Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Hard Squares with Negative Activity on Cylinders with Odd Circumference. | Hard Squares with Negative Activity on Cylinders with Odd Circumference Jakob Jonsson Department of Mathematics KTH Stockholm Sweden jakobj@ Submitted Sep 30 2008 Accepted Mar 13 2009 Published Mar 23 2009 Mathematics Subject Classification 05A15 05C69 52C20 Dedicated to Anders Bjorner on the occasion of his 60th birthday. Abstract Let Cmn be the graph on the vertex set 1 . m X 0 . n 1 in which there is an edge between a b and c d if and only if either a b c d 1 or a b c 1 d where the second index is computed modulo n. One may view Cmn as a unit square grid on a cylinder with circumference n units. For odd n we prove that the Euler characteristic of the simplicial complex Am n of independent sets in Cmn is either 2 or 1 depending on whether or not gcd m 1 n is divisble by 3. The proof relies heavily on previous work due to Thapper who reduced the problem of computing the Euler characteristic of sm n to that of analyzing a certain subfamily of sets with attractive properties. The situation for even n remains unclear. In the language of statistical mechanics the reduced Euler characteristic of Am n coincides with minus the partition function of the corresponding hard square model with activity 1. 1 Introduction An independent set in a simple and loopless graph G is a subset of the vertex set of G with the property that no two vertices in the subset are adjacent. The family of independent sets in G forms a simplicial complex the independence complex E G of G. The purpose of this paper is to analyze the independence complex of square grids with cylindrical boundary conditions. Specifically define Cm to be the graph with vertex set Research financed by the Swedish Research Council. Part of this research was carried out at the Erwin Schrodinger International Institute for Mathematical Physics in Vienna within the programme Combinatorics and Statistical Physics. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R5 1 m X Z and with an edge between a b and c d if and
