Complementary edge ideals

TL;DR

This paper studies the algebraic and homological properties of a special class of squarefree monomial ideals called complementary edge ideals, linking their properties to graph combinatorics and analyzing their powers' resolutions.

Contribution

It introduces the concept of complementary edge ideals, establishes a correspondence between algebraic invariants and graph properties, and characterizes when their powers have linear resolutions.

Findings

Characterization of Cohen-Macaulay and Gorenstein properties via graph structure.

Determination of regularity of powers in terms of graph invariants.

Identification of conditions for linear resolutions of powers.

Abstract

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as established in [11], in this work we investigate the algebraic and homological properties of $I$ and its powers. To this end, we introduce the complementary edge ideal of a finite simple graph $G$ as the ideal $$I_c(G)=((x_1\cdots x_n)/(x_ix_j):\{i,j\}\in E(G)) $$ of $S$, where $V(G)=\{1,\ldots,n\}$ and $E(G)$ is the edge set of $G$. By interpreting any squarefree monomial ideal $I$ generated in degree $n-2$ as the complementary edge ideal of a graph $G$, we establish a correspondence between algebraic invariants of $I$ and combinatorial properties of $G$. More precisely, we characterize sequentially Cohen-Macaulay, Cohen-Macaulay, Gorenstein, nearly Gorenstein and matroidal complementary edge ideals. Moreover, we determine the regularity of powers of $I$ in terms of combinatorial invariants of the graph $G$ and obtain that $I^k$ has linear resolution or linear quotients for some $k$ (equivalently for all $k\geq 1$) if and only if $G$ has only one connected component with at least two vertices.