On \(L(d,1)\)-labelings of graphs (Q1567607)

For a graph \(G\) and positive integer \(d\) an \(L(d,1)\)-labeling of \(G\) is defined as a function \(f: V_G\to {\mathbb{Z}}^{i\geq 0}\) such that \(|f(u)-f(v)|\geq d\) if \(uv\in E_G\) and \(|f(u)-f(v)|\geq 1\) if \(uv\not\in E_G\) and \(u,v\) are connected by a path. The \(L(d,1)\)-number of \(G\), \(\lambda_d(G)\), is a minimal number \(m\) such that there exists an \(L(d,1)\)-labeling \(f\) of \(G\) with the image \(\{1,2,\dots,m\}\). The authors extend the study of the number \(\lambda_d(G)\) from the extensively studied case \(d=2\) to arbitrary \(d\geq 2\). The main result is an upper bound \(\lambda_d(G)\leq \Delta^2 + (d-1)\Delta\) for any graph with maximum degree \(\Delta\). More specific results (lower and upper bounds) are obtained for chordal graphs and their various subclasses, in particular trees. It is shown that the bounds for trees are attainable.