An upper bound of the basis number of the semi-strong product of bipartite graphs
Mohammed M. M. Jaradat · SUT Journal of Mathematics · 2005
A basis of the cycle space, C(G), of a graph G is called a d-fold if each edge of G occurs in at most d cycles of the basis. 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)≤2. Schmeichel showed that for n≥5, b(Kn•P2)≤1+b(Kn) Ali proved that for n,m≥5, b(Kn•Km)≤3+b(Km)+b(Kn). Jaradat proved that for any two bipartite graphs G and H, b(GΛH) ≤ 5 + b(G) + b(H). In this paper we give an upper bound of the basis number of the semi-strong product of bipartite graphs. Also, we give an example where the bound is achieved.