Flows and colorings

Abstract

Tutte first introduced the dichromate of a graph in large part motivated by the fact that it contained the flow polynomial and chromatic polynomial as univariate specializations. The latter receives attention in Chapter 11. In this chapter we consider flows. • Flows of graphs taking values in a finite abelian group, colorings and tensions (dual to flows). • The flow polynomial, chromatic polynomial and dichromatic polynomial as specializations of the Tutte polynomial. • Coloring-flow convolution formulas for the Tutte polynomial. • “A-bicycles” (called bicycles when A = Z2), the Tutte polynomial evalu ated at (−1, −1), (e2πi/3, e4πi/3) and (i, −i). • Tutte’s flow conjectures.