Distance Labelings of Graphs

Abstract

An \(L(2,1)\)-labelling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x) – f(y)| \geq 2\) if \(d(x,y) = 1\) and \(|f(x) – f(y)| \geq 1\) if \(d(x,y) = 2\), where \(d(x,y)\) denotes the distance between \(x\) and \(y\) in \(G\). The \((2,1)\)-labelling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has an \(L(2,1)\)-labelling with \(\max\{f(v) : v \in V(G)\} = k\). Griggs and Yeh conjecture that \(\lambda(G) \leq \Delta^2\) for any simple graph with maximum degree \(\Delta \geq 2\). This article considers the graphs formed by the cartesian product of \(n\) (\(n \geq 2\) graphs. The new graph satisfies the above conjecture (with minor exceptions). Moreover, we generalize our results in [19].