The v -number of monomial ideals

We show that the v -number of an arbitrary monomial ideal is bounded below by the 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 v -number v (I(G)) of the edge ideal I(G), the induced matching number im (G) and the regularity reg (R/ I(G)) of a graph G, satisfy v (I(G)) ≤ im (G) ≤ reg (R/ I(G)) , where G is either a bipartite graph, or a (C<inf>4</inf>, C<inf>5</inf>) -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 v (I) ≤ reg (R/ I) + 1 , for any square-free monomial ideal I. We show that v (I(G)) > reg (R/ I(G)) + 1 , for a disconnected graph G. We derive some inequalities of v -numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v (I(G)) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that reg (R/ I(G)) can be arbitrarily larger than v (I(G)). Also, we try to see how the v -number is related to the Cohen–Macaulay property of square-free monomial ideals.