The maximum forcing number of polyomino

The forcing number of a perfect matching $M$ of a graph $G$ is the cardinality of the smallest subset of $M$ that is contained in no other perfect matchings of $G$. For a planar embedding of a 2-connected bipartite planar graph $G$ which has a perfect matching, the concept of Clar number of hexagonal system had been extended by Abeledo and Atkinson as follows: a spanning subgraph $C$ of is called a Clar cover of $G$ if each of its components is either an even face or an edge, the maximum number of even faces in Clar covers of $G$ is called Clar number of $G$, and the Clar cover with the maximum number of even faces is called the maximum Clar cover. It was proved that if $G$ is a hexagonal system with a perfect matching $M$ and $K'$ is a set of hexagons in a maximum Clar cover of $G$, then $G-K'$ has a unique 1-factor. Using this result, Xu {\it et. at.} proved that the maximum forcing number of the elementary hexagonal system are equal to their Clar numbers, and then the maximum forcing number of the elementary hexagonal system can be computed in polynomial time. In this paper, we show that an elementary polyomino has a unique perfect matching when removing the set of tetragons from its maximum Clar cover. Thus the maximum forcing number of elementary polyomino equals to its Clar number and can be computed in polynomial time. Also, we have extended our result to the non-elementary polyomino and hexagonal system

Ground-State Roughness of the Disordered Substrate and Flux Line in d=2

We apply optimization algorithms to the problem of finding ground states for crystalline surfaces and flux lines arrays in presence of disorder. The algorithms provide ground states in polynomial time, which provides for a more precise study of the interface widths than from Monte Carlo simulations at finite temperature. Using $d=2$ systems up to size $420^2$, with a minimum of $2 \times 10^3$ realizations at each size, we find very strong evidence for a $\ln^2(L)$ super-rough state at low temperatures.Comment: 10 pages, 3 PS figures, to appear in PR

Trees and Matchings

In this article, Temperley's bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph G can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related planar graph H. One special case of this result is a bijection between perfect matchings of the hexagonal honeycomb lattice and directed spanning trees of a triangular lattice. Another special case gives a correspondence between perfect matchings of the ``square-octagon'' lattice and directed weighted spanning trees on a directed weighted version of the cartesian lattice. In conjunction with results of Kenyon, our main theorem allows us to compute the measures of all cylinder events for random spanning trees on any (directed, weighted) planar graph. Conversely, in cases where the perfect matching model arises from a tree model, Wilson's algorithm allows us to quickly generate random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1

A note on dimer models and McKay quivers

We give one formulation of an algorithm of Hanany and Vegh which takes a lattice polygon as an input and produces a set of isoradial dimer models. We study the case of lattice triangles in detail and discuss the relation with coamoebas following Feng, He, Kennaway and Vafa.Comment: 25 pages, 35 figures. v3:completely rewritte