Totally frustrated states in the chromatic theory of gain graphs

Details Totally frustrated states in the chromatic theory of gain graphs Totally frustrated states in the chromatic theory of gain graphs

2006-09-01

arxiv.org/abs/math/0609048v1

European J. Combinatorics, 30 (2009), 133-156.

Mathematics

Combinatorics

14 pages, 2 figures

Scientific paper

We generalize proper coloring of gain graphs to totally frustrated states, where each vertex takes a value in a set of `qualities' or `spins' that is permuted by the gain group. (An example is the Potts model.) The number of totally frustrated states satisfies the usual deletion-contraction law but is matroidal only for standard coloring, where the group action is trivial or nearly regular. One can generalize chromatic polynomials by constructing spin sets with repeated transitive components.

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