The $\mathrm{v}$-number of Monomial Ideals

Abstract

We generalize some results of v-number for arbitrary monomial ideals by showing that the v-number of an arbitrary monomial ideal is the same as the v-number of its polarization. We prove that the vnumber v(I(G)) of the edge ideal I(G), the induced matching number im(G) and the regularity reg, 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 [16], 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.

The $v$-number and Castelnuovo-Mumford regularity of graphs

“…For every n ě 3, notice the following isomorphisms (11) G (11) together with the induction that regpG n ´a1 q " regpG n´1 q `2 ď 2pn ´1q `1 `2 " 2n `1, regpG n ´NGn ra 1 sq " 2n. Thus, we conclude that regpG n q ď 2n `1 for each n ě 2 by Lemma 3.…”

“…The subject of two recent papers [7,11] is the comparison of the v-number and the regularity of graphs. Under suitable restrictions, it is proved that the v-number of a graph G provides a lower bound to regpGq.…”