The $\v$-number of generalized binomial edge ideals of some graphs

Abstract: Let (G) be a finite connected simple graph, and let (\calJ_{K_m,G}) denote its generalized binomial edge ideal. By investigating the colon ideals of (\calJ_{K_m,G}), we derive a formula for the local (\v)-number of (\calJ_{K_m,G}) with respect to the empty cut set. Furthermore, we classify graphs for which this generalized binomial edge ideal has (\v)-numbers 1 or 2. When (G) is a connected closed graph, we compute the local (\v)-number of (\calJ_{K_2,G}) by generalizing the work of Dey et al. Additionally, under the condition that (G) is Cohen--Macaulay, we derive formulas for the (\v)-number of (\calJ_{K_m,G}) and (\calJ_{K_2,G}^k), and show that the (\v)-number of (\calJ_{K_2,G}^k) is a linear function of (k).

• The paper presents explicit formulas for the v-number of generalized binomial edge ideals, linking it to minimal vertex completions and connected domination numbers.
• It develops combinatorial methods to compute colon ideals and characterizes graphs with global v-number 1 or 2, extending known m=2 cases.
• The study shows linear growth of the v-number for powers of binomial edge ideals in Cohen–Macaulay closed graphs, offering insights for applications in algebraic statistics and coding theory.

The $\v$-Number of Generalized Binomial Edge Ideals of Some Graphs

Introduction and Context

The paper "The $\v$-number of generalized binomial edge ideals of some graphs" (2603.29516) provides a thorough investigation of the v-number for generalized binomial edge ideals associated to certain classes of graphs. The v-number, an invariant originating with Vasconcelos, encodes minimal generator degrees for colon ideals and plays a key role in the asymptotic behavior of related homological invariants, with connections to projective Reed-Muller codes and the geometry of algebraic varieties defined by (generalized) binomial edge ideals.

Binomial edge ideals, initially introduced for simple graphs, are extended in this work to the generalized setting, where one considers the ideal generated by all $2$-minors corresponding to edge pairs from a fixed complete graph $K_m$ and a given graph $G$. While the classical v-number for monomial and binomial edge ideals has received substantial attention, the theory for the generalized variant, particularly for its powers, was heretofore undeveloped. This paper systematically addresses this gap.

Definitions and Main Objects

Given a finite connected simple graph $G$ and an integer $m \geq 2$, the authors focus on the generalized binomial edge ideal $J_{K_m,G}$ in the polynomial ring $S = \mathbb{K}[x_{i,j} : i \in [m], j \in [n]]$. The ideal is generated by $2$-minors corresponding to pairs of edges from $\v$0 and $\v$1. The set of minimal associated primes is described via cut sets and completions of the graph.

The local v-number $\v$2 of a homogeneous ideal $\v$3 at an associated prime $\v$4 is defined as the minimal degree $\v$5 such that there is a homogeneous $\v$6 of degree $\v$7 with $\v$8. The global v-number is the minimal such value over all associated primes.

Key Results: Structure and Main Theorems

Colon Ideals and Reductions

The paper establishes explicit formulas for colon ideals of $\v$9 with respect to variables and binomials, exploiting the combinatorial structure of the underlying graph. In particular, the colon of $2$0 with respect to a variable $2$1 is shown to be $2$2, where $2$3 is the completion of $2$4 at vertex $2$5. This observation enables an inductive approach to the computation of local v-numbers and underpins later classifications.

v-Number Formulas for Generalized Binomial Edge Ideals

A central result is an exact formula for the local v-number at the prime corresponding to the empty cut set. Specifically,

$2$6

where $2$7 is the minimal cardinality of a set of vertices whose completion yields a union of cliques, and $2$8 is a variant of the connected domination number. Notably, this invariance under $2$9 fails for non-empty cut sets, a point which is clarified through combinatorial constructions.

Classification for v-Numbers 1 and 2

The authors give a full combinatorial characterization of graphs $K_m$0 for which the global v-number $K_m$1 is 1 or 2. For $K_m$2, $K_m$3 must be a cone over a non-complete graph; for $K_m$4, they classify graphs in terms of connected domination and combinatorial structure of the cut sets, extending known results in the $K_m$5 case.

Local v-Numbers for Connected Closed Graphs

A substantial part of the paper is dedicated to the computation of local v-numbers for $K_m$6 when $K_m$7 is a connected closed graph, with and without the Cohen–Macaulay property. Through a sequence of combinatorial constructions involving partitions of the clique complex and associated graphs $K_m$8 tied to cut sets, the authors determine the necessary minimal degrees by linking them to determinants of appropriate submatrices of variables, dictated by $K_m$9-compatible partitions.

A key claim is that the computation of the v-number for generalized binomial edge ideals becomes strictly more involved for $G$0. Contrary to what might be expected, the v-number can depend intricately on $G$1, the combinatorics of maximal cliques, and positioning of cut vertices in $G$2.

Explicit Closed Formulas in the Cohen–Macaulay Case

For connected Cohen–Macaulay closed graphs with $G$3 maximal cliques, the v-number is determined explicitly as

$G$4

where $G$5 are defined in terms of the Euclidean division of $G$6 relative to $G$7. For $G$8, this recovers the formula $G$9, in agreement with and extending previously established results for binomial edge ideals.

v-Number of Powers

A particularly significant finding is that, for $G$0 a connected Cohen–Macaulay closed graph, the v-number of powers of the binomial edge ideal $G$1 satisfies

$G$2

demonstrating that the $G$3-number grows linearly with respect to the exponent $G$4. This is rigorously established using reduction to spine graphs (paths in the clique complex), explicit analysis of initial ideals, and careful control of the combinatorics of cut sets under the operation of taking powers.

Implications and Theoretical Significance

The results presented construct a comprehensive theory for the v-number of generalized binomial edge ideals, integrating combinatorial characterizations with homological invariants. The explicit formulas yield new insights into the relationship between the algebraic structure of binomial ideals and the fine combinatorics of the underlying graphs, particularly for closed and Cohen–Macaulay structures.

The identification that the $G$5-number for powers of $G$6 is always a linear function of the exponent, with precise slope and intercept, offers an exact parallel to the well-studied asymptotic behavior of regularity. These findings provide new pathways for further study of asymptotic and extremal algebraic properties of ideals associated to graphs and suggest analogues of known invariants in broader contexts.

Practically, these results impact computations in algebraic statistics (e.g., conditional independence ideals), coding theory (invariants of Reed-Muller codes), and the construction of algebraic varieties from graph-theoretical data.

Future Directions

Several questions remain open following this work. While formulas are provided for $G$7 and for the Cohen–Macaulay closed graph case, the general structure of the v-number for higher $G$8 or for arbitrary graphs remains complicated. Extensions to non-closed graphs, higher symbolic or analytic powers, and deeper connections with other homological invariants (such as Waldschmidt constants or Hilbert functions) are natural directions.

Furthermore, understanding the relationship between local and global v-numbers in more general settings, as well as the impact on the geometry and combinatorics of secant and multi-graded Rees algebras associated to such ideals, appears promising.

Conclusion

This paper delivers a rigorous and detailed structural analysis of the v-number for generalized binomial edge ideals, providing both theoretical foundations and explicit computational results across a variety of graph classes. The interplay between the combinatorics of cut sets, the algebra of colon ideals, and homological invariants is laid bare through explicit formulas and constructions. The characterization of $G$9 as a linear function in the Cohen–Macaulay closed graph setting is a notable and robust feature, closing gaps between combinatorics and commutative algebra for this family of binomial ideals.