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Cohen-macaulay bipartite graphs

Research paper by Mario Estrada, Rafael H. Villarreal

Indexed on: 01 Apr '97Published on: 01 Apr '97Published in: Archiv der Mathematik



Abstract

Let G be a graph on the vertex set V={x1, ..., xn}. Let k be a field and let R be the polynomial ring k[x1, ..., xn]. The graph idealI(G), associated to G, is the ideal of R generated by the set of square-free monomials xixj so that xi, is adjacent to xj. The graph G is Cohen-Macaulay over k if R/I(G) is a Cohen-Macaulay ring.Let G be a Cohen-Macaulay bipartite graph. The main result of this paper shows that G{v} is Cohen-Macaulay for some vertex v in G. Then as a consequence it is shown that the Reisner-Stanley simplicial complex of I(G) is shellable. An example of N. Terai is presented showing these results fail for Cohen-Macaulay non bipartite graphs.