Radio number for total graph of paths (Q1952708)

Summary: A radio labeling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of nonnegative integers such that \(|f(u) - f(v)| \geq \text{diam}(G) + 1 - d_G(u, v)\), where \(\operatorname{diam}(G)\) and \(d_G(u, v)\) are diameter and distance between \(u\) and \(v\) in graph \(G\), respectively. The radio number \(\operatorname{rn}(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has radio labeling with \(\max\{f(v) : v \in V(G)\} = k\). We investigate the radio number for the total graphs of paths.