On regularity of symbolic Rees algebras and symbolic powers of vertex cover ideals of graphs.

In this work, we study the bigraded regularities of the symbolic Rees algebras R_s(J(G)), R_s(I(G)), of the vertex cover ideal J(G) and the edge ideal I(G), of a graph G respectively. We give combinatorial upper bounds for the (1,0)-regularities of R_s(J(G)) and R_s(I(G)). By using this upper bounds, we give general linear upper bounds for reg(J(G)^{(k)}), reg(I(G)^{(k)}) for any k\geq 1. Let G be a graph on n vertices and \deg (J(G)) be the maximum degree of minimal generators of J(G). We show that if G is a non-bipartite graph, then \begin{equation*} k \deg (J(G)) \!\leq \! reg(J(G)^{(k)})\!\leq \! k \deg (J(G))+ \alpha _0(G)-1+|A_0\cup \{x_{i_1}, \ldots, x_{i_r}\}|-r, \end{equation*} for all k \geq 1, where \alpha _0(G) denotes the vertex cover number of G, A_0 is a maximal independent set in G of maximal cardinality, and r is the number of 0-covers that are present in an irreducible representation of the affine cone associated with the irreducible covers of G. Also if G is a non-bipartite perfect graph, then \begin{equation*} 2k \leq reg(I(G)^{(k)})\leq 2k+n-r+1, \end{equation*} for all k \geq 1, where r is the number of 0-covers of \Gamma (G) that are present in an irreducible representation of the affine cone associated with the irreducible covers of \Gamma (G). [ABSTRACT FROM AUTHOR]