The v-Number of Binomial Edge Ideals

The invariant \(\textrm{v}\)-number was introduced very recently in the study of Reed-Muller-type codes. Jaramillo and Villarreal (J. Combin. Theory Ser. A 177:105310, 2021) initiated the study of the \(\textrm{v}\)-number of edge ideals. Inspired by their work, we take the initiation to study the \(\textrm{v}\)-number of binomial edge ideals in this paper. We discuss some properties and bounds of the \(\textrm{v}\)-number of binomial edge ideals. We explicitly find the \(\textrm{v}\)-number of binomial edge ideals locally at the associated prime corresponding to the cutset \(\emptyset \). We show that the \(\textrm{v}\)-number of Knutson binomial edge ideals is less than or equal to the \(\textrm{v}\)-number of their initial ideals. Also, we classify all binomial edge ideals whose \(\textrm{v}\)-number is 1. Moreover, we try to relate the \(\textrm{v}\)-number with the Castelnuvo-Mumford regularity of binomial edge ideals and give a conjecture in this direction.