标题： 无间隙图和具有线性商的边理想幂 显示英文标题

标题： Gapfree graphs and powers of edge ideals with linear quotients

摘要： 设$I(G)$为无间隙图$G$的边理想。 Nevo 和 Peeva 的一个开放猜想指出，$I(G)^q$对于$q\gg 0$有线性分解。 我们通过研究线性商的更强性质，提出了一个解决这个具有挑战性猜想的有希望的方法。 具体来说，我们提出一个猜想，如果对于某个整数$q\geq 1$，$I(G)^q$具有线性商，则对于所有$s\geq q$，$I(G)^{s}$具有线性商。我们对该猜想给出部分解，并确定仅需检查有限多个幂的条件。 已知如果$G$不包含蟋蟀、钻石或$C_4$，则$I(G)^q$对$q \geq 2$有线性分解。 我们构造了一族不含间隙的图$G$，其中包含蟋蟀、钻石、$C_4$以及$C_5$作为$G$的诱导子图，对于$I(G)^q$在$q \ge 2$下具有线性商。 显示英文摘要

摘要： Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if $I(G)^q$ has linear quotients for some integer $q\geq 1$, then $I(G)^{s}$ has linear quotients for all $s\geq q$. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if $G$ does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has linear resolution for $q \geq 2$. We construct a family of gapfree graphs $G$ containing cricket, diamond, $C_4$ together with $C_5$ as induced subgraphs of $G$ for which $I(G)^q$ has linear quotients for $q \ge 2$.