Dirac's theorem and multigraded syzygies

TL;DR

This paper explores algebraic properties of homological shift ideals of edge ideals in graphs, linking graph-theoretic structures to algebraic resolutions, and proves specific cases of a conjecture on polymatroidal ideals.

Contribution

It establishes that the first homological shift ideal of any monomial ideal with linear quotients has linear quotients, and proves the polymatroidal conjecture for degree-two ideals.

Findings

$ ext{HS}_1(I)$ has linear quotients for monomial ideals with linear quotients.

$ ext{HS}_k(I(G))$ has linear quotients for certain graph classes like proper interval graphs and forests.

The conjecture that polymatroidal ideals have polymatroidal homological shifts is proved for degree-two cases.

Abstract

Let $G$ be a simple finite graph. A famous theorem of Dirac says that $G$ is chordal if and only if $G$ admits a perfect elimination order. It is known by Fr\"oberg that the edge ideal $I(G)$ of $G$ has a linear resolution if and only if the complementary graph $G^c$ of $G$ is chordal. In this article, we discuss some algebraic consequences of Dirac's theorem in the theory of homological shift ideals of edge ideals. Recall that if $I$ is a monomial ideal, $\mbox{HS}_k(I)$ is the monomial ideal generated by the $k$th multigraded shifts of $I$. We prove that $\mbox{HS}_1(I)$ has linear quotients, for any monomial ideal $I$ with linear quotients generated in a single degree. For and edge ideal $I(G)$ with linear quotients, it is not true that $\mbox{HS}_k(I(G))$ has linear quotients for all $k\ge0$. On the other hand, if $G^c$ is a proper interval graph or a forest, we prove that this is the case. Finally, we discuss a conjecture of Bandari, Bayati and Herzog that predicts that if $I$ is polymatroidal, $\mbox{HS}_k(I)$ is polymatroidal too, for all $k\ge0$. We are able to prove that this conjecture holds for all polymatroidal ideals generated in degree two.