\(P_{4}\)-Factorization of Cartesian Product of Complete Graphs

In 1996, Muthusamy and Paulraja conjectured that for \( k \geq 3 \), the Cartesian product \( K_m \Box K_n \) has a \( P_k \)-factorization if and only if \( mn \equiv 0 \mod k \) and \( 2(k-1) | k(m+n-2) \). Recently, Chitra and Muthusamy have partially settled this conjecture for \( k = 3 \). In this paper, it is shown that for \(k = 4\) the above conjecture is true if \((m\mod 12, n \mod 12)\) \in \(\{(0,2), (2,0), (0,8), (8,0), (2,6), (6,2), (6,8), (8,6), (4,4)\}\). The left over cases for \(k = 4\) are \((m\mod 12, n\mod 12) \)\in \(\{(0,5), (5,0), (0,11), (11,0), (1,4), (4,1), (3,8), (8,3), (4,7), (7,4), (4,10), (10,4), (8,9), (9,8), (10,10)\}\).