Minimal rankings and the arank number of a path

Given a graph G, a function f : V (G) {1, 2, . . . , 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 r(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular, we determine r(Pn), a problem posed by Laskar and Pillone in 2000.

• Type: Journal
• Year: 2006
• Where: DM
• Authors: Victor Kostyuk, Darren A. Narayan, Victoria A. Williams
• Keywords: Arank Number, DM 2006, Label, Ranking Property