The basis number of the Cartesian product of different ladders. (Q2918612)

The regular ladder with \(n\) steps (\(2n\) vertices and \(3n - 2\) edges) will be denoted by \(L_n\). Containing this ladder \(L_n\) are the Möbius ladder \(ML_n\) (with \(2n\) vertices and \(3n\) edges) and the cycle ladder \(CL_n\) (with \(2n\) vertices and \(3n\) edges). It is shown that the cycle basis number (\(b\)) of the cartesian product of any pair of the ladders \(L_m, ML_n, CL_p\) for (\(3 \leq m, n, p\)) is at most \(5\). More specifically, it is shown that the cycle basis is \(4\) for the cartesian product of a cycle ladder with any of the other ladders (Möbius, cycle, regular), and also \(b(L_n \times ML_m) = b(ML_n \times ML_m) = 4\).