Hankel edge ideals of trees and (semi-)Hamiltonian graphs

TL;DR

This paper investigates algebraic properties of Hankel edge ideals associated with various classes of graphs, including Hamiltonian, semi-Hamiltonian, and rooted trees, providing characterizations of their prime ideals and intersection properties.

Contribution

It offers new characterizations of Hankel edge ideals for specific graph classes, including conditions for being a complete intersection and descriptions of their prime ideals.

Findings

Minimal prime ideals of Hankel edge ideals are determined for Hamiltonian and semi-Hamiltonian graphs.

Conditions under which Hankel edge ideals are complete intersections are characterized for rooted trees.

The properties of initial ideals with respect to reverse lexicographic order are analyzed for these graph classes.

Abstract

In this paper, we study the Hankel edge ideals of graphs. We determine the minimal prime ideals of the Hankel edge ideal of labeled Hamiltonian and semi-Hamiltonian graphs, and we investigate radicality, being a complete intersection, almost complete intersection and set theoretic complete intersection for such graphs. We also consider the Hankel edge ideal of trees with a natural labeling, called rooted labeling. We characterize such trees whose Hankel edge ideal is a complete intersection, and moreover, we determine those whose initial ideal with respect to the reverse lexicographic order satisfies this property.