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

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) <= 2. Schmeichel showed that for n >= 5, b(K center dot P2) <= 1 + b(K). Ali proved that for n, m >= 5, b(K center dot Km) <= 3 + b(K) b(Km). In this paper, we give an upper bound for the basis number of the semi-strong product of a bipartite graph with a cycle.