Factoriality 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 factoriality, of \(R_G(d)\) and combinatorial invariants of the graph \(G\). In particular, we prove that if \(d \geq pmd(G)+k(G)+1\), then \(R_G(d)\) is UFD. Here, \(pmd(G)\) is the positive matching decomposition number of \(G\) and \(k(G)\) is its degeneracy number.