for each squarefree monomial ideal I ⊂ S = k[x₁, …, x_n], we associate a simple finite graph G_I by using the first linear syzygies of I. The nodes of G_I 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. In the cases, where G_I is a cycle or a tree, we show that I has a linear resolution if and only if I has linear quotients and if and only if I is variable-decomposable. In addition, with the same assumption on G_I, we characterize all squarefree monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension 2 monomial ideals with a linear resolution. As another application of our results, we also characterize all Cohen-Macaulay simplicial complexes in the case, where \({G_{\rm{\Delta }}} \cong {G_{{I_{\rm{\Delta }}} \vee }}\) is a cycle or a tree.

对于每个无平方单项式理想I ⊂ S = k [ x ₁ , …, x _n ]，我们通过使用_I的第一个线性合子关联一个简单的有限图G I。的节点ģ_我是发电机予，和两个顶点ü_我和ü _Ĵ如果存在变量是相邻的X，Y，使得许_我=宇_Ĵ。在G _I是环或树的情况下，我们证明I具有线性分辨率当且仅当I具有线性商且当且仅当I是可变可分解的。此外，在对G _I的相同假设下，我们用线性分辨率表征所有无平方单项式理想。使用我们的结果，我们以线性分辨率表征所有 Cohen-Macaulay 共维 2 单项式理想。作为我们结果的另一个应用，我们还刻画了这种情况下的所有 Cohen-Macaulay 单纯复形，其中\({G_{\rm{\Delta }}} \cong {G_{{I_{\rm{\Delta }}} \vee }}\)是一个循环或一棵树。