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A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

“…We also expect that the maximum of the background distribution may be shifted further to the right by requiring the graphs to be two-or three-connected, and so presumably the peaks at the first few B k should stand out more clearly. We plan to investigate this and further issues in more detail elsewhere [37].…”

“…It is clearly visible from figure 7 that both the mean and worst-case n max exhibit a slower than linear growth with N. Assuming that GreedyFillIn yields always an n max which is of the order of the true tree width, we would have n max cst √ N , and the data of figure 7 appear to be compatible with such a scaling. We defer the further investigation of the asymptotic behaviour of GreedyFillIn to future work [37].…”

“…In particular, we have adapted (but not yet implemented) our algorithm to counting versions of the following problems: Hamiltonian walks and cycles, longest path, vertex cover, dominating set, feedback vertex set, and minimum maximal independent set. We plan to report on these problems elsewhere [37].…”