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 ψr(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 ψr(Pn), a problem posed by Laskar and Pillone in 2000 (Refer to PDF file for exact formulas).

Publication Date



This is a post-print or an article published by Elsevier. Copyright 2006 Elsevier Science B.V., Amsterdam. The final published version is located here: https://doi.org/10.1016/j.disc.2006.01.027

Research travel support provided by JetBlue Airways, Kay & Tony Carlisi, and Timothy Gilbert. Partially partially supported by a RIT COS Dean’s Summer Research Fellowship Grant.ISSN:0012-365X

Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2014.

Document Type


Department, Program, or Center

School of Mathematical Sciences (COS)


RIT – Main Campus