, $m\;{\geq}\;5,\;b(K_n\;{\bullet}\;K_m)\;{\leq}\;3\;+\;b(K_n)\;+\;b(K_m)$. In this paper, we give an upper bound for the basis number of the semi-strong product of a bipartite graph with a cycle.

키워드

참고문헌

  1. Ars Combin. v.28 The basis number of complete multipartite graphs Ali, A.A.
  2. J. Indian Math. Soc. (NS) v.58 no.1-4 The basis number of cartesian product of some graphs Ali, A.A.;Marougi, G.T.
  3. Graph theory with applications Bondy, J.A.;Murty, U.S.
  4. Graph theory Harary, F.
  5. Australasian Journal of Combinatorics v.27 On the basis number of the direct product of graphs Jaradat, M.M.
  6. Fundamenta Math. v.28 A combinatorial condition for planar graphs MacLane, S.
  7. J. Combin. Theory Ser. B v.30 no.2 The basis number of a graph Schmeichel, E.F.