标题： 稳定集多面体的代码数和正则性 显示英文标题

标题： Codegree and regularity of stable set polytopes

摘要： 码度${\rm codeg}(P)$的格子多面体$P$是离散几何中的基本不变量。 在本文中，我们研究与图$G$相关的稳定集多面体$P_G$的码度。 具体来说，我们建立了不等式\[ \omega(G) + 1 \leq {\rm codeg}(P_G) \leq \chi(G) + 1, \]，其中$\omega(G)$和$\chi(G)$分别表示 G 的团数和色数。 此外，当G是线图或$h$-完美图时，给出了${\rm codeg}(P_G)$的显式公式。 最后，作为这些结果的应用，我们提供了与$P_G$相关的环的正则性的上下界。 显示英文摘要

摘要： The codegree ${\rm codeg}(P)$ of a lattice polytope $P$ is a fundamental invariant in discrete geometry. In the present paper, we investigate the codegree of the stable set polytope $P_G$ associated with a graph $G$. Specifically, we establish the inequalities \[ \omega(G) + 1 \leq {\rm codeg}(P_G) \leq \chi(G) + 1, \] where $\omega(G)$ and $\chi(G)$ denote the clique number and the chromatic number of G, respectively. Furthermore, an explicit formula for ${\rm codeg}(P_G)$ is given when G is either a line graph or an $h$-perfect graph. Finally, as an application of these results, we provide upper and lower bounds on the regularity of the toric ring associated with $P_G$.