UNMIXED EDGE IDEALS AND THE
COHEN-MACAULAY CONDITION
GREG BURNHAM
1. Introduction
To every simple graph G on n vertices we associate an ideal I(G)
in the polynomial ring K[x1 , . . . , xn ], where K is any field. Certain
algebraic properties of this edge ideal can be read off from purely combinatorial information about the graph.
In Section 2 we explore one such connection between the minimal
primary decompositions of the edge ideal on the algebraic side, and
the minimal vertex covers of G on the combinatorial side. This allows
us to state in purely graph-theoretic terms precisely when an edge ideal
is unmixed of a particular dimension.
In Section 3 we use this characterization along with a theorem of
Eagon and Reiner to give, in graph-theoretic terms, a necessary condition for an edge ideal to be Cohen-Macaulay.
The author would like to acknowledge Jessica Sidman for on-going
and unflagging support and advice, as well as Adam Van Tuyl and Tai
Huy Ha for several indispensable suggestions and references.
2. Unmixed Edge Ideals
Definition 2.1. Suppose G is a graph with vertices x1 , . . . , xn . The
edge ideal of G, denoted I(G), is the ideal of K[x1 , . . . , xn ] with generators specified as follows: xi xj is a generator of I(G) if and only if xi
is connected to xj in G.
Example 2.2. The graph G shown below
x3
x2
x4
x1
x5
G
This work was partially completed at the Mount Holyoke College REU, which is
partially funded by NSF grant DMS-0849637.
1
2
GREG BURNHAM
has edge ideal:
I(G) = !x1 x2 , x1 x3 , x2 x3 , x3 x4 , x4 x5 ".
Definition 2.3. Let I be an ideal in K[x1 , . . . , xn ]. A prime ideal that
contains I and is minimal—with respect to inclusion—among all such
prime ideals is called a minimal prime ideal of I.
Fact 2.4. An ideal I ⊆ K[x1 , . . . , xn ] is radical if and only if it is equal
to the intersection of its minimal prime ideals. If I is radical, then I
cannot be written as the intersection of prime ideals in any other way.
For a proof, see Chapter 4 of [1]. Since an edge ideal I(G) is generated by square-free monomials, it is radical. Fact 2.4 thus shows
that an edge ideal is the intersection of its minimal prime ideals. We
refer to the expression of I(G) in this way as the minimal primary
decomposition of I(G).
Caution 2.5. In general, an ideal I ⊂ K[x1 , . . . , xn ] may be expressible
as several different ideal-theoretic intersections, all bearing the name
“minimal primary decomposition.” The reader should bear in mind
that our definitions here accord with broader terminology only in the
case of radical ideals.
Example 2.6. The minimal primary decomposition of the edge ideal
I(G) from Example 2.2 is:
I(G) = !x1 , x2 , x4 " ∩ !x1 , x3 , x4 " ∩ !x1 , x3 , x5 " ∩ !x2 , x3 , x4 " ∩ !x2 , x3 , x5 ".
We have not yet developed a way to prove that these really are the
minimal prime ideals of I(G). We now collect the necessary tools.
Lemma 2.7. Let I(G) ⊂ K[x1 , . . . , xn ] be an edge ideal. Then, every
minimal prime ideal of I is generated by variables.
Proof. By Proposition 9.1 in [2], the variety of any ideal that is
generated by monomials is a finite union of coordinate subspaces of
K n . Thus, the variety defined by I(G)—call it V (G)—is a finite union
of coordinate subspaces of K n .
Now, a coordinate subspace is also the variety of an ideal generated
by variables—specifically those variables corresponding to axes not in
the subspace. Note that such ideals are prime and hence radical, and
recall that taking radicals commutes with taking intersections. Thus,
V (G) is the variety of some radical ideal that can be expressed as the
intersection of ideals generated by variables.
Let K be the algebraic closure of K. In K[x1 , . . . , xn ], the Nullstellensatz gives that there is only one radical ideal which gives rise to
UNMIXED EDGE IDEALS AND THE COHEN-MACAULAY CONDITION
3
V (G). Thus, I(G) itself can be expressed as the intersection of ideals
generated by variables, in the ring K[x1 , . . . , xn ]. But this relationship
will hold regardless of ground field, so we have in addition that I(G)
can be expressed as the intersection of ideals generated by variables in
the ring K[x1 , . . . , xn ].
Since I(G) is radical, we have from Fact 2.4 that this must be the
minimal primary decomposition of I(G). Thus, all of the minimal
primes of I(G) are generated by variables. !
Definition 2.8. Let G be a graph with vertices V = {x1 , . . . , xn }. A
subset A ⊆ V is a vertex cover of G if every edge in G is incident to
some vertex in A. A vertex cover A is minimal if no proper subset of
A is a vertex cover.
Example 2.9. In the graph G below, the set {x1 , x2 , x3 , x5 } is a vertex
cover, but it is not minimal because the set {x1 , x3 , x5 } is also a vertex
cover.
x3
x2
x4
x1
x5
G
Proposition 2.10. (Villarreal, 6.2.5) Let G be a graph with vertices
x1 , . . . , xn . Then, an ideal generated by variables is a minimal prime
of I(G) if and only if the corresponding vertices are a minimal vertex
cover of G.
Proof. Let P = !xi1 , . . . , xis ". Note that P ⊃ I(G) if and only
if P is a vertex cover of G. This small observation is crucial: it relates a combinatoric property to an algebraic property. Accordingly,
if {xi1 , . . . , xis } is a minimal vertex cover of G, then, by Lemma 2.7,
!x1 , . . . , xr " is a minimal prime of I(G).
Conversely, if P is a minimal prime of I(G) then, by Lemma 2.7, P
is generated by variables, and so the corresponding vertices must be a
vertex cover. Since the ideal is a minimal prime, the vertices must be
a minimal cover. !
Definition 2.11. An edge ideal is unmixed of dimension r if all of
its minimal prime ideals are generated by n − r variables. A graph is
unmixed of dimension r if its edge ideal is unmixed of dimension r.
4
GREG BURNHAM
Example 2.12. We may now check that Example 2.6 gives the minimal
primary decomposition of the edge ideal from Example 2.2.
x3
x2
x4
x1
x5
G
G has precisely five minimal vertex covers, corresponding to the five
ideals in what was claimed to be the primary decomposition of I(G):
I(G) = !x1 , x2 , x4 " ∩ !x1 , x3 , x4 " ∩ !x1 , x3 , x5 " ∩ !x2 , x3 , x4 " ∩ !x2 , x3 , x5 ".
Thus, in addition, I(G) and G are unmixed of dimension 5 − 3 = 2.
Proposition 2.10 allows us to describe the minimal primes of I(G) in
terms of the the minimal vertex covers of G. It is useful to restate this
in a slightly different way.
Definition 2.13. The complement of a graph G, denoted Gc , is the
graph with the same vertex set of G but the “opposite” edge set: an
edge is in Gc if and only if that same edge is not in G. A complete
graph is a graph where each vertex is connected to every other vertex.
A complete graph on n vertices is denoted Kn . A clique of G is a
subgraph of G that is isomorphic to a complete graph.
Example 2.14. On the left is a graph G and on the right is its complement Gc . The largest clique in G is a K3 , and the largest clique in
Gc is a five-way tie among the edges, all of which are K2 ’s.
x3
x3
x2
x4
x1
x5
G
x2
x4
x1
x5
Gc
Proposition 2.15. Let G be a graph with vertex set V = {x1 , . . . , xn }.
Then the ideal in K[x1 , . . . , xn ] generated by the variables corresponding
to a set A ⊆ V is a minimal prime of I(G) if and only if the subgraph
of Gc on the vertices V − A is a maximal clique in Gc .
Suppose that A ⊆ V is any vertex cover of G. Label the vertices
of G so that A = {x1 , . . . , xs } and V − A = {y1 , . . . , yt }. Note that
since A is a vertex cover, no two of the yj are connected by an edge:
UNMIXED EDGE IDEALS AND THE COHEN-MACAULAY CONDITION
5
if this were so, this edge would not be “covered” by the vertices in A.
Therefore, the vertices in V − A are mutually disjoint—such a set of
vertices is called an independent set—and hence the subgraph of Gc on
these same vertices is a clique. Conversely, if V − A is a clique in Gc
then, there are no edges between the yj in G, and hence all edges are
incident to at least one of the xi . Thus A is a vertex cover.
We have shown that the set A is a vertex cover of G if and only if
V − A is a clique in Gc . That is, vertex covers of G are in one-to-one
correspondence with cliques of Gc . Ordering by inclusion, it follows
that A is a minimal vertex cover of G if and only if V − A is a maximal
clique in Gc . Proposition 2.10 now gives the desired conclusion. !
Example 2.16. Recal that for G shown below we have:
I(G) = !x1 , x2 , x4 " ∩ !x1 , x3 , x4 " ∩ !x1 , x3 , x5 " ∩ !x2 , x3 , x4 " ∩ !x2 , x3 , x5 "
x3
x3
x2
x4
x1
x5
x2
x4
x1
x5
c
G
G
As predicted by Proposition 2.15, the five ideals in the minimal primary
decomposition of I(G) correspond to the five edges of Gc .
Corollary 2.17. An edge ideal I(G) is unmixed of dimension r if and
only if all the maximal cliques of Gc have size r. !.
Example 2.18. Corollary 2.17 shows that I(G) is unmixed of dimension 2 if and only if Gc contains no isolated vertices (cliques of size 1)
and no triangles (cliques of size 3). It is computationally easy to check
these conditions.
Let Mc denote the adjacency matrix of Gc . It is well known that
(Mck )i,j is the number of walks of length k that start at the ith vertex
and and end at the j th vertex. (The proof of this is a straightforward
induction argument.) In particular, the ith entry on the diagonal of
Mck is the number of walks of length k that start and end at the ith
vertex of Gc .
Now, a vertex v is isolated if and only if there are no walks of length
2 that start and end at v. Thus, Gc contains no isolated vertices if
and only if every entry on the diagonal of Mc2 is non-zero. Similarly, a
vertex v is in a triangle if and only if there is a walk of length 3 that
starts and ends at v. Thus, Gc contains no triangles if and only if every
6
GREG BURNHAM
entry on the diagonal of Mc3 is zero. Therefore, it is possible to read
off from Mc2 and Mc3 whether I(G) is unmixed of dimension 2.
For a specific example, consider the graph G and its complement Gc
below:
x3
x3
x2
x4
x1
x2
x5
x1
and

2
2

Mc2 = 
1
0
0
2
2
1
0
0

0
0

Mc = 
0
1
1
1
1
1
0
0
x5
Gc
G
In this case, we have:
x4
0
0
0
2
2
0
0
0
1
1
0
0
0
0
1

0
0

0

2
3
1
1
0
0
0

1
1

1

0
0

0
0

Mc3 = 
0
4
5
0
0
0
4
5
0
0
0
2
3
4
4
2
0
0

5
5

3

0
0
Since the diagonal of Mc2 contains only non-zero entries and the diagonal of Mc3 contains only 0’s, this confirms that I(G) is unmixed of
dimension 2.
3. Cohen-Macaulay Edge Ideals
Definition 3.1. Suppose I(G) ⊂ K[x1 , . . . , xn ] is an edge ideal with
minimal primes P1 , . . . , Pk . Let mi be the monomial given by the product of the variables that generate Pi . Then, the Alexander dual of I(G),
denoted I(G)∨ , is the ideal of K[x1 , . . . , xn ] generated by the mi .
Example 3.2. Suppose I(G) is the edge ideal for the graph G below.
x3
x2
x4
x1
x5
G
UNMIXED EDGE IDEALS AND THE COHEN-MACAULAY CONDITION
7
We saw in Example 2.12 that I(G) has minimal primary decomposition:
I(G) = !x1 , x2 , x4 " ∩ !x1 , x3 , x4 " ∩ !x1 , x3 , x5 " ∩ !x2 , x3 , x4 " ∩ !x2 , x3 , x5 ".
Therefore,
I(G)∨ = !x1 x2 x4 , x1 x3 x4 , x1 x3 x5 , x2 x3 x4 , x2 x3 x5 ".
Definition 3.3. The edge ideal of a hypergraph is the ideal generated
by products of variables corresponding to the hypergraph’s hyperedges.
Thus, for instance, the edge ideal of a 3-uniform hypergraph with s
edges is generated by s square-free cubic monomials.
Definition 3.4. Let G be a graph with vertex set V = {x1 , . . . , xn }
and let A1 , . . . , Ak ⊂ V be the minimal vertex covers of G. Define the
vertex cover hypergraph, denoted H(G), to be the hypergraph on V
with hyperedges precisely A1 , . . . , Ak . For an arbitrary hypergraph H,
we say that H is a vertex cover hypergraph if there is some G such that
H = H(G).
Proposition 3.5. Let G be a graph. Then, the edge ideal of H(G) is
the Alexander dual of I(G).
Proof. The statement follows immediately from Proposition 2.10 and
the definitions of H(G) and the Alexander dual. !
If H = H(G) for some G then the edges of H must correspond to
the complements of the maximal cliques in Gc . Knowing the maximal
cliques of Gc entirely determines Gc , and hence G, so every graph can
be paired with a unique vertex cover hypergraph.
Many hypergraphs encode a graph in the same way a vertex hypergraph does: the maximal cliques in the complement of the graph are
specified by the edges of the hypergraph. However, two hypergraphs
can correspond to the same graph in this way, as the following example
shows. Thus, not all hypergraphs are vertex cover hypergraphs.
Example 3.6. Consider the graph G depicted below.
1
2
3
4
5
G
6
G has eight minimal vertex covers so H(G) has eight edges. Specifically,
E(H(G)) = { {1, 2, 3}, {1, 2, 6}, {1, 3, 5}, {1, 5, 6},
{2, 3, 4}, {2, 4, 6}, {3, 4, 5}, {4, 5, 6} }.
8
GREG BURNHAM
However, the hypergraph H " with edge set
E(H " ) = { {1, 2, 3}, {4, 5, 6}, {1, 2, 6}, {3, 4, 5},
{1, 3, 5}, {2, 4, 6} }
gives rise to the same graph, if the hyperedges are taken to encode the
maximal cliques in Gc .
By Proposition 2.10, the Alexander dual of I(G), denoted I(G)∨ , is
simply the edge ideal of H(G). It is also clear that H(G) is (n − r)uniform if and only if I(G) is unmixed of dimension r.
One problem of interest is classifying which graphs have CohenMacaulay edge ideals. The following allows us to restrict our attention
to unmixed graphs.
Proposition 3.7. If I(G) is Cohen-Macaulay then I(G) is unmixed.
Proof. See 6.2.15 in [6]. !
In particular, we may restrict our attention to graphs G with H(G)
uniform. The following theorem allows us to phrase the Cohen-Macaulay
condition entirely in terms of H(G):
Theorem 3.8. (Eagon-Reiner) An edge ideal I(G) is Cohen-Macaulay
if and only if the syzygies of I(G)∨ are all linear.
Proof. See 5.56 in [5]. !
Since H(G) = I(G)∨ , Theorem 3.8 then gives the following.
Corollary 3.9. Determining which graphs are Cohen-Macaulay is equivalent to determining which uniform vertex cover hypergraphs have linear resolutions. !
There has been some progress toward solving the larger problem of
determining which uniform hypergraphs have linear resolutions. Fröberg
completely answered the question in the case of 2-uniform hypergraphs,
[3], and Van Tuyl and Ha determined when a properly-connected uniform hypergraph has linear first syzygies, [4]. The author does not
know whether, by restricting attention to uniform vertex cover hypergraphs, the question of linear resolutions becomes any easier.
References
[1] M. Atiyah, I. MacDonald. Introduction to Commutative Algebra. AddisonWesley, 1969.
[2] D. Cox, J. Little, D. O’Shea. Ideals, Algorithms, and Varieties. SpringerVerlag, 1997.
[3] R. Fröberg. “On Stanley-Reisner rings”, Topics in algebra 26:2 (1990), 57–70.
UNMIXED EDGE IDEALS AND THE COHEN-MACAULAY CONDITION
9
[4] H. T. Ha, A. Van Tuyl. “Monomial Ideals, Edge Ideals, and Their Graded Betti
Numbers”, Journal of Algebraic Combinatorics 27 (2008), 215–245.
[5] E. Miller, B. Sturmfels. Combinatorial Commutative Algebra. Springer-Verlag,
2005.
[6] R. Villarreal. Combinatorial Optimization Methods in Commutative Algebra.
Preprint, 2009.