Invariant chains of graphs

Abstract: We initiate a systematic study of Inc-invariant chains of graphs, the combinatorial counterparts of Inc-invariant chains of edge ideals arising in the theory of equivariant Noetherianity. Such a chain consists of graphs on growing vertex sets whose edge sets are compatible with the action of the monoid of strictly increasing maps on the positive integers. We show that several associated combinatorial invariants exhibit rigid asymptotic behavior. The independence number eventually stabilizes, and every fixed entry of the $f$-vector and the $h$-vector of the independence complex is eventually linear. For clique complexes, every fixed entry of the $f$-vector is eventually polynomial, whereas the entries of the $h$-vector are eventually quasi-polynomial. Moreover, the clique and chromatic numbers are eventually quasi-linear, and their difference is eventually at most one. We also prove that the matching number eventually attains the maximal value $\lfloor n/2\rfloor$. Finally, admissible and minimal paths eventually have lengths at most $3$ and $5$, respectively, and their maximal lengths stabilize. These results reveal strong asymptotic regularity in graph families governed by increasing symmetry.

• The paper develops structural tools based on missing edges, sparsity, cardinal moves, and deletion isomorphisms to analyze graph invariants across Inc-invariant chains.
• Independence-complex data eventually becomes linear, while clique-complex face numbers become polynomial or quasi-polynomial, with growth controlled by the sparsity index.
• Clique and chromatic numbers share eventual period and slope 1/s, matching numbers reach floor(n/2), and admissible and minimal path lengths stabilize, yielding bounded-degree Gröbner bases.

Context and motivation

The paper studies $\Inc$-invariant chains of graphs: sequences $\mathcal{G}=(G_n)_{n\ge1}$ of simple graphs on growing vertex sets $[n]$ such that every strictly increasing relabeling of $G_n$ into $[m]$ embeds as a subgraph of $G_m$. Such chains are the combinatorial counterparts of $\Inc$-invariant chains of edge ideals, whose asymptotic behavior has been a central theme in equivariant Noetherianity since Cohen's theorem that the infinite polynomial ring is Noetherian up to the action of the monoid $\Inc$ of strictly increasing maps on $\mathbb{N}$. Prior work established that for such chains of edge ideals, Castelnuovo–Mumford regularity is eventually constant (equal to 2 or 3) and asymptotic depth is determined; these proofs exposed rigid combinatorial structure in the underlying graphs, e.g., induced matching numbers eventually bounded by 2. The present work takes this combinatorial structure as its object of study directly, asking (Problem 1.1) how standard graph-theoretic invariants behave along an arbitrary $\Inc$-invariant chain.

The framework rests on two numerical parameters attached to a chain with stability index $\mathcal{G}=(G_n)_{n\ge1}$0: the number of missing edges $\mathcal{G}=(G_n)_{n\ge1}$1, measuring distance from the complete-graph chain, and the sparsity index $\mathcal{G}=(G_n)_{n\ge1}$2, the smallest gap between endpoints of an edge of $\mathcal{G}=(G_n)_{n\ge1}$3. The sparsity index governs nearly all growth rates below. A key structural fact is that edges of $\mathcal{G}=(G_n)_{n\ge1}$4 correspond to lattice points in isosceles right triangles determined by edges of $\mathcal{G}=(G_n)_{n\ge1}$5, which yields "cardinal move" lemmas allowing edges to be shifted within $\mathcal{G}=(G_n)_{n\ge1}$6; these moves drive most of the arguments.

Independence complexes

For the independence complex $\mathcal{G}=(G_n)_{n\ge1}$7, whose Stanley–Reisner ring is $\mathcal{G}=(G_n)_{n\ge1}$8, the main theorem establishes eventual linearity across the board. For $\mathcal{G}=(G_n)_{n\ge1}$9 where $[n]$0: the independence number $[n]$1 stabilizes to a constant; each entry $[n]$2 satisfies the exact increment $[n]$3, hence is eventually linear in $[n]$4; each $[n]$5 satisfies $[n]$6 with $[n]$7, so it too is eventually linear; and the Euler characteristic changes by exactly 1 per step when $[n]$8 and is eventually constant when $[n]$9. The proof proceeds by induction on $G_n$0, using a deletion formula for Hilbert series and an induced-chain construction that lowers the missing-edge count while preserving the sparsity index.

A companion result shows the facet count is eventually exactly linear: $G_n$1 for all $G_n$2. The argument decomposes facets into central intervals $G_n$3 contained in $G_n$4—each of which is provably a facet—plus facets supported near the boundary, and uses deletion isomorphisms between consecutive members of the chain. This confirms a conjecture attributed to Hop D. Nguyen.

Clique complexes

Clique complexes exhibit a different regime: dimensions grow rather than stabilize. With $G_n$5, the clique number satisfies $G_n$6 for all large $G_n$7, i.e., it is quasi-linear of period $G_n$8. The $G_n$9-vectors obey the recurrence $[m]$0, from which the authors deduce that $[m]$1 is eventually a polynomial of degree $[m]$2 with leading coefficient $[m]$3, while $[m]$4 is eventually a quasi-polynomial of quasiperiod $[m]$5 with leading term $[m]$6. Euler characteristics satisfy matching recurrences. These results follow from a Hilbert series recurrence for complements, again proved by induction on missing edges, together with a general characterization of eventually quasi-linear functions via recurrences of the form $[m]$7.

In the special case $[m]$8, the behavior sharpens considerably: the last entry of the $[m]$9-vector is constant, the $G_m$0-polynomial is independent of $G_m$1, and both Euler characteristics stabilize ($G_m$2, $G_m$3). Moreover, minimal nonfaces of $G_m$4 biject with those of $G_m$5, meaning the Stanley–Reisner rings differ only by adjoining one variable up to relabeling. The paper notes that $G_m$6 need not be pure, so constancy of the top $G_m$7-entry and of the facet count are genuinely distinct statements—an example chain with five facets but top $G_m$8-entry 2 is computed with Macaulay2. The asymptotic facet count for general $G_m$9 is left open.

Chromatic numbers

The chromatic number $\Inc$0 is shown to be eventually quasi-linear with the same period $\Inc$1 and slope $\Inc$2 as the clique number. The upper bound $\Inc$3 is obtained constructively: color classes of $\Inc$4 are transported forward via explicit deletion isomorphisms, leaving only one interval uncovered, which is itself independent. Combined with the clique-number recurrence and the abstract quasi-linearity criterion, this yields the result.

More strikingly, the paper proves $\Inc$5 for all $\Inc$6. This contrasts sharply with arbitrary graphs, where Mycielski's construction makes both the difference and ratio $\Inc$7 unbounded. The proof shows that deleting any independent interval $\Inc$8 from $\Inc$9 yields a weakly chordal graph, hence a perfect graph by the Strong Perfect Graph Theorem; coloring that perfect subgraph optimally and adding one color for $\Inc$0 gives the bound. The weak chordality argument bounds induced cycles in the complement to length at most $\Inc$1 via monotonicity of induced paths in the middle region.

Matching numbers

Every graph on $\Inc$2 vertices has matching number at most $\Inc$3. The paper proves this bound is attained: $\Inc$4 for all $\Inc$5. The proof establishes an augmentation property—if $\Inc$6 then $\Inc$7—via an alternating-path argument exploiting the neighborhood structure of the newest vertex, then runs an induction showing $\Inc$8.

This result determines the eventual dimension of the matching complex: $\Inc$9 for $\mathbb{N}$0. It also stands in deliberate contrast to the induced matching number, which prior work showed is eventually constant in $\mathbb{N}$1: ordinary matchings grow extremally while induced matchings do not grow at all. Finer asymptotics of the matching complexes are posed as an open problem.

Admissible and minimal paths

Admissible and minimal paths control Gröbner bases and generic initial ideals of binomial edge ideals. The paper proves uniform length bounds: for $\mathbb{N}$2, every admissible path in $\mathbb{N}$3 has length at most 3 and every minimal path has length at most 5, and both maxima stabilize for $\mathbb{N}$4. The bounds are sharp—explicit chains realizing admissible paths of length 3 and minimal paths of length 5 are given. The stabilization argument locates an interval of $\mathbb{N}$5 consecutive integers disjoint from any short path and uses the deletion isomorphism induced by $\mathbb{N}$6 to transfer paths between consecutive chain members in both directions.

An immediate algebraic consequence: for suitable term orders, the initial ideal $\mathbb{N}$7 is generated in degrees at most 4 and the generic initial ideal $\mathbb{N}$8 in degrees at most 6, with maximal generator degrees eventually independent of $\mathbb{N}$9.

Limitations and open questions

Several restrictions qualify the results. All main theorems are stated under a reduced-form assumption (vertex set $\Inc$0 at stability, no isolated endpoints), justified by a reduction procedure that applies when edges exist eventually; chains with finitely many nonempty members fall outside the framework. The facet-count formula for independence complexes holds only for $\Inc$1, and the chromatic-clique inequality requires $\Inc$2, a quadratic threshold; whether these thresholds are optimal is not addressed. The asymptotic behavior of facet counts of clique complexes for general sparsity index, of chromatic polynomials $\Inc$3, and of matching complexes $\Inc$4 are explicitly posed as open problems. The paper also leaves open whether the eventual linear/quasi-polynomial descriptions can be made effective with explicit constants depending only on $\Inc$5 and $\Inc$6.

Conclusion

The paper demonstrates that $\Inc$7-invariance imposes strong asymptotic regularity on graph invariants: stabilization (independence number, path lengths), eventual linearity ($\Inc$8- and $\Inc$9-vectors of independence complexes, facet counts), polynomial and quasi-polynomial growth (face data of clique complexes), quasi-linear growth with common period equal to the sparsity index (clique and chromatic numbers), and extremal saturation (matching numbers). The techniques—Hilbert series recursions combined with induction on missing-edge count, cardinal-move lemmas, and deletion isomorphisms—provide a reusable toolkit, and the results supply the combinatorial substrate underlying the previously obtained algebraic regularity and depth theorems for invariant chains of edge ideals.