Castelnuovo–Mumford regularity of the closed neighborhood ideal of a graph

Abstract

Let G be a finite simple graph, and let NI(G) denote the closed neighborhood ideal of G in a polynomial ring R. We show that if G is a forest, then the Castelnuovo–Mumford regularity of R/NI(G) is the same as the matching number of G, thus proving a conjecture of Sharifan and Moradi in the affirmative. We also show that the matching number of G provides a lower bound for the Castelnuovo–Mumford regularity of R/NI(G) for any G. Furthermore, we prove that if G contains a simplicial vertex, then NI(G) admits a Betti splitting, and consequently, we show that the projective dimension of R/NI(G) is also bounded below by the matching number of G, if G is a forest or a unicyclic graph. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.