Combinatorial bounds for the v-number of binomial edge ideals

We study the local $\mathrm{v}$-numbers of binomial edge ideals from a combinatorial perspective. For each minimal prime $P_{\mathcal{S}}$ of $J_G$, indexed by a cutset $\mathcal{S}$ of $G$, we consider two combinatorial invariants, ${\mathrm{v}}_{\mathcal{S}}^{-}(G)$ and ${\mathrm{v}}_{\mathcal{S}}^{+}(G)$, defined in terms of transversals of suitable hypergraphs, and show that the local $\mathrm{v}$-number ${\mathrm{v}}_{P_{\mathcal{S}}}(J_G)$ satisfies ${\mathrm{v}}_{\mathcal{S}}^{-}(G)\le {\mathrm{v}}_{P_{\mathcal{S}}}(J_G)\le {\mathrm{v}}_{\mathcal{S}}^{+}(G)$. Under a suitable hypothesis on the connected components, we provide explicit graph-theoretic descriptions of these invariants and analyze when they coincide, giving a combinatorial way to determine the local $\mathrm{v}$-numbers. As an application, we derive a sufficient condition for this coincidence, which applies to a broad class of graphs. Furthermore, we identify a constant that controls the gap between ${\mathrm{v}}_{\mathcal{S}}^{-}(G)$ and ${\mathrm{v}}_{\mathcal{S}}^{+}(G)$, thereby showing that the local $\mathrm{v}$-number lies within a bounded interval. Applications to cycles and trees illustrate both the equality criterion and the resulting bounds.