Lambda number for the direct product of some family of graphs

Abstract

An L(2, 1)-labeling for a graph $$G=(V,E)$$G=(V,E) is a function f on V such that $$|f(u)-f(v)|\ge 2$$|f(u)-f(v)|ź2 if u and v are adjacent and f(u) and f(v) are distinct if u and v are vertices of distance two. The L(2, 1)-labeling number, or the lambda number $$\lambda (G)$$ź(G), for G is the minimum span over all L(2, 1)-labelings of G. When $$P_{m}\times C_{n}$$Pm×Cn is the direct product of a path $$P_m$$Pm and a cycle $$C_n$$Cn, Jha et al. (Discret Appl Math 145:317---325, 2005) computed the lambda number of $$P_{m}\times C_{n}$$Pm×Cn for $$n\ge 3$$nź3 and $$m=4,5$$m=4,5. They also showed that when $$m\ge 6$$mź6 and $$n\ge 7$$nź7, $$\lambda (P_{m}\times C_{n})=6$$ź(Pm×Cn)=6 if and only if n is the multiple of 7 and conjectured that it is 7 if otherwise. They also showed that $$\lambda (C_{7i}\times C_{7j})=6$$ź(C7i×C7j)=6 for some i, j. In this paper, we show that when $$m\ge 6$$mź6 and $$n\ge 3$$nź3, $$\lambda (P_m\times C_n)=7$$ź(Pm×Cn)=7 if and only if n is not a multiple of 7. Consequently the conjecture is proved. Here we also provide the conditions on m and n such that $$\lambda (C_m\times C_n)\le 7$$ź(Cm×Cn)≤7.