Tension-flow polynomials on graphs

An orientation of a graph is acyclic (totally cyclic) if and only if it is a “positive orientation” of a nowhere-zero integral tension (!ow). We unify the notions of tension and !ow and introduce the so-called tension-!ows so that every orientation of a graph is a positive orientation of a nowhere-zero integral tension-!ow. Furthermore, we introduce an (integral) tension-!ow polynomial, which generalizes the (integral) tension and (integral) !ow polynomials. For every graph G, the tension-!ow polynomial FG(k1; k2) on G and the Tutte polynomial TG(k1; k2) on G satisfy FG(k1; k2)6 TG(k1−1; k2−1). We also characterize the graphs for which the inequality is sharp. c © 2003 Elsevier B.V. All rights reserved.