Maximum Minimal \(k\)-rankings of Cycles

Abstract

Given a graph \(G\), a function \(f: V(G) \to \{1, 2, \ldots, k\}\) is a \(k\)-ranking of \(G\) if \(f(u) = f(v)\) implies every \(u-v\)
path contains a vertex \(w\) such that \(f(w) > f(u)\). A \(k\)-ranking is minimal if the reduction of any label greater
than \(1\) violates the described ranking property.The \(arank\) number of a graph, denoted \(\psi_r(G)\),
is the maximum \(k\) such that \(G\) has a minimal \(k\)-ranking.We establish new properties for minimal rankings and present
new results for the \(arank\) number of a cycle.