Ordered alternating paths and the depth of symbolic powers of cover ideals of graphs

Abstract: Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ in $|V(G)|$ variables. For a matching $M$ of $G$, we denote by $\ell(M)$ the length of the longest $M$-alternating path in $G$. We define $α_t(G)$ to be the maximum size of an ordered matching $M$ of $G$ such that $\ell(M) \le 2t-1$. We then prove that $$\operatorname{depth}(S/J(G)^{(t)}) \le |V(G)| - 1 - α_t(G)$$ for all $t \ge 1$, where $J(G)<sup>{(t)}$ denotes the $t$-th symbolic power of $J(G)$, and that equality holds when $G$ is a forest.

• The paper introduces αₜ(G), a new combinatorial invariant to bound the depth of symbolic powers of cover ideals.
• It leverages ordered matchings and alternating paths to connect graph properties with algebraic depth measurements.
• Results yield explicit depth formulas for forests and open avenues for similar bounds in weakly chordal and other graph classes.

Ordered Alternating Paths, Symbolic Powers, and the Depth of Cover Ideals

Introduction

The study of symbolic powers of monomial ideals, particularly those associated with graphs, bridges commutative algebra and discrete mathematics. This paper investigates a fundamental homological invariant—the depth—of the $t$-th symbolic power, $J(G)^{(t)}$, of the cover ideal $J(G)$ of a simple graph $G$. Specifically, it introduces the combinatorial invariant $\alpha_t(G)$, defined via ordered matchings and alternating path lengths, to sharply bound and, in certain cases, exactly compute the depth of $S/J(G)^{(t)}$ in terms of graph-theoretical data. Notably, the paper establishes an upper bound for the depth and proves that equality is attained for forests, thus giving a comprehensive algebraic-combinatorial formula in this setting.

Ordered Matchings, Alternating Paths, and $\alpha_t(G)$

The crux of the paper is a new combinatorial invariant $\alpha_t(G)$, defined for a simple graph $G$ and integer $t \geq 1$ as the maximal size of an ordered matching $J(G)^{(t)}$0 of $J(G)^{(t)}$1 such that the maximal $J(G)^{(t)}$2-alternating path length $J(G)^{(t)}$3 does not exceed $J(G)^{(t)}$4.

An ordered matching is a specific form of matching where edges are indexed and constrained so that:

• The collection $J(G)^{(t)}$5 of "free parameter vertices" (first vertex in each edge) forms an independent set.
• If $J(G)^{(t)}$6 is an edge in $J(G)^{(t)}$7, then $J(G)^{(t)}$8 according to the order.

This construction facilitates direct combinatorial control on the behavior of symbolic powers of cover ideals, as the depth can be linked to the maximal length of such alternating paths enumerated by $J(G)^{(t)}$9, and the cardinality $J(G)$0 reflecting the matching's size.

Main Results

The central result is encapsulated in the following inequality, holding for all $J(G)$1:

$J(G)$2

Equality holds when $J(G)$3 is a forest, thus yielding an explicit depth formula in this case.

The proof strategy leverages previous reductions of depth calculations for symbolic powers of cover ideals to the computation of regularities of edge ideals of $J(G)$4-admissible subgraphs (Dung et al., 5 May 2026). It establishes that such admissible subgraphs correspond to matchings with restricted alternating path lengths, with induced matchings forming ordered matchings within the original graph.

For forests, every matching is an ordered matching, enabling an exact correspondence and concluding the equality. The result also recovers and refines earlier formulas for the asymptotic behavior of the depth, where for large $J(G)$5, $J(G)$6, the ordered matching number.

Technical Innovations

Key innovations include:

• Introduction of $J(G)$7: This unifies and extends previous invariants, enabling finer granularity for all $J(G)$8, not only for $J(G)$9 asymptotically large.
• Reduction to Admissible Subgraphs: The analysis builds on translating depth computations to regularity computations for edge ideals of $G$0-admissible subgraphs, leveraging the interplay between combinatorial admissibility and homological algebra (Dung et al., 5 May 2026).
• Explicit Construction of Admissibility Certificates: The paper gives explicit constructions of exponent vectors certifying $G$1-admissibility for ordered matchings bounded by alternating path lengths, yielding the required depth bounds.

Numerical Results and Claims

For forests, the depth satisfies

$G$2

where $G$3 is computed over all matchings $G$4 in $G$5 with $G$6. The result is both optimal and constructive. For general graphs, the upper bound may be strict, yet aligns for classes such as weakly chordal graphs, for which the paper conjectures equality for all $G$7.

The paper supports its claims by referencing explicit instances for paths, cycles, and Ferrers graphs, where recent work has provided calculations in line with the new bounds (Dung et al., 5 May 2026, Dung et al., 5 May 2026, Hoang et al., 30 May 2026).

Implications and Future Directions

The algebraic-combinatorial formula for the depth of symbolic powers of cover ideals advances the understanding of how combinatorial properties of graphs influence the behavior of symbolic powers in commutative algebra. Practically, for forests and potentially larger graph classes, it allows for formulaic computation of depth, supporting algorithmic approaches and further explorations into regularity and other homological invariants.

From a theoretical perspective, the introduction of $G$8 sharpens the landscape of combinatorial invariants governing symbolic power behavior. The conjectured equality for weakly chordal graphs, supported by computational evidence and partial results, offers a concrete avenue for future research. Exploration into classes of graphs where equality holds, and potential extensions to other classes of monomial or squarefree monomial ideals, are natural next steps.

Furthermore, the combinatorial framework laid out by ordered matchings and bounded alternating paths may inspire analogous invariants in other algebraic or topological settings associated with discrete structures.

Conclusion

This work provides a rigorous connection between the depth of symbolic powers of cover ideals and well-structured combinatorial data drawn from ordered matchings and alternating paths. By characterizing the depth exactly for forests—and more generally bounding it for arbitrary graphs—it enriches the toolkit for analyzing monomial ideals associated with graphs, expanding both theoretical understanding and computational capability in combinatorial commutative algebra. The conjecture regarding weakly chordal graphs signals fertile ground for further advances at the intersection of algebra and graph theory.