The $$\mathrm {v}$$-number of monomial ideals

Abstract

We show that the \(\mathrm {v}\)-number of an arbitrary monomial ideal is bounded below by the \(\mathrm {v}\)-number of its polarization and also find a criteria for the equality. By showing the additivity of associated primes of monomial ideals, we obtain the additivity of the v-numbers for arbitrary monomial ideals. We prove that the \(\mathrm {v}\)-number \(\mathrm {v}(I(G))\) of the edge ideal I(G), the induced matching number \(\mathrm {im}(G)\) and the regularity \(\mathrm {reg}(R/I(G))\) of a graph G, satisfy \(\mathrm {v}(I(G))\le \mathrm {im}(G)\le \mathrm {reg}(R/I(G))\), where G is either a bipartite graph, or a \((C_{4},C_{5})\)-free vertex decomposable graph, or a whisker graph. There is an open problem in Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021), whether \(\mathrm {v}(I)\le \mathrm {reg}(R/I)+1\), for any square-free monomial ideal I. We show that \(\mathrm {v}(I(G))>\mathrm {reg}(R/I(G))+1\), for a disconnected graph G. We derive some inequalities of \(\mathrm {v}\)-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect \(\mathrm {v}(I(G))\) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that \(\mathrm {reg}(R/I(G))\) can be arbitrarily larger than \(\mathrm {v}(I(G))\). Also, we try to see how the \(\mathrm {v}\)-number is related to the Cohen–Macaulay property of square-free monomial ideals.