On the basis number of the strong product of graphs

105  On the Basis Number of Some Special Graphs for Mycielski’s Graph Raghad A. Mustafa  Ghassan T. Marougi  Raghad.Math@uomosul.edu.iq   College of Computer Sciences and Mathematics  University of Mosul, Iraq  Accepted on: 16/05/2011       Received on: 26/12/2010  ABSTRACT The basis number b(G) ,of a graph G is defined to be the least positive integer k such that G has a k-fold basis for it’s cycle space. In this research paper,we have studied the basis number of some special graphs for Mycielski’s graph .It is proved that

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On the Basis Number of Some Special Graphs for Mycielski’s Graph

The basis number, b(G) ,of a graph G is defined to be the smallest positive integer k such that G has a k-fold basis for its cycle space. We investigate an upper bound for ) ( 4 3 2 1 G G G G b + + + .It is proved that, if 3 2 1 , , G G G and 4 G are connected vertex-disjoint graphs and each has a spanning tree of vertex degree not more than 4, then }. 1 ) b(G 2, ) b(G 2, ) b(G 1, ) b(G , 4 max{ ) ( 4 3 2 1 4 3 2 1 + + + +  + + + G G G G b The basis number of quadruple join of paths, are studied. It is proved that . 6 , 5 t , , 4 ) ( m    = + + + p n and m P P P P b t p n

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The Basis Number of Quadruple Join of Graphs

In (8) M. Y. Alzoubi and M. M. Jaradat studied the basis number of the com- position of paths and cycles with Ladders, Circular ladders and Mobius 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 inves- tigate the basis number of the lexicographic product of the dierent kinds of ladders with paths and cycles. In fact, we prove that the basis number of almost all of these graphs is 4.

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Reem Rafe' Ayed Al-Ta'Ani

Papers

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