Tutte's dichromate for signed graphs

We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero 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 trivariate 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 nowhere-zero flows of a signed graph. Therefore the ``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function). The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set -- for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant 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 was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.Comment: 53 pp. 9 figure

Rapid Mixing for Lattice Colorings with Fewer Colors

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA

List Distinguishing Parameters of Trees

A coloring of the vertices of a graph G is said to be distinguishing} provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, chi_D(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring. Recently the notion of a distinguishing coloring was extended to that of a list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of lists of available colors to the vertices of G, we say that G is (properly) L-distinguishable if there is a (proper) distinguishing coloring f of G such that f(v) is in L(v) for all v. The list distinguishing number of G, D_l(G), is the minimum integer k such that G is L-distinguishable for any list assignment L with |L(v)| = k for all v. Similarly, the list distinguishing chromatic number of G, denoted chi_{D_l}(G) is the minimum integer k such that G is properly L-distinguishable for any list assignment L with |L(v)| = k for all v. In this paper, we study these distinguishing parameters for trees, and in particular extend an enumerative technique of Cheng to show that for any tree T, D_l(T) = D(T), chi_D(T)=chi_{D_l}(T), and chi_D(T) <= D(T) + 1.Comment: 10 page