Bicoms: Bi-Cohen–Macaulay Graphs

• Bicoms are bi-Cohen–Macaulay graphs defined by finite simple graphs with edge ideals and Alexander duals that are both Cohen–Macaulay and admit linear resolutions.
• They leverage techniques like Alexander duality, Hilbert–Burch theory, and Artinian reduction to establish precise homological and combinatorial criteria.
• Explicit classifications in bipartite and chordal cases connect bicoms to unique tree structures and inseparable models, offering actionable insights in commutative algebra and graph theory.

In commutative algebra and graph theory, “bicoms” informally denotes bi-Cohen–Macaulay graphs: finite simple graphs $G$ with no isolated vertices such that the edge ideal $I_G$ and its Alexander dual $I_G^\vee$ are both Cohen–Macaulay. For graphs on $[n]=\{1,\dots,n\}$, with edge ideal

$$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$

this places the subject at the intersection of edge ideals, Stanley–Reisner theory, Alexander duality, and linear resolutions. A central theme is that the bi-Cohen–Macaulay condition is far more rigid than ordinary Cohen–Macaulayness: in the bipartite and chordal cases it yields explicit classifications, and in general every bi-CM graph is controlled, up to separation, by a unique inseparable model attached to a tree (Herzog et al., 2015).

1. Definition and algebraic framework

Let $G$ be a finite simple graph on vertex set $[n]=\{1,\dots,n\}$, with no isolated vertices. Its edge ideal in

$S=K[x_1,\dots,x_n]$

is

$I_G=(x_ix_j:\{i,j\}\in E(G)).$

This is a squarefree monomial ideal, and the corresponding simplicial complex is the independence complex of $G$, whose faces are the independent sets of $I_G$0. Following Fløystad–Vatne, a squarefree monomial ideal $I_G$1 is called bi-Cohen–Macaulay if both $I_G$2 and $I_G$3 are Cohen–Macaulay; accordingly, $I_G$4 is bi-Cohen–Macaulay (bi-CM) over $I_G$5 if $I_G$6 is bi-Cohen–Macaulay (Herzog et al., 2015).

For a squarefree monomial ideal

$I_G$7

with $I_G$8 monomial prime ideals, the Alexander dual is

$I_G$9

For $I_G^\vee$0, the minimal primes correspond to minimal vertex covers, so $I_G^\vee$1 is the vertex cover ideal of $I_G^\vee$2. The graph-theoretic dictionary used throughout is standard: vertex covers $I_G^\vee$3 correspond to monomial prime ideals

$I_G^\vee$4

$I_G^\vee$5 is a minimal prime of $I_G^\vee$6 iff $I_G^\vee$7 is a minimal vertex cover, and independent sets are complements of vertex covers.

If $I_G^\vee$8 denotes the independence number of $I_G^\vee$9, then

$[n]=\{1,\dots,n\}$0

A useful structural consequence is that every bi-CM graph is connected. This is not a superficial property: if $[n]=\{1,\dots,n\}$1 were disconnected, then $[n]=\{1,\dots,n\}$2 would decompose in disjoint variables, and the tensor-product resolution would force degree $[n]=\{1,\dots,n\}$3 syzygies, so $[n]=\{1,\dots,n\}$4 would not have a linear resolution. This already indicates that bi-CM graphs are not simply Cohen–Macaulay graphs with an added duality statement; they are Cohen–Macaulay graphs subject to a strong homological constraint.

2. Equivalent criteria and homological signatures

The decisive equivalence is furnished by the Eagon–Reiner theorem: a squarefree monomial ideal $[n]=\{1,\dots,n\}$5 is Cohen–Macaulay iff $[n]=\{1,\dots,n\}$6 has a linear resolution. Applied to graphs, this yields

$[n]=\{1,\dots,n\}$7

Equivalently, if $[n]=\{1,\dots,n\}$8, then

$[n]=\{1,\dots,n\}$9

Thus the additional condition distinguishing bi-CM graphs from CM graphs is precisely the existence of a linear resolution for the edge ideal (Herzog et al., 2015).

A more concrete formulation is given by reduction modulo a maximal regular sequence of linear forms. If $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$0 has independence number $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$1 and $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$2 is infinite, then $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$3 is bi-CM iff $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$4 is CM and, for a maximal regular sequence of linear forms on $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$5, the Artinian reduction is

$$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$6

where $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$7 is a polynomial ring in $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$8 variables and $$I_G=(x_ix_j:\{i,j\}\in E(G)) \subset S=K[x_1,\dots,x_n],$$9 is its graded maximal ideal. This makes the bi-CM condition computationally transparent: after a regular sequence reduction, the ideal must become the square of the maximal ideal.

This reduction produces a package of equivalent numerical criteria. Let $G$0 be on $G$1 vertices with independence number $G$2. Then the following are equivalent: $G$3

$G$4

$G$5

and

$G$6

These are exactly the Betti numbers of $G$7, obtained via the Eagon–Northcott resolution. In practice, this means that a CM graph is bi-CM precisely when its number of edges, its minimal vertex covers, and the graded Betti table of its edge ideal match those of the square of a maximal ideal in $G$8 variables.

3. Explicit classifications in the bipartite and chordal cases

The bipartite case is completely rigid. Let $G$9 be a bipartite graph with bipartition

$[n]=\{1,\dots,n\}$0

Then $[n]=\{1,\dots,n\}$1 is bi-CM iff

$[n]=\{1,\dots,n\}$2

After suitable labeling, this is the staircase Ferrers graph in which $[n]=\{1,\dots,n\}$3 is adjacent to $[n]=\{1,\dots,n\}$4. The proof uses the known classification of CM bipartite graphs via posets: a CM bipartite graph is of the form $[n]=\{1,\dots,n\}$5 for a poset $[n]=\{1,\dots,n\}$6, with

$[n]=\{1,\dots,n\}$7

and the bi-CM condition forces $[n]=\{1,\dots,n\}$8 to be a chain. Thus, among bipartite graphs, bicoms are exactly the chain graphs of this staircase form (Herzog et al., 2015).

The chordal case is broader but still explicit. Let $[n]=\{1,\dots,n\}$9 be a chordal graph on $S=K[x_1,\dots,x_n]$0, and let

$S=K[x_1,\dots,x_n]$1

be the facets of the clique complex of $S=K[x_1,\dots,x_n]$2, equivalently the maximal cliques. Then $S=K[x_1,\dots,x_n]$3 is bi-CM iff either $S=K[x_1,\dots,x_n]$4, or $S=K[x_1,\dots,x_n]$5 and the following hold: $S=K[x_1,\dots,x_n]$6 with this union disjoint; each $S=K[x_1,\dots,x_n]$7 has exactly one free vertex $S=K[x_1,\dots,x_n]$8; and the restriction of $S=K[x_1,\dots,x_n]$9 to

$I_G=(x_ix_j:\{i,j\}\in E(G)).$0

is a clique. If $I_G=(x_ix_j:\{i,j\}\in E(G)).$1, $I_G=(x_ix_j:\{i,j\}\in E(G)).$2 is simply the complete graph.

This description is best understood as a “center-plus-free-vertices” decomposition. The nonfree vertices form a complete induced subgraph, called the center of $I_G=(x_ix_j:\{i,j\}\in E(G)).$3, while each maximal clique contributes exactly one free vertex attached to an appropriate subset of the center. A plausible implication is that, in the chordal world, the bi-CM condition selects those Cohen–Macaulay graphs whose facet structure is as compressed as possible while still permitting a linear resolution.

4. Relation trees and generic bi-CM graphs

The general classification is organized through the Alexander dual

$I_G=(x_ix_j:\{i,j\}\in E(G)).$4

For a bi-CM graph, $I_G=(x_ix_j:\{i,j\}\in E(G)).$5 is a codimension $I_G=(x_ix_j:\{i,j\}\in E(G)).$6 Cohen–Macaulay monomial ideal with linear resolution, so Hilbert–Burch theory applies. If $I_G=(x_ix_j:\{i,j\}\in E(G)).$7 has minimal generators

$I_G=(x_ix_j:\{i,j\}\in E(G)).$8

a relation matrix $I_G=(x_ix_j:\{i,j\}\in E(G)).$9 for $G$0 has size $G$1, and because the resolution is linear, the first syzygies can be chosen of binomial type

$G$2

Each row of $G$3 therefore has exactly two nonzero entries, variables with opposite signs. From such a matrix one defines a graph $G$4: its vertices are $G$5, and $G$6 is an edge of $G$7 iff some row of $G$8 has nonzero entries in columns $G$9 and $I_G$00. This graph is always a tree, called a relation tree of $I_G$01 (Herzog et al., 2015).

Conversely, starting with any tree $I_G$02 on $I_G$03, one constructs a generic Hilbert–Burch matrix. If the edges of $I_G$04 are

$I_G$05

and $I_G$06 with $I_G$07, define the $I_G$08-th row of an $I_G$09 matrix $I_G$10 by

$I_G$11

Let $I_G$12 be the ideal of maximal minors of $I_G$13. By Hilbert–Burch, $I_G$14 is codimension $I_G$15, Cohen–Macaulay, with linear resolution. The associated graph $I_G$16 is then defined by

$I_G$17

For any tree $I_G$18, the graph $I_G$19 is bi-CM.

The structure of $I_G$20 is completely explicit. If $I_G$21 is the unique path from $I_G$22 to $I_G$23 in $I_G$24, define

$I_G$25

Then the vertices of $I_G$26 are the oriented edges of $I_G$27: $I_G$28 Two vertices $I_G$29 and $I_G$30 form an edge of $I_G$31 iff there exists a path from $I_G$32 to $I_G$33 in $I_G$34 such that

$I_G$35

Thus each unoriented edge of $I_G$36 gives two vertices of $I_G$37, and the path combinatorics of the tree determines all adjacencies. If $I_G$38 has $I_G$39 vertices, then $I_G$40 has

$I_G$41

vertices and

$I_G$42

5. Separation, inseparability, and the classification up to trees

To classify all bi-CM graphs, the paper uses separation. Let $I_G$43 be a squarefree monomial ideal minimally generated by $I_G$44, and let $I_G$45 be a new indeterminate. A monomial ideal $I_G$46 is a separation of $I_G$47 for the variable $I_G$48 if $I_G$49 is the image of $I_G$50 under the specialization $I_G$51, both $I_G$52 and $I_G$53 divide some minimal generator of $I_G$54, and $I_G$55 is a non-zerodivisor on $I_G$56. An ideal is separable if it admits a separation, and inseparable otherwise. For graphs, if $I_G$57, a separation is again an edge ideal

$I_G$58

for some graph $I_G$59 with one extra vertex; the original graph is recovered by identifying the new vertex with the old one (Herzog et al., 2015).

Separation preserves the relevant homological data: $I_G$60 is bi-CM iff any separation $I_G$61 is bi-CM. This makes inseparable models canonical representatives. The graph-theoretic criterion for inseparability is elegant. For a vertex $I_G$62, let $I_G$63 be its neighborhood, and let $I_G$64 be the complement of the induced subgraph on $I_G$65. Then

$I_G$66

Equivalently, $I_G$67 is disconnected iff

$I_G$68

such that every vertex of $I_G$69 is adjacent in $I_G$70 to every vertex of $I_G$71.

The main classification theorem has three parts. First, for any tree $I_G$72, the generic graph $I_G$73 is an inseparable bi-CM graph. Second, for any inseparable bi-CM graph $I_G$74, there exists a unique tree $I_G$75 such that

$I_G$76

Third, for any bi-CM graph $I_G$77, there exists a tree $I_G$78 such that $I_G$79 is an inseparable model of $I_G$80. This is the paper’s classification “up to separation”: every bicon is obtained from a unique tree-shaped generic model by finitely many specializations, and the inseparable bicoms are in bijection with trees.

6. Examples, non-examples, and conceptual significance

Several standard families illustrate the theory sharply (Herzog et al., 2015). Every complete graph $I_G$81 is bi-CM: it is chordal with one facet, its independence number is $I_G$82, and

$I_G$83

Among bipartite graphs, the staircase graph with edges

$I_G$84

is bi-CM, while $I_G$85 is not. Among chordal graphs, a triangle with one pendant edge is not bi-CM because its maximal cliques are not disjoint, and a star $I_G$86 with $I_G$87 is not bi-CM for the same structural reason.

The examples attached to trees are more revealing. The paper gives a bi-CM graph on $I_G$88 with edges

$I_G$89

whose Alexander dual has generators

$I_G$90

This graph has multiple relation trees, showing that relation trees need not be unique for general bi-CM graphs. By contrast, uniqueness is recovered precisely in the inseparable case. Another example compares the triangle $I_G$91 with the path on $I_G$92 vertices: the path is an inseparable model of the triangle, obtained by separating one vertex.

Conceptually, bicoms are important because they identify the graphs for which the edge ideal is simultaneously Cohen–Macaulay and as homologically simple as possible, namely $I_G$93-linear after Alexander duality. In the language of simplicial complexes, they are exactly the independence complexes whose Stanley–Reisner ideal and Alexander dual are both Cohen–Macaulay. A plausible implication is that their rigidity makes them unusually well suited as test objects at the interface of graph theory and commutative algebra: explicit enough to classify, but rich enough to encode Hilbert–Burch matrices, Alexander duality, and tree combinatorics in a single framework.