Codegree and regularity of stable set polytopes

The codegree $\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{g}(\mathcal{P})$ of a lattice polytope $\mathcal{P}$ is a fundamental invariant in discrete geometry. In the present paper, we investigate the codegree of the stable set polytope ${\mathcal{P}}_G$ associated with a simple graph $G$. Specifically, we establish the inequalities $$\omega (G)+1\le \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{g}({\mathcal{P}}_G)\le \chi (G)+1,$$ where $\omega (G)$ and $\chi (G)$ denote the clique number and the chromatic number of $G$, respectively. Furthermore, an explicit formula for {\rm codeg}($\mathcal{P}$_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 ${\mathcal{P}}_G$.