Codegree and regularity of stable set polytopes

The codegree ${\rm codeg}(\mathcal{P})$ 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 $$\omega(G) + 1 \leq {\rm codeg}(\mathcal{P}_G) \leq \chi(G) + 1,$$ where $\omega(G)$ and $\chi(G)$ denote the clique number and the chromatic number of $G$, respectively. Furthermore, an explicit formula for {\rm codeg}(\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 $\mathcal{P}_G$.