L(2,1)-Labeling of the Direct Product of a Path and a Complete Graph

In order to obtain the L(2,1)-labeling of the direct product P_m×K_n of a pathP_m and a complete graph K_n, by means of induction, conjecture and classified discussion, the L(2,1)-labeling number of P_m×K₃ is determined as 6 if m=3 or 4, and the L(2,1)-labeling number of P_m×K_n is determined as 7 if m≥5.The upper bound of the L(2,1)-labeling number of P_m×K_n is shown at most m≥n-2 if m≥n-2 and n≥3