Normality, factoriality and strong $F$-regularity of Lovász-Saks-Schrijver rings

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 normality, factoriality and strong $F$-regularity of $R_G(d)$ and combinatorial invariants of the graph $G$. In particular we prove that if $d\ge \text{pmd}(G)+k(G)$ then $R_G(d)$ is $F$-regular in finite characteristic and rational singularity in characteristic $0$ and furthermore if $d\ge \text{pmd}(G)+k(G)+1$ 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.