Homological shifts of powers a complementary edge ideal

TL;DR

This paper explores the homological properties and projective dimensions of powers of complementary edge ideals in graphs, providing explicit formulas and structural descriptions for bipartite graphs, trees, and cycles.

Contribution

It introduces new explicit formulas for the projective dimension of powers of complementary edge ideals and characterizes their homological shift algebras for specific graph classes.

Findings

Projective dimension of $I^s$ increases strictly with $s$ in bipartite graphs.

Homological shift algebra generators are bounded in degree by $i$ for certain ideals.

Homological shift ideals of trees relate to Veronese-type ideals.

Abstract

The homological shift algebra and the projective dimension function of complementary edge ideals are investigated. Let $G$ be a connected graph, and let $I$ be its complementary edge ideal. For bipartite graphs $G$, we show that the projective dimension of $I^s$ increases strictly with $s$ until reaching its maximum value. For trees and cycles, explicit expressions for the projective dimension of $I^s$ are provided, along with detailed descriptions of their homological shift algebras. In particular, it is shown that the $i$-th homological shift algebra of such ideals is generated in degree at most $i$. Additionally, we prove that if $G$ is a tree, then the homological shift ideal $\mathrm{HS}_i(I^i)$, when divided by a suitable monomial, becomes a Veronese-type ideal, and every Veronese-type ideal arises in this manner.