2023/08/19 by Saha, Kamalesh
#05C69 #05E40 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13F20 #Secondary 13C70
paper · doi:10.48550/arxiv.2308.09996
The v-number of a graded ideal I⊆ R, denoted by v(I), is the minimum degree of a polynomial f for which I:f is a prime ideal. Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021) studied the v-number of edge ideals. In this paper, we study the v-number of the cover ideal J(G) of a graph G. The main result shows that v(J(G))≤ reg(R/J(G)) for any simple graph G, which is quite surprising because, for the case of edge ideals, this inequality does not hold. Our main result relates v(J(G)) with the Cohen-Macaulay property of R/I(G). We provide an infinite class of connected graphs, which satisfy v(J(G))=reg(R/J(G)). Also, we show that for every positive integer k, there exists a connected graph Gk such that reg(R/J(Gk))-v(J(Gk))=k. Also, we explicitly compute the v-number of cover ideals of cycles.