On \(L(d, 1)\)-Labelings of the Cartesian Product of Two Cycles

Abstract

A \(k\)-L\((d, 1)\)-labeling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to \(\{0, 1, \ldots, k\}\) such that \(|f(u) – f(v)| > 1\) if \(d(u,v) = 2\) and \(|f(u) – f(v)| \geq d\) if \(d(u,v) = 1\). The L\((d,1)\)-labeling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has a \(k\)-L\((d, 1)\)-labeling. In this paper, we show that \(2d+2 \leq \lambda(C_m \square C_n) \leq 2d+4\) if either \(m\) or \(n\) is odd. Furthermore, the following cases are determined: (a) \(\lambda_d(C_3 \square C_n)\) and \(\lambda_d(C_4 \square C_n)\) for \(d \geq 3\), (b) \(\lambda_d(C_m \square C_n)\) for some \(m\) and \(n\), (c) \(\lambda_d(C_{2m} \square C_{2n})\) for \(d \geq 5\) when \(m\) and \(n\) are even.