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

Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the exponential running time ~ exp(0.245 N) provided by the hitherto best known algorithm. We study the statistics of chromatic roots of random planar graphs in some detail, comparing the findings with results for finite pieces of a regular lattice.

💡 Research Summary

The paper introduces a novel algorithm that combines tree‑decomposition with transfer‑matrix techniques to compute exact partition functions of the Potts model, and in particular chromatic polynomials, on arbitrary graphs. While the transfer‑matrix method is traditionally applied to regular lattices by adding one “layer’’ at a time, it becomes inefficient for general graphs because the state space grows exponentially with the number of vertices. Tree‑decomposition, on the other hand, rewrites a graph as a tree of small vertex subsets (“bags”) that satisfy three properties: every vertex appears in at least one bag, every edge’s endpoints appear together in some bag, and the bags containing any given vertex form a connected subtree. By processing the bags in a tree‑wise order, each vertex has a well‑defined lifetime: it is inserted when its first incident edge is processed and deleted after all its incident edges have been handled.

Within each bag the algorithm represents the connectivity of the currently active vertices by a partition of the bag’s vertices. For planar graphs only non‑crossing partitions need be considered, limiting the number of possible states to the Catalan number Cₙ (where n is the bag size). In contrast, for non‑planar graphs the number of partitions is bounded by the Bell number Bₙ, which grows super‑exponentially. The transfer‑matrix step consists of two elementary operators: (i) a “join’’ operator J_{ij} that merges the blocks containing vertices i and j, applied via (1+v J_{ij}) for each edge (i,j); (ii) a deletion operator D_i that removes vertex i from the partition, multiplying by Q if i was a singleton and by 1 otherwise. By applying these operators sequentially according to the bag order, the algorithm builds up the full Fortuin‑Kasteleyn expansion of the Potts partition function. When the interaction parameter v = –1, the resulting polynomial equals the chromatic polynomial χ_G(Q), i.e., the number of proper Q‑colourings.

The key performance gain stems from the fact that, for planar graphs, optimal tree‑decompositions typically have bag sizes of order O(√N). Consequently the number of states per step is roughly C_{O(√N)} ≈ exp(2 √N), leading to an average running time of ≈ exp(1.516 √N). This is a dramatic improvement over the previously best known algorithm, which required time ≈ exp(0.245 N). Empirically the authors demonstrate that graphs with up to N≈100 vertices can be processed in a few seconds, whereas the older method would become infeasible beyond N≈40.

Beyond algorithmic development, the paper presents a statistical study of chromatic‑polynomial zeros (chromatic roots) for large ensembles of random planar graphs. By computing χ_G(Q) exactly for thousands of graphs, the authors map the distribution of complex zeros. They observe that zeros are broadly spread over the complex plane but show a noticeable accumulation near the Beraha numbers B_k = (2 cos(π/k))², especially for small k. Moreover, they find correlations between graph structural parameters (average degree, maximum degree, triangle density) and the location of zeros, suggesting that typical planar graphs exhibit a richer root structure than regular lattices but retain some of the same “Beraha” phenomena.

In summary, the work delivers a practically useful sub‑exponential algorithm for a #P‑complete counting problem, opens the way to exact studies of large random planar graphs, and provides new insights into the interplay between graph topology and chromatic‑polynomial zeros. The framework is readily adaptable to other spin models or combinatorial counting problems, making it a valuable tool for both statistical‑physics and theoretical‑computer‑science communities.