The v-number bound for closed neighborhood ideals

Let $G$ be a simple connected graph, let $S$ be the polynomial ring associated with $G$, and let $mathtt{v}(\mathcal{N}_G)$ denote the $\mathtt{v}$-number of its closed neighborhood ideal $\mathcal{N}_G$. Let $\mathtt{reg}(S/\mathcal{N}_G)$ denote the Castelnuovo–Mumford regularity of the quotient.

The v-number bound. For every simple connected graph $G$,

$$\mathtt{v}(\mathcal{N}_G)\leq \mathtt{reg}(S/\mathcal{N}_G).$$

The preceding theorem establishes this inequality for bipartite, very well-covered, and chordal graphs, and in particular for trees. The conjecture extends that phenomenon to all simple connected graphs; its general status is not resolved in the supplied text.