Max-optimal and sum-optimal labelings of graphs

Abstract

Given a graph G, a function $f:V(G)\to \{1,2,\dots ,k\}$ is a k-ranking of G if $f(u)=f(v)$ implies that 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.

We consider two norms for minimal rankings. The max-optimal norm $\Vert f(G){\Vert }_{\infty }$ is the smallest k for which G has a minimal k-ranking. This value is also referred to as the rank number ${\chi }_r(G)$. In this paper we introduce the sum-optimal norm $\Vert f(G){\Vert }_1$ which is the minimum sum of all labels over all minimal rankings. We investigate similarities and differences between the two norms. In particular we show rankings for paths and cycles that are sum-optimal are also max-optimal.