Powers of the vertex cover ideals (Q486331)

Let \(G\) be a graph with the vertex set \(V(G)=\{1,\ldots,n\}\), \(\Delta(G)\) its clique complex and \(I(G)\subseteq k[x_1,\ldots,x_n]\) its vertex cover ideal. The author shows that if \(G\) has enough free vertices (that are vertices that belongs to one clique only) then all the powers of the vertex cover ideal \(I(G)\) have linear quotients, therefore they are componentwise linear. For the class of cactus graphs, the author completely characterises those graphs which are sequentially Cohen-Macaulay. The characterization is expressed in terms of the combinatorics of the graph.