k-neighborhood ideals of graphs

In this paper, we introduce and investigate the $\textbf{k}$-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let $G$ be a simple graph on the vertex set $[n]$, and let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$. For a vector $\textbf{k}=(k_1,\ldots,k_n)\in \mathbb{N}^n$ satisfying $1\leq k_i\leq \textrm{deg}_G(i)+1$ for all $i$, the $\textbf{k}$-neighborhood ideal of $G$ is defined as the squarefree monomial ideal $$\textrm{NI}_{\textbf{k}}(G)=\sum_{i=1}^n\, (\textbf{x}_W:\, W\subseteq N_G[i],\, |W|=k_i)$$ of $S$, where $\textbf{x}_W=\prod_{i\in W} x_i$. We study homological invariants and properties of $\textrm{NI}_{\textbf{k}}(G)$ focusing on its Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness. Special attention is devoted to the case where the vector ${\textbf{k}}$ is the degree-vector of the graph, i.e., $k_i=\textrm{deg}_G(i)$ for all vertices $i$, and to the case where $\textrm{NI}_{\textbf{k}}(G)$ coincides with the edge ideal of a graph. In these settings, we provide combinatorial characterizations and bounds for the regularity and projective dimension of $\textrm{NI}_{\textbf{k}}(G)$ for several classes of graphs, and further investigate the Cohen-Macaulay property of these ideals.