A Join-Matching Theorem for Squarefree Powers of Edge Ideals, with Applications to Wheel and Related Graphs

Abstract: For $q\ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ of the edge ideal of a graph $G$ is generated by the squarefree monomials supported on $q$-matchings of $G$; it is the Stanley--Reisner ideal of the complex $Δ_q(G)={F\subseteq V(G):ν(G[F])<q}$, where $ν$ denotes matching number. We prove a general formula for the matching number of an arbitrary graph join, [ ν(G\ast H) = \min\Big(ν(G)+|V(H)|,\ \ ν(H)+|V(G)|,\ \ \Big\lfloor\tfrac{|V(G)|+|V(H)|}{2}\Big\rfloor\Big), ] via the Tutte--Berge formula, and use it to decompose $Δ_q(G\ast H)$ for arbitrary graphs $G,H$. Specializing to the wheel graph $\mathcal{W}_n = \mathcal{C}_n\ast\mathcal K_1$, we determine the Krull dimension and height of $R/I(\mathcal{W}_n)^{[q]}$ exactly for all $n\ge 3$, $1\le q\le\lfloor n/2\rfloor$, and -- combining our matching-number computations with a recent Tutte-type Cohen-Macaulayness criterion of Ficarra and Moradi -- prove that at the \emph{top} squarefree power $q=ν(\mathcal{W}_n)=\lceil n/2\rceil$, the ideal $I(\mathcal{W}_n)^{[ν(\mathcal{W}_n)]}$ is literally the squarefree Veronese ideal, so that $R/I(\mathcal{W}_n)^{[ν(\mathcal{W}_n)]}$ is Cohen-Macaulay with [ {\rm dim} = {\rm depth} = {\rm reg}\big(R/I(\mathcal{W}_n)^{[ν(\mathcal{W}_n)]}\big) = 2\Big\lceil\frac n2\Big\rceil-1. ] This resolves all four classical invariants at the top power, and confirms there the pattern depth$(R/I(\mathcal{W}_n)^{[q]}) = 2q-1$ that our computational data (now extended to $n\le13$, every valid $q$) suggests holds throughout. We prove a general depth formula for squarefree powers of cone graphs, via a Betti-splitting exact sequence, that reduces this pattern to two more tractable statements about the underlying cycle alone; both are verified computationally in every case checked but left open in general.

• The paper introduces a join-matching theorem that computes the matching number for joins of graphs and applies it to squarefree powers of edge ideals.
• It derives explicit formulas for invariants such as Krull dimension, depth, and regularity in wheel, cone, fan, and friendship graphs.
• The work bridges combinatorial graph structures with homological algebra, simplifying classical extremal problems and conjectures in commutative algebra.

Join-Matching Theorem for Squarefree Powers of Edge Ideals: Structural Analysis and Applications to Wheel, Cone, and Related Graphs

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Introduction and Context

The paper establishes a general graph-theoretic formula for the matching number of a graph join, and leverages this to systematically analyze the algebraic invariants of squarefree powers of edge ideals indexed by q-matchings. The formalism connects combinatorial data from graph theory (matchings/joins) to homological invariants of monomial ideals associated with graphs. The authors focus on wheel graphs—and extend to cone, fan, multi-hub wheel, complete split, and friendship graphs—to provide explicit decompositions and closed-form formulas for Krull dimension, height, depth, and regularity of the corresponding ideals.

Edge ideals $I(G)$ of a graph $G$ play a central role in combinatorial commutative algebra, encoding the graph's structure as a monomial ideal in the polynomial ring $R = \mathbb{k}[x_1, ..., x_n]$. The squarefree powers $I(G)^{[q]}$ generalize ordinary powers by restricting to monomials supported on $q$-matchings, and their minimal generating sets are in bijection with facets of a combinatorial complex defined by matching constraints.

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Join-Matching Formula: Main Theorem

The paper presents a general formula for the matching number $\nu(G \ast H)$ of the join of any two graphs $G$ and $H$: $$\nu(G\ast H) = \min \Big( \nu(G) + |V(H)|,\ \nu(H) + |V(G)|,\ \Big\lfloor \frac{|V(G)| + |V(H)|}{2} \Big\rfloor \Big)$$ This formula is derived via the Tutte–Berge characterization of matching number, employing an exhaustive casewise maximization over all vertex subsets. Its strength is the abstraction: it reduces the computation for complicated graph joins to three easily accessible numerical invariants. The authors further specialize this to cones and joins with complete (and edgeless) graphs, providing explicit formulas for several families.

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Decomposition of Squarefree Powers and Complexes

By translating the join-matching theorem into a decomposition of the Stanley-Reisner complex $\Delta_q(G\ast H)$, whose facets encode the constraints on the matching number, the authors give: $G$0 This decomposition allows for a precise combinatorial description of the generating sets for $G$1, and thus facilitates the calculation of Krull dimension, height, and depth.

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Applications to Wheel Graphs: Explicit Invariants

For wheel graphs $G$2 (where $G$3 is a cycle), the authors provide explicit closed-form formulas:

• Matching number: $G$4.
• Krull dimension: $G$5 for $G$6.
• Height: $G$7.

A notable result is that at the top squarefree power ($G$8), the ideal $G$9 is precisely the squarefree Veronese ideal $R = \mathbb{k}[x_1, ..., x_n]$0, and the quotient ring $R = \mathbb{k}[x_1, ..., x_n]$1 is Cohen–Macaulay. The authors confirm all four classical invariants at this top power: $R = \mathbb{k}[x_1, ..., x_n]$2 This exact identification is validated via a recent combinatorial Cohen–Macaulayness criterion (Ficarra–Moradi).

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Depth Formulas and Betti Splitting for Cones

A general depth formula for cone graphs is derived using Betti-splitting exact sequences and auxiliary ideals to relate the depth of $R = \mathbb{k}[x_1, ..., x_n]$3 to those of $R = \mathbb{k}[x_1, ..., x_n]$4 and of associated ideals. The authors provide reduction steps for wheel graphs, showing that conjectured patterns (e.g., $R = \mathbb{k}[x_1, ..., x_n]$5) can be formulated as more tractable combinatorial statements concerning cycles and explicit auxiliary ideals.

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Extensions to Other Graph Classes

The join-matching and decomposition results are further applied to:

• Fan graphs ($R = \mathbb{k}[x_1, ..., x_n]$6): Dimension formula $R = \mathbb{k}[x_1, ..., x_n]$7.
• Multi-hub wheels ($R = \mathbb{k}[x_1, ..., x_n]$8, $R = \mathbb{k}[x_1, ..., x_n]$9): Explicit decomposition, matching number formulas, closed-form dimension for $I(G)^{[q]}$0.
• Complete split graphs: Dimension formula $I(G)^{[q]}$1 for $I(G)^{[q]}$2.
• Friendship graphs ($I(G)^{[q]}$3): Dimension formula $I(G)^{[q]}$4.

The authors highlight open extremal problems for multi-hub wheels and note that, for several graph classes, the combinatorial join-matching framework reduces the algebraic analysis to explicit extremal problems.

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Computational Results and Conjectures

Computational evidence is presented for $I(G)^{[q]}$5, an empirically observed phenomenon for wheels and cycles up to $I(G)^{[q]}$6. The paper connects these computations to conjectures on depth, regularity, and projective dimension—reducing them to cycle graphs and auxiliary ideals and investigating them via direct combinatorial and homological methods.

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Conclusion

The paper provides a general and reusable combinatorial tool for analyzing matchings in joins, with direct implications for the algebraic invariants of edge ideals and their squarefree powers. The explicit formulas and decompositions for wheel and related graphs offer systematic ways to resolve classical invariants and connect homological algebra with graph theory. The practical implication is a significant simplification and unification of the computation of invariants for several important monomial ideals indexed by graph families, including precise Cohen–Macaulayness criteria for wheels and structured patterns in depth and dimension. Theoretical implications include new reduction methods for depth and dimension conjectures, and the identification of combinatorial extremal problems which, if further developed, could lead to resolution of broader conjectures in commutative algebra and combinatorics.