Let \(G_{n,m} = C_n \times P_m\), be the cartesian product of an \(n\)-cycle \(C_n\) and a path \(P_m\) of length \(m-1\). We prove that \(\chi'(G_{n,m}) = \chi'(G_{n,m}) = 4\) if \(m \geq 3\), which implies that the list-edge-coloring conjecture (LLECC) holds for all graphs \(G_{n,m}\).