Graphs whose \(l_p\)-optimal rankings are \(l_{\infty}\) Optimal

A ranking on a graph G is a function f : V ( G ) → { 1 , 2 , … , k } with the following restriction: if f ( u ) = f ( v ) for any u , v ∈ V ( G ) , then on every u v path in G , there exists a vertex w with f ( w ) > f ( u ) . The optimality of a ranking is conventionally measured in terms of the l ∞ norm of the sequence of labels produced by the ranking. In \cite{jacob2017lp} we compared this conventional notion of optimality with the l p norm of the sequence of labels in the ranking for any p ∈ [ 0 , ∞ ) , showing that for any non-negative integer c and any non-negative real number p , we can find a graph such that the sets of l p -optimal and l ∞ -optimal rankings are disjoint. In this paper we identify some graphs whose set of l p -optimal rankings and set of l ∞ -optimal rankings overlap. In particular, we establish that for paths and cycles, if p > 0 then l p optimality implies l ∞ optimality but not the other way around, while for any complete multipartite graph, l p optimality and l ∞ optimality are equivalent.