Codegree and regularity of stable set polytopes

TL;DR

This paper explores the codegree of stable set polytopes in graphs, establishing bounds related to graph invariants and providing explicit formulas for certain graph classes, with applications to toric ring regularity.

Contribution

It introduces bounds for the codegree of stable set polytopes based on graph invariants and derives explicit formulas for line and h-perfect graphs, linking polytope geometry to algebraic properties.

Findings

Bounds on codegree in terms of clique and chromatic numbers.

Explicit formula for codegree in line and h-perfect graphs.

Bounds on the regularity of the associated toric ring.

Abstract

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 $\mathcal{P}_G$.