Polynomials associated with nowhere-zero flows

Abstract

In this paper we study relations between nowhere-zero ℤk- and integer-valued flows in graphs and the functions FG(k) and IG(k) evaluating the numbers of nowhere-zero ℤk- and k-flows in a graph G, respectively. It is known that FG(k) is a polynomial for k > 0. We show that IG(k) is also a polynomial and that 2m(G)FG(k) ≥ IG(k) ≥ (m(G)+1) FG(k), where m(G) is the rank of the cocycle matroid of G. Finally we prove that FG(k+1) ≥ FG(k)·k/(k-1) and IG(k+1) ≥ IG(k)·k/(k-1) for every k > 1. © 2002 Elsevier Science (USA).