Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles

TL;DR

This paper confirms a conjecture by precisely calculating the Castelnuovo-Mumford regularity of squarefree powers of edge ideals in whiskered cycles, revealing a specific formula for all relevant powers.

Contribution

It provides the exact regularity values for squarefree powers of edge ideals in whiskered cycles, confirming a previously conjectured formula.

Findings

Confirmed the conjectured regularity formula for whiskered cycles.

Derived explicit regularity values for all squarefree powers up to the matching number.

Enhanced understanding of algebraic invariants in graph theory contexts.

Abstract

Let $G$ be a finite simple graph and let $I(G)$ denote its edge ideal. For $q \ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by squarefree monomials corresponding to matchings of size $q$ in $G$. We denote by $\operatorname{reg}(-)$ the Castelnuovo-Mumford regularity. Das, Roy, and Saha conjectured that if $G = W(C_n)$ is a whiskered cycle, then \[ \operatorname{reg}\big(I(G)^{[q]}\big) = 2q + \left\lfloor \frac{n - q - 1}{2} \right\rfloor ~ \text{for all } 1 \le q \le \nu(G), \] where $\nu(G)$ denotes the matching number of $G$. In this paper, we confirm this conjecture by determining the exact value of $\operatorname{reg}(I(G)^{[q]})$.