COHEN-MACAULAY BIPARTITE GRAPHS

Let G be a graph on the vertex set V = {x(1),...,x(n)}. Let k be a fie ld and let R be the polynomial ring k[x(1),...,x(n)]. The graph ideal I(G), associated to G, is the ideal of R generated by the set of squar e-free monomials x(i)x(j) so that x(i) is adjacent to x(j). The graph G is Cohen-Macaulay over k if R/I(G) is a Cohen-Macaulay ring. Let G b e a Cohen-Macaulay bipartite graph. The main result of this paper show s that G\{nu} is Cohen-Macaulay for some vertex nu in G. Then as a con sequence it is shown that the Reisner-Stanley simplicial complex of I( G) is shellable. An example of N. Terai is presented showing these res ults fail for Cohen-Macaulay non bipartite graphs.