Complementary Edge Ideal in Graph-Theoretic Algebra

• Complementary edge ideal is a squarefree monomial ideal associated with a graph, where each generator is the product of all variables except two corresponding to an edge.
• It bridges graph theory and commutative algebra by linking duality principles with classifications such as Cohen–Macaulay, Gorenstein, and sequentially Cohen–Macaulay properties.
• Studies of its powers, minimal resolutions, and Rees algebras reveal precise regularity patterns, linearity conditions, and persistence behaviors influenced by specific graph structures.

to=arxiv_search 大发快三开奖 天天中彩票不中返json {"query":"\"complementary edge ideal\" arXiv", "max_results": 10, "sort_by": "submittedDate"}ถวายสัตย์ฯ A complementary edge ideal is, in current graph-theoretic commutative algebra, the squarefree monomial ideal

$$I_c(G)\;=\;\Bigl(\,\frac{x_1x_2\cdots x_n}{x_i\,x_j}\;:\;\{i,j\}\in E(G)\Bigr)\subset K[x_1,\dots,x_n]$$

attached to a finite simple graph $G$ on $[n]$. Its generators all have degree $n-2$, and every squarefree monomial ideal generated in degree $n-2$ arises in this form for a suitable graph (Ficarra et al., 14 Aug 2025). A nearby, but distinct, usage appears for a $1$-dimensional flag simplicial complex $\Delta$, where the Stanley–Reisner ideal

$$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$

is described as the complementary edge ideal of $\Delta$ because it is the edge ideal of the graph-theoretic complement of $\Delta$ (Navarra et al., 1 Jul 2026). The modern literature develops both usages, with the $G$0-generated ideal $G$1 serving as the central object in a broad classification program.

1. Definition, realization, and duality

For a graph $G$2 with vertex set $G$3, the complementary edge ideal $G$4 is generated by the squarefree monomials whose support is the complement of one edge of $G$5. If $G$6 has an isolated vertex $G$7, then $G$8 does not divide any generator, and one has

$G$9

directly from the definition (Ficarra, 2 Mar 2026).

A fundamental realization theorem states that every squarefree monomial ideal generated in degree $[n]$0 is a complementary edge ideal. Concretely, if

$[n]$1

then each $[n]$2 has the form $[n]$3 for a unique pair $[n]$4, hence $[n]$5 for the graph whose edges are those pairs (Ficarra et al., 14 Aug 2025). This gives a graph-theoretic model for the entire class of squarefree degree-$[n]$6 ideals.

The construction is closely tied to Alexander duality. One formulation identifies $[n]$7 with the Alexander dual of the usual edge ideal of the complement graph $[n]$8, up to the obvious identification of variables (Ficarra, 2 Mar 2026). A more explicit description writes

$[n]$9

where $n-2$0 is the clique complex of $n-2$1 and $n-2$2 is its pure $n-2$3-skeleton (Hibi et al., 13 Aug 2025). This dual viewpoint underlies the Cohen–Macaulay, sequentially Cohen–Macaulay, and linearity criteria proved for $n-2$4.

2. Basic structural classifications

The algebraic properties of $n-2$5 are governed by explicit graph classes. When $n-2$6 has no isolated vertices, the principal classifications are as follows.

Property of $n-2$7	Graph-theoretic characterization
Sequentially Cohen–Macaulay	$n-2$8 is chordal
Cohen–Macaulay	$n-2$9 is a forest or a complete graph
Gorenstein	$n-2$0
Matroidal	$n-2$1 is a complete multipartite graph

These equivalences are established in the recent systematic treatment of complementary edge ideals (Ficarra et al., 14 Aug 2025). The matroidal criterion is also equivalent to $n-2$2 being $n-2$3-free, or equivalently a disjoint union of cliques (Ficarra et al., 14 Aug 2025).

A finer nearly Gorenstein classification is also available. Under the no-isolated-vertices hypothesis, $n-2$4 is nearly Gorenstein exactly when

$n-2$5

with the first two cases nearly Gorenstein but not Gorenstein, and the last four Gorenstein (Ficarra et al., 14 Aug 2025).

Linearity properties for the ideal itself admit a particularly sharp criterion. One treatment proves that $n-2$6 has a linear resolution if and only if it has linear quotients if and only if it is linearly related if and only if $n-2$7 is connected (Hibi et al., 13 Aug 2025). The same source shows that $n-2$8 has a pure minimal free resolution if and only if $n-2$9 is connected or a disjoint union of edges, and that $1$0 is level if and only if $1$1 is complete, or a tree, or a disjoint union of edges (Hibi et al., 13 Aug 2025).

The minimal-prime structure is also described combinatorially. One statement gives

$1$2

so the minimal primes are indexed by induced triangles or induced pairs of nonadjacent vertices (Hibi et al., 13 Aug 2025). Correspondingly, $1$3 is unmixed if and only if $1$4 is complete or triangle-free (Hibi et al., 13 Aug 2025).

For the minimal free resolution of $1$5 itself, the only nonzero graded Betti numbers occur in bidegrees

$1$6

and one has

$1$7

for the base ideal $1$8 (Hibi et al., 13 Aug 2025).

3. Powers, regularity, associated primes, and the $1$9-function

The powers of a complementary edge ideal exhibit a rigid piecewise-linear regularity pattern. If $\Delta$0 denotes the number of connected components of $\Delta$1 of size $\Delta$2, then for every $\Delta$3,

$\Delta$4

Moreover, the depth function $\Delta$5 is non-increasing (Ficarra et al., 14 Aug 2025). The same large-degree framework shows more generally that if a squarefree monomial ideal is generated in degrees $\Delta$6, then its depth stabilizes by $\Delta$7, and for complementary edge ideals the Betti numbers of $\Delta$8 depend only on the combinatorics of $\Delta$9 and not on the base field $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$0 (Ficarra, 2 Mar 2026).

The criterion for linear powers is exact. The following are equivalent: $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$1 has a linear resolution for some $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$2; $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$3 has linear quotients for some $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$4; and $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$5. Equivalently, the same holds for all $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$6 (Ficarra et al., 14 Aug 2025). Thus a connected graph of size at least $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$7 has linear quotients in every power, whereas disconnectedness among nontrivial components forces failure of linearity in higher powers.

The associated primes of all powers are explicitly determined. Writing

$$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$8

one has for every $$I_\Delta=(x_i x_j:\{i,j\}\notin \Delta)=I((\Delta)^c)$$9,

$\Delta$0

and each such $\Delta$1 appears already for all powers $\Delta$2 (Ficarra, 2 Mar 2026). In particular, complementary edge ideals satisfy the persistence property: $\Delta$3 for $\Delta$4 (Ficarra, 2 Mar 2026).

The $\Delta$5-function is equally explicit. If $\Delta$6, then for every $\Delta$7,

$\Delta$8

and in all cases

$\Delta$9

If $\Delta$0 has linear powers, then $\Delta$1 for all $\Delta$2 (Ficarra, 2 Mar 2026). The cycle $\Delta$3 provides a concrete model: $\Delta$4 for every $\Delta$5, while

$\Delta$6

(Ficarra, 2 Mar 2026).

4. Rees algebras, fiber cones, and depth stability

The Rees algebra

$\Delta$7

admits a combinatorial description through the defining ideal $\Delta$8, where $\Delta$9 and $G$00 for the complementary edge generators $G$01. A structural theorem states that every primitive binomial in $G$02 can be chosen so as to have $G$03-degree at most $G$04. Equivalently, $G$05 has a Gröbner basis consisting of binomials

$G$06

with $G$07 of degree at most $G$08 (Ficarra et al., 22 Sep 2025).

This immediately yields bounds on the $G$09-regularity: $G$10 while asymptotic regularity gives the lower bound

$G$11

(Ficarra et al., 22 Sep 2025). The same paper proves large normality and Cohen–Macaulayness results: if $G$12 is bipartite, or if $G$13 is connected unicyclic, then $G$14 is a normal Cohen–Macaulay domain (Ficarra et al., 22 Sep 2025).

Koszulness is known in two further cases. If $G$15 is a tree, then $G$16 has a quadratic Gröbner basis and $G$17 is Koszul. The same holds for a connected unicyclic graph whose unique cycle has length $G$18 or $G$19 (Ficarra et al., 22 Sep 2025). The fiber cone $G$20 is normal exactly when $G$21 satisfies the odd-cycle condition (Ficarra et al., 22 Sep 2025).

Asymptotic depth is controlled by the number $G$22 of bipartite connected components of $G$23, with isolated vertices counted as bipartite components. The analytic spread is

$G$24

and the limit depth formula is

$G$25

(Ficarra et al., 22 Sep 2025). The index of depth stability satisfies

$G$26

and equality holds for path graphs. More precisely, if $G$27, then

$G$28

so $G$29 (Ficarra et al., 22 Sep 2025).

5. Licci behavior, random graphs, and homological shifts

The liaison-theoretic classification is especially rigid. Localizing at the homogeneous maximal ideal $G$30, one has: $G$31 (Lama, 17 Mar 2026). The proof combines the Cohen–Macaulay criterion $G$32 Cohen–Macaulay $G$33 $G$34 is a forest or a complete graph with the Huneke–Ulrich numerical obstruction for licci ideals; among complete graphs, only $G$35 survives (Lama, 17 Mar 2026).

This licci criterion ties directly to standard homological invariants. If $G$36 is licci and $G$37 is a connected forest, then $G$38 and $G$39. If $G$40 is a disconnected forest, then $G$41 and $G$42 (Lama, 17 Mar 2026). The random-graph consequence is asymptotic: for the Erdős–Rényi graph $G$43,

$G$44

This follows from the equivalence between non-licci behavior and the presence of a cycle of length $G$45 (Lama, 17 Mar 2026).

A separate homological line studies powers via homological shift ideals and algebras. For a monomial ideal $G$46, the $G$47-th homological shift ideal $G$48 is generated by the multidegrees appearing in the $G$49-th free module of a minimal multigraded free resolution. For complementary edge ideals of trees and cycles, this structure can be computed explicitly (Lu et al., 17 Nov 2025).

Projective dimension of powers is particularly transparent in several families. If $G$50 is a connected bipartite graph on $G$51 vertices, then

$G$52

and thereafter $G$53 (Lu et al., 17 Nov 2025). For a tree on $G$54 vertices,

$G$55

For an even cycle of length $G$56,

$G$57

while for an odd cycle of length $G$58,

$G$59

These formulas show that powers of $G$60 can have large projective dimension even when the base ideal has $G$61 (Lu et al., 17 Nov 2025).

For trees, the homological shift algebras are especially structured: $G$62 is generated in degree $G$63, and $G$64, after dividing by a suitable monomial, is exactly a Veronese-type ideal. Conversely, every Veronese-type ideal arises in this way from a caterpillar tree (Lu et al., 17 Nov 2025).

6. The $G$65-dimensional flag-complex usage and squarefree powers

In the simplicial-complex literature, a $G$66-dimensional flag simplicial complex $G$67 is exactly a graph with no triangles, and its Stanley–Reisner ideal is

$G$68

In this setting $G$69 is often called the complementary edge ideal of $G$70 because it is the edge ideal of the complement graph of $G$71 (Navarra et al., 1 Jul 2026). This usage differs from the $G$72-generated ideal $G$73, but it is algebraically adjacent: both are squarefree monomial ideals controlled by forbidden induced subgraphs.

For an arbitrary graph $G$74, the $G$75-th squarefree power

$G$76

encodes matchings of size $G$77. The central linearity theorem states that for $G$78,

$G$79

which is disconnected and satisfies $G$80 (Navarra et al., 1 Jul 2026).

Transferred to a $G$81-dimensional flag complex $G$82, this becomes a forbidden-subgraph criterion: $G$83 (Navarra et al., 1 Jul 2026). The Betti table then has a highly constrained form. There is always a linear strand in bidegrees $G$84. If $G$85 contains an induced even-bipartite $G$86 with $G$87 even, then

$G$88

indeed

$G$89

and no other nonzero extra Betti numbers occur (Navarra et al., 1 Jul 2026).

If $G$90 contains no such even complete bipartite graph but does contain a crown graph $G$91, then all first syzygies remain linear, $G$92, but nonlinearity first appears at

$G$93

(Navarra et al., 1 Jul 2026). Consequently,

$G$94

nor any induced crown graph $G$95 (Navarra et al., 1 Jul 2026).

The model examples isolate the two obstruction types. If $G$96, then $G$97 has $G$98, coming from the unique induced $G$99. If $[n]$00, then $[n]$01 is linearly related but not linearly resolved, with first failure at $[n]$02 (Navarra et al., 1 Jul 2026). These examples complete the combinatorial picture of the complementary-edge-ideal terminology in the triangle-free setting.

The two strands of the subject therefore share a common pattern: a squarefree monomial ideal attached to complementary graph data, with homological behavior governed by induced subgraphs, duality, and explicit graph classes. In the $[n]$03-generated theory this leads to classifications of Cohen–Macaulayness, Gorensteinness, licci behavior, Rees algebras, and asymptotic invariants; in the $[n]$04-dimensional flag-complex theory it leads to exact forbidden-subgraph criteria for linear relations and linear resolutions of squarefree powers.