Lambda number for the direct product of some family of graphs

Abstract

An L(2, 1)-labeling for a graph $G=(V,E)$ is a function f on V such that $|f(u)-f(v)|\ge 2$ 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 $\lambda (G)$, for G is the minimum span over all L(2, 1)-labelings of G. When $P_{m}\times C_{n}$ is the direct product of a path $P_m$ and a cycle $C_n$, Jha et al. (Discret Appl Math 145:317–325, 2005) computed the lambda number of $P_{m}\times C_{n}$ for $n\ge 3$ and $m=4,5$. They also showed that when $m\ge 6$ and $n\ge 7$, $\lambda (P_{m}\times C_{n})=6$ if and only if n is the multiple of 7 and conjectured that it is 7 if otherwise. They also showed that $\lambda (C_{7i}\times C_{7j})=6$ for some i, j. In this paper, we show that when $m\ge 6$ and $n\ge 3$, $\lambda (P_m\times C_n)=7$ 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 $\lambda (C_m\times C_n)\le 7$.