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 a-rank number of G, denoted u,(G) equals the largest k such that G has a minimal k-ranking. We establish new results involving minimal rankings of paths and in particular we determine u(Pn), a problem suggested by Laskar and Pillone in 2000. We show u(Pn) = [log2 (n + 1)] + [log2(n + 1 - (2^([log2n]-1))] (Refer to PDF file for exact formulas).

Publication Date



Presentation at the 34th Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Florida Atlantic University, Boca Raton, FL, March 2003. Research partially supported by JetBlue Airways, Kay & Tony Carlisi, and Timothy Gilbert. Partially supported by a 2002 RIT COS Dean’s Summer Research Fellowship Grant 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