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

Abstract:

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 $\Gamma (G)$.