Toric rings arising from vertex cover ideals (Q6056549)

\textit{B. Sturmfels} [Gröbner bases and convex polytopes. Providece, RI: AMS, American Mathematical Society (1996; Zbl 0856.13020)] considered a \textit{sortable} set of monomials in a polynomial ring \(S=K[x_1, \ldots, x_n]\) for a set of monomials of the same degree, which is a powerful property in the study of toric rings. The main aim of this work is to study the concept of sortability for a set of monomials in arbitrary degrees and also to explore the toric rings of the form \(A = K[u_1t, \ldots, u_mt]\), where \(\{u_1, \ldots, u_m\}\) is a sortable set of monomials. Moreover, the authors try to find Koszul and normal Cohen-Macaulay toric rings, whose generators are in arbitrary degrees. In particular, they show the following result: {Theorem 2.1.} Let \(G\) be a proper interval graph on \([n]\) and \(I_G\) be the cover ideal of \(G\). Then \(I_G\) is a sortable ideal. The authors, in addition, focus on some families of graphs so that their vertex cover ideals have componentwise linear powers, to see some well-known classes of these graphs, the reader can refer to [\textit{N. Erey}, J. Pure Appl. Algebra 223, No. 7, 3071--3080 (2019; Zbl 1431.05154); \textit{A. Kumar} and \textit{R. Kumar}, J. Pure Appl. Algebra 226, No. 1, Article ID 106808, 10 p. (2022; Zbl 1468.13026)], and [\textit{F. Mohammadi}, Commun. Algebra 39, No. 10, 3753--3764 (2011; Zbl 1236.13001)]. In particular, the authors attach to an arbitrary graph \(G\) a family of graphs called \textit{clique multi-whiskerings} of \(G\) and prove the following results: {Lemma 3.2.} The ideal \(J_{G^{\pi}(r_1, \ldots, r_m)}\) satisfies the \(x\)-condition with respect to \(<\). {Theorem 3.4.} Let \(G\) be a graph, \(\pi\) be an arbitrary clique partition of \(G\) (to \(m\) parts) and \(r_1, \ldots, r_m\) be positive integers. Then all powers of the vertex cover ideal \(I_{G^{\pi}(r_1, \ldots, r_m)}\) have linear quotients and hence are componentwise linear.