The arithmetical rank of the edge ideals of cactus graphs

TL;DR

This paper investigates the arithmetical rank of edge ideals in cactus graphs, establishing bounds and conditions under which these ideals are set-theoretic complete intersections, advancing understanding in combinatorial commutative algebra.

Contribution

It provides a new upper bound for the arithmetical rank of edge ideals in cactus graphs and characterizes when these ideals are set-theoretic complete intersections.

Findings

Bound on arithmetical rank involving cycles and primes

Sharpness of the bound demonstrated

Conditions for edge ideals to be set-theoretic complete intersections

Abstract

We prove that, for the edge ideal of a cactus graph, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes. The bound is sharp, but in many cases it can be improved. Moreover, we show that the edge ideal of a Cohen-Macaulay graph that contains exactly one cycle or is chordal or has no cycles of length 4 and 5 is a set-theoretic complete intersection.