On Linear Quotients of Squarefree Monomial Ideals

Let I subset of K[x(1),..., x(n)] be a squarefree monomial ideal generated in one degree. Let G(I) be the graph whose nodes are the generators of I, and two vertices u(i) and u(j) are adjacent if there exist variables x, y such that xu(i) = yu(j). We show that if I is (i) G(I) is a connected graph; (ii) I has a (n - 2)-linear resolution; (iii) I has linear quotients; (iv) I is a variable-decomposable ideal. We also prove that if I has linear relations and (G(I)) over bar is chordal, then I has linear quotients.