论文标题
关于Eliahou和Villarrear的猜想，关于共同图形的投影维度
On the Eliahou and Villarreal conjecture about the projective dimension of co-chordal graphs
论文作者
论文摘要
令$ i（g）$成为图形$ g $的边缘理想，$ | v（g）| = n $和$ r = \ mathbb {k} [x \ mid x \ in v（g）] $是$ n $变量的多项式戒指。在本文中，我们对Eliahou和Villarreal的猜想感兴趣，该猜想指出，$ \ text {pdim}（r/i（g））= \ max_ {1 \ leq i \ leq i \ leq n} \ lest \ left \ left \ left \ left {\ text {deg} _ {g} _ {g} _ {__i）\ right text \ right right tectect and $ right right y $ g $ n $ n $ n $ n $ n $ n $ s $我们表明，这个猜想一般不正确。实际上，我们表明$ \ text {pdim}（r/i（g））$和$ \ max_ {1 \ leq i \ leq i \ leq n} \ left \ left \ {\ text {deg} _ {g} _ {g}（x_i）\ right \} $之间的差异。对于任何图形$ g $，我们证明$ \ max_ {1 \ leq i \ leq n} \ left \ {\ text {deg} _ {g}（x_i）（x_i）\ right \} \ leq \ leq \ leq \ text {pdim}（pdim}（r/i（g））$。对于正整数的非插入顺序$（d_1，d_2，\ dots，d_q）$，我们定义$（d_1，d_2，\ dots，d_q）$ - 树图。我们表明，这类树的补充的独立性复合物是顶点分解和准森林。最后，我们证明，当$ g $的补充是$（d_1，d_2，\ dots，d_q）$ - 树或$ g $时，猜想是有效的。据我们所知，本文中的结果概括了猜想是正确的所有现有图形类别。
Let $I(G)$ be the edge ideal of a graph $G$ with $|V(G)|=n$ and $R=\mathbb{K}[x\mid x\in V(G)]$ be a polynomial ring in $n$ variables over a field $\mathbb{K}$. In this paper we are interested in a conjecture of Eliahou and Villarreal which states that $\text{pdim}(R/I(G))=\max_{1\leq i \leq n}\left\{\text{deg}_{G}(x_i)\right\}$ when $G$ is connected and co-chordal. We show that this conjecture is not true in general. In fact we show that the difference between $\text{pdim}(R/I(G))$ and $\max_{1\leq i \leq n}\left\{\text{deg}_{G}(x_i)\right\}$ is not necessarily bounded. For any graph $G$ we prove that $\max_{1\leq i \leq n}\left\{\text{deg}_{G}(x_i)\right\}\leq \text{pdim}(R/I(G))$. For a non-increasing sequence of positive integers $(d_1,d_2,\dots,d_q)$, we define the $(d_1,d_2,\dots,d_q)$-tree graphs. We show that the independence complex of the complements of these type of trees is vertex decomposable and quasi-forest. Finally we show that the conjecture is valid when the complement of $G$ is a $(d_1,d_2,\dots,d_q)$-tree or $G$ has a full-vertex. To our knowledge the results in this paper generalise all the existing classes of graphs for which the conjecture is true.
