$$L_p$$-Optimal Rankings and Max-Optimal Rankings Are Different

A k-ranking of a graph G is a function \(f: V(G) \rightarrow \left\{ 0, 1, \ldots , k-1 \right\} \) such that if \(f(u)=f(v)\) for any \(u, v \in V(G)\), then on every uv path in G, there exists a vertex w with \(f(w) > f(u)\). The optimality of a k-ranking is measured in terms of the \(l_{\infty }\) norm of the sequence of labels produced by the k-ranking (max optimality). We study the optimality of rankings of graphs under all \(l_p\) norms. In particular, we show that there exist graphs where no max-optimal rankings are \(l_p\) optimal for any non-negative finite real number p, and vice versa.