### Annals of Combinatorics 3 (1999) 417-429

Forests, Colorings and Acyclic Orientations of the Square Lattice

C. Merino and D.J.A. Welsh

Mathematical Institute, 24--26 St. Giles, Oxford University, OX1 3LB, UK
{merino, dwelsh}@maths.ox.ac.uk

AMS Subject Classification: 05C99

Abstract. There is no known polynomial time algorithm which generates a random forest or counts forests or acyclic orientations in general graphs. On the other hand, there is no technical reason why such algorithms should not exist. These are key questions in the theory of approximately evaluating the Tutte polynomial which in turn contains several other specializations of interest to statistical physics, such as the Ising, Potts, and random cluster models. Here, we consider these problems on the square lattice, which apart from its interest to statistical physics is, as we explain, also a crucial structure in complexity theory. We obtain some asymptotic counting results about these quantities on the n×n section of the square lattice together with some properties of the structure of the random forest. There are, however, many unanswered questions.

Keywords: square lattice, Tutte polynomial, forest, acyclic orientation, chromatic polynomial

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