A \(\lambda\) -vertex coloring of a graph G is an assignment of \(\lambda\) colors to the vertices of G such that adjacent vertices have different colors. We say that two \(\lambda\) -vertex colorings f and g of G are called distinct if there exists a vertex assigned different colors. The chromatic polynomial of G on variable \(\lambda\) , denoted by \(\chi (G;\lambda )\) , is the number of \(\lambda\) -vertex colorings of G. The cartesian product of two simple graphs G and H is the graph \(G \square H\) , whose vertex set is \(V(G) \times V(H)\) and whose edge set consists of all pairs \((u_1,v_1)(u_2,v_2)\) such that either \(u_1u_2 \in E(G)\) and \(v_1 = v_2\) , or \(v_1v_2 \in E(H)\) and \(u_1 = u_2\) . In 2008, Pfaff and Walker determined the chromatic polynomial of \(C_3 \square P_n\) using the edge deletion-contraction theorem. However, this result is a recursive expression in two variables, making its computation extremely costly for large n. In 2024, Yadav et al. applied the transfer matrix method to compute the chromatic polynomials of \(P_m \square P_n\) for \(n \in \mathbb {N}\) and \(m \in \{1,2,3\}\) . In this paper, we determine a closed-form formula for the chromatic polynomial of \(C_3 \square P_n\) such that n is a positive integer, given by \(\chi (C_3 \square P_n; \lambda ) =\) \(\lambda (\lambda -1)(\lambda -2) (\lambda ^3 - 6 \lambda ^2 + 14 \lambda - 13)^{n - 1}.\)
