S. Selvaraja, Joseph W. Skelton, Componentwise Linearity of Powers of Cover Ideals, Journal of Algebraic Combinatorics Vol 57 (2023)

• Let G be a finite simple graph and J(G) denote its vertex cover ideal in a polynomial ring over a field. Assume that J(G)^(k) is its k-th symbolic power. In this paper, we give a criteria for cover ideals of vertex decomposable graphs to have the property that all their symbolic powers are not componentwise linear. Also, we give a necessary and sufficient condition on G so that J(G)^(k) is a componentwise linear ideal for some (equivalently, for all) k≥2 when G is a graph such that G∖ N[A] has a simplicial vertex for any independent set A of G. Using this result, we prove that J(G)^(k) is a componentwise linear ideal for several classes of graphs for all k≥2. In particular, if G is a bipartite graph, then J(G) is a componentwise linear ideal if and only if J(G)^k is a componentwise linear ideal for some (equivalently, for all) k≥2.
  
  
  

Yan Gu, Huy Tài Hà, Joseph W. Skelton, Symbolic Powers of Cover Ideals of Graphs and Koszul Property, International J. of Alg. and Comp. Vol 31 (2021) 865 - 881

• We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

Yan Gu, Huy Tài Hà, Jonathan O'Rourke, Joseph W. Skelton, Symbolic powers of edge ideals of graphs, Communications in Algebra, Vol 48 (2020) 3743-3760 

• Let G be a graph and let I = I(G) be its edge ideal. When G is unicyclic, we give a decomposition of symbolic powers of I in terms of its ordinary powers. This allows us to explicitly compute the Waldschmidt constant and the resurgence number of I. When G is an odd cycle, we explicitly compute the regularity of I^(s) for all s ∈ N. In doing so, we also give a natural lower bound for the regularity function reg I^(s), for s ∈ N, for an arbitrary graph G.