Báo cáo toán học: "Gorenstein polytopes obtained from bipartite graphs"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Gorenstein polytopes obtained from bipartite graphs. | Gorenstein polytopes obtained from bipartite graphs Makoto Tagami Graduate School of Science Tohoku University Aoba Sendai JAPAN 980-8578 tagami@ Submitted Jun 29 2009 Accepted Dec 14 2009 Published Jan 5 2010 Mathematics Subject Classification 52B20 Abstract Beck et al. characterized the grid graphs whose perfect matching polytopes are Gorenstein and they also showed that for some parameters perfect matching polytopes of torus graphs are Gorenstein. In this paper we complement their result that is we characterize the torus graphs whose perfect matching polytopes are Gorenstein. Beck et al. also gave a method to construct an infinite family of Goren-stein polytopes. In this paper we introduce a new class of polytopes obtained from graphs and we extend their method to construct many more Gorenstein polytopes. Keywords Gorenstein polytopes Perfect matching polytopes Torus graphs Bipartite graphs. 1 Introduction Lattice polytopes are polytopes whose vertices all are lattice points. N denotes the set of positive integers. For S c Rn and t G N we put tS tx x G S and LS t tS n Zn . Ehrhart 6 proved that for a d-dimensional lattice polytope P LP t is always a polynomial of degree d in t. LP t is called the Ehrhart polynomial of P. Also the formal power series EhrP z 1 y . LP t z is called the Ehrhart series of P. Since LP t is a polynomial of degree d the Ehrhart series of P can be written as the rational function E z r 7 where s d. s and r d 1 s are called the degree and codegree of P respectively. The polynomial of the numerator is called the h -polynomial of P. It is well-known that h0 1 and the codegree r is equal to the minimal integer t for which tP contains a lattice point and hs rP n Zn . Here for S c Rn S denotes the relative interior THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R8 1 of S. As a general reference on the Ehrhart theory of lattice polytopes we refer to the recent book of Matthias Beck and Sinai Robins 3 and the references within.

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