$L(1,1)$-labelling of the direct product of a complete graph and a cycle

Abstract

An $L(j,k)$-labeling of a graph is a vertex labeling such that the difference of the labels of any two adjacent vertices is at least $j$ and that of any two vertices of distance $2$ is at least $k$. The minimum span of all $L(j,k)$-labelings of $G$ is denoted by $\lambda _k^j(G)$. Lin and Lam (Discret Math 308:3805–3815, 2008) provided an upper bound of $\lambda _1^2(K_m \times C_n)$ when $K_m \times C_n$ is the direct product of a complete graph $K_m$ and a cycle $C_n$. And they found the exact value of $\lambda _1^2(K_m \times C_n)$ for some $m$ and $n$. In this paper, we obtain an upper bound and a lower bound of $\lambda _1^1(K_m \times C_n)$. As a consequence we compute $\lambda _1^1(K_m \times C_n)$ when $n$ is even or $n\ge 4m+1$.