Colorations , Orthotopes , and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs Version of May 12 , 2007 David Forge

A gain graph is a graph whose edges are labelled invertibly from a group. A weighted gain graph is a gain graph with vertex weights from a semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and tree-expansion polynomials that are Tutte invariants (they satisfy Tutte’s deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky’s for gain graphs, Noble and Welsh’s for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is unusual in sometimes having uncountably many variables, in contrast to other known Tutte invariants that have at most countably many variables, and in not being itself a universal Tutte invariant of weighted gain graphs; that remains to be found. An evaluation of our polynomial counts proper colorations of the gain graph when the vertex weights are lists of permissible colors from a color set with a gain-group action. When the gain group is Zd, the lists are order ideals in the integer lattice Zd, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function, of degree d|V |, of the upper bounds. This example leads to graphtheoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n×d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations. Mathematics Subject Classifications (2000): Primary 05C22; Secondary 05C15, 11P21.