Combinatorial bounds for the v-number of binomial edge ideals

@misc{liwski2026combinatorial,
  title = {{Combinatorial bounds for the v-number of binomial edge ideals}},
  author = {Emiliano Liwski},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18213},
  eprint = {2610.18213},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18213},
}

TY  - UNPB
TI  - Combinatorial bounds for the v-number of binomial edge ideals
AU  - Emiliano Liwski
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.18213
UR  - https://arxiv.org/abs/2610.18213
AB  - We study the local v-numbers of binomial edge ideals from a combinatorial perspective. For each minimal prime P_𝒮 of J_G, indexed by a cutset 𝒮 of G, we consider two combinatorial invariants, v^-_𝒮(G) and v^+_𝒮(G), defined in terms of transversals of suitable hypergraphs, and show that the local v-number v_P_𝒮(J_G) satisfies v^-_𝒮(G)≤ v_P_𝒮(J_G)≤ v^+_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 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 v^-_𝒮(G) and v^+_𝒮(G), thereby showing that the local v-number lies within a bounded interval. Applications to cycles and trees illustrate both the equality criterion and the resulting bounds.
ER  -