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 _nof a pathP _mand a complete graph K _n, by means of induction, conjecture and classified discussion, the L(2,1)-labeling number of P _m×K ₃is determined as 6 if m=3 or 4, and the L(2,1)-labeling number of P _m×K _nis determined as 7 if m≥5.The upper bound of the L(2,1)-labeling number of P _m×K _nis shown at most m≥n-2 if m≥n-2 and n≥3