Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles

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 ν(G), ] where $ν(G)$ denotes the matching number of $G$. In this paper, we confirm this conjecture by determining the exact value of $\operatorname{reg}(I(G)^{[q]})$.

• The paper confirms a conjecture by Das, Roy, and Saha, providing an explicit formula for the regularity of squarefree powers in whiskered cycles.
• It employs recursive decomposition, even-connection analysis, and precise generator ordering to bridge lower and upper bounds.
• These results deepen understanding of algebraic invariants in combinatorial commutative algebra and may extend to broader graph classes.

Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles

Introduction

This paper addresses the Castelnuovo-Mumford regularity of the $q$-th squarefree power of the edge ideal $I(G)$ when $G$ is the whiskered cycle $W(C_n)$. The regularity of powers and squarefree powers of monomial edge ideals is a central invariant in combinatorial commutative algebra, closely linked to the syzygetic complexity of the ideal and the combinatorics of the underlying graph. The authors confirm a conjecture of Das, Roy, and Saha [DRS24], giving an explicit formula for this regularity as a function of $n$ and $q$.

Mathematical Context and Definitions

Let $G$ be a finite simple graph on $n$ vertices and $I(G)\subseteq R = K[x_1,\dots,x_n]$ the corresponding edge ideal. The $q$-th squarefree power $I(G)$0 is generated by all squarefree monomials corresponding to size-$I(G)$1 matchings in $I(G)$2. For an ideal $I(G)$3, its Castelnuovo-Mumford regularity, denoted $I(G)$4, is the maximal degree difference $I(G)$5 such that a graded Betti number $I(G)$6 is nonzero.

The whiskered cycle $I(G)$7 is formed from the $I(G)$8-cycle $I(G)$9 by joining a new leaf (a "whisker") to each cycle vertex.

Recent work characterizes $G$0 in various classes, motivates lower and upper bounds, and suggests explicit formulas when $G$1 is well-covered, block, or whiskered graph. In particular, Das, Roy, and Saha conjectured:

$G$2

The present work settles this conjecture by precisely computing the invariant for all $G$3 and $G$4.

Structural and Technical Framework

The paper uses several important theoretical components:

• Matching and Induced Matching Numbers: The matching number $G$5 and induced matching number $G$6 play a crucial role in both lower and upper bounds for regularity of (squarefree) powers.
• Even-Connections and Colon Ideals: The concept of even-connection, as introduced by Banerjee, is central for understanding the generators of the colon ideals $G$7 and for recursive computations of regularity. For a given matching, a corresponding auxiliary graph $G$8 is defined via even-connections.
• Recursive Decomposition: Regularity is estimated by recursive decomposition, inductively passing to induced subgraphs and their whiskered structure, and decomposing along connected components created by removing certain vertices.
• Linear Quotients and Ordering: The authors impose an explicit ordering on the minimal generators of the squarefree powers for control over linear quotients and to facilitate induction. This ordering enables them to analyze the colon ideals as sums of explicitly tractable ideals.

Main Results

The principal claim is the following closed formula for regularity:

Theorem: For $G$9 and any $W(C_n)$0,

$W(C_n)$1

The proof combines:

• Sharp Lower and Upper Bounds: Previously established lower bounds and an initial upper bound are bridged using recursive analysis, Betti splitting, and structural decomposition of the whiskered cycle.
• Reduction to Paths and Cycles: By a careful analysis of how matchings and whiskers interact, the authors reduce the estimation of regularity on the whiskered cycle to that on whiskered paths and their matching structures, for which explicit bounds are available.
• Colon Ideal Structure: By an intricate study of the order of generators and the structure of colon ideals, the regularity is shown not to exceed the claimed value at each induction stage.
• Inductive Arguments and Bounding via Component Graphs: The induction involves bounding the regularity of sums and decompositions of whiskered path graphs, and showing that the regularity behaves additively except for specific controlled overlaps.

Notably, this formula not only tightens previously known upper and lower bounds but also provides exact values in a class where combinatorial and homological complexity is nontrivial.

Implications and Theoretical Significance

This result contributes a concrete case to the general program of relating combinatorial properties of graphs (here, via cycles and whiskering operations) to homological invariants of associated ideals. The explicit formula clarifies the impact of adding whiskers to cycles on regularity phenomena, subsuming cases for paths, cycles, and their whiskered analogues.

The methodology involving even-connections, explicit generator orderings, and the recursive splitting of the graph/ideal pairs is extensible, potentially informing more general results for matched or whiskered graphs, higher-dimensional simplicial complexes, or powers of other related monomial ideals.

Further, the techniques and formula are relevant in the study of sequentially Cohen-Macaulay properties, regularity stabilization, and Betti table phenomena for edge ideals and their squarefree powers. The result may also provide structural insight useful in the study of algebraic invariants for classes of graphs relevant in algebraic statistics, coding theory, or network science.

Conclusion

This work confirms the explicit regularity formula conjectured for squarefree powers of edge ideals of whiskered cycles, establishing

$W(C_n)$2

for all eligible $W(C_n)$3 and $W(C_n)$4. The proof leverages structural decompositions, even-connection theory, and a fine-grained analysis of generator orderings. These methods reinforce the fundamental relationship between graph-theoretical constructions and the homological algebra of monomial ideals, and position whiskered cycles as a key testing ground for new conjectures regarding regularity of symbolic and squarefree powers. The approach is broadly applicable and provides a foundation for investigations into more complex or less symmetric graph classes.

Reference:

"Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles" (2604.17100)