Keywords

• simple graph
• orthogonal representation
• Lovász-Saks-Schrijver ring

ABSTRACT Every simple finite graph $G$ has an associated Lovász-Saks-Schrijver ring $R_G(d)$ that is related to the $d$-dimensional orthogonal representations of $G$. The study of $R_G(d)$ lies at the intersection between algebraic geometry, commutative algebra, and combinatorics. We find a link between algebraic properties, such as factoriality, of $R_G(d)$ and combinatorial invariants of the graph $G$. In particular, we prove that if $d \geqslant \text{pmd}(G)+k(G)+1$, then $R_G(d)$ is UFD. Here, $\text{pmd}(G)$ is the positive matching decomposition number of $G$ and $k(G)$ is its degeneracy number.