The crossing number \(\textrm{cr}(G)\) of a graph G is defined as the smallest number of crossings in any drawing of G. In this paper, we first derive upper and lower bounds on \(\textrm{cr}(K_{3,3,n})\) for general \(n\ge 1\) , and prove that \(\textrm{cr}(K_{3,3,3})=15\) . Next, we introduce a contracting operation on the complete bipartite graph \(K_{3,3}\) , which enables us to establish a relationship between the crossing number of the capped Cartesian product \(K_{3,3}\Box _L T\) and the Cartesian product \(K_{3,3}\Box T\) for any tree T. Finally, combining these results with known properties of the zip product, we establish lower bounds on \(\textrm{cr}(K_{3,3}\Box T)\) for any tree T and determine the exact values for subcubic trees.
