1
$\begingroup$

Let $G$ be a finite simple graph on the vertex set $\{x_1, \ldots, x_n\}$ and $I(G) := (\{x_ix_j \mid \{i,j\} \in E(G)\}) \subset R=K[x_1, \ldots, x_n]$ be the edge ideal corresponding to the graph $G$, where $K$ is a field. The Castelnuovo–Mumford regularity (or simply, regularity) $reg(I(G))$ of $I(G)$ is defined as $$reg(I(G))=\max\{j-i \mid Tor^R_i(I(G),K) \neq 0\}$$

(I) no independent set in $G\setminus N_G[x]$ is a maximal independent set in $G \setminus x$.

A vertex $x$ which satisfies Condition (I) is called a shedding vertex.

Suppose $x$ is a shedding vertex. Is $reg(I(G \setminus N_G[x]))+1 \leq reg(I(G))$?

$\endgroup$
2
$\begingroup$

In the case of a shedding vertex $v$, $\operatorname{reg} I(G) = \max \{ \operatorname{reg} I(G \setminus N[v]) + 1, \operatorname{reg} I(G\setminus v) \}$, by a theorem of myself and Tài Hà. So your inequality is true. See Theorem 1.5 of the following.

Hà, Huy Tài; Woodroofe, Russ, Results on the regularity of square-free monomial ideals, Adv. Appl. Math. 58, 21-36 (2014). ZBL1299.13017.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.