Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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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)
  • Received : 2003.12.08
  • Published : 2005.03.23
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Abstract

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.

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References

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