An upper bound for the regularity of symbolic powers of edge ideals of chordal graphs
Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$variables over $\mathbb{K}$. Let $G$ be a finite simple graph with $V(G)=\{x_1,\dots,x_n\}$ and edge set $E(G)$ and let $I=I(G)$ be the edge ideal defined by $I(G)=(x_ix_j: x_ix_j\in E(G))\subseteq S$. Let $M=\{\{a_i,b_i\}| 1\leq i\leq r\}$ be a nonempty matching of $G$. $M$ is an ordered matching of $G$ if the following hold: \begin{itemize} \item[(1)] $A=\{a_1,\dots,a_r\}\subseteq V(G)$ is a set of independent vertices of $G$; and \item[(2)] $\{a_i,b_i\}\in E(G)$ implies that $i\leq j$. \end{itemize} The ordered matching number of $G$, denoted by ord-match $(G)$, is defined to be ord-match$(G)=\max\{|M|~| M\subseteq E(G)$ is an ordered matching of $G\}$. \par In this paper under review, the author shows that if $G$ is a chordal graph, then the Castelnuovo-Mumford regularity $\mathrm{reg}(I(G)^{(s)})\leq 2s$+ ord-match$(G)-1$. As a consequence, he determines the regularity of symbolic powers of edge ideals of chordal Cameron-Walker graphs.
scientific article; zbMATH DE number 7049110
scientific article