Type: Article

Cohen-Macaulayness of triangular graphs

de Alba, Hernán; Carballosa, Walter; Duarte, Daniel; Rivera, Luis Manuel;

Abstract:

Let R=K[x1,…,xN] be the polynomial ring over K, where K is any field. Let G be a simple graph with vertex set V(G)={v1,…,vN} and edge set E(G). The edge ideal I(G) of G is the ideal ?xixj:{vi,vj}?E(G)?. The graph G is called Cohen-Macaulay over K if R/I(G) is a Cohen-Macaulay ring. The triangular graph Tn is the simple graph whose vertices are the 2-subsets of an n-set, n?2. In the paper under review, the authors study the Cohen-Macaulayness of triangular graphs Tn. They show that T2, T3 and T5 are Cohen-Macaulay graphs, and that T4, T6, T8 and Tn are not Cohen-Macaulay graphs, for n?10. In addition, the authors prove that over fields of characteristic zero, T7 and T9 are Cohen-Macaulay.