Tutte's dichromate for signed graphs
Abstract
We introduce a trivariate polynomial invariant for signed graphs, which we call the signed Tutte polynomial, and show that it contains among its evaluations the number of proper colorings and the number of nowherezero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowherezero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the signed Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowherezero flows of a signed graph. Therefore the "dichromate" for signed graphs (our signed Tutte polynomial) differs from the dichromatic polynomial (the ranksize generating function). The signed Tutte polynomial is a special case of a trivariate "joint Tutte polynomial" of ordered pairs of matroids on a common ground set  for a signed graph, the cycle matroid of its underlying graph and its signedgraphic matroid form the relevant pair of matroids. This is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and contains as a specialization the Tutte polynomial of a matroid perspective defined by Las Vergnas.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.07548
 Bibcode:
 2019arXiv190307548G
 Keywords:

 Mathematics  Combinatorics;
 05C22;
 05C31
 EPrint:
 37 pp. 2 figures