Unmixed Edge Ideals and the Cohen-Macaulay Condition

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 offfrom 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 gener- ators speciﬁed as follows: xixj 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)= x x ,x x ,x x ,x x ,x x . ! 1 2 1 3 2 3 3 4 4 5" 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)= x ,x ,x x ,x ,x x ,x ,x x ,x ,x x ,x ,x . ! 1 2 4" ∩ ! 1 3 4" ∩ ! 1 3 5" ∩ ! 2 3 4" ∩ ! 2 3 5" 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 Kn. Thus, the variety defined by I(G)—call it V (G)—is a finite union of coordinate subspaces of Kn. 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 Nullstel- lensatz 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 = xi , . . . , xi . Note that P I(G) if and only ! 1 s " ⊃ if P is a vertex cover of G. This small observation is crucial: it re- lates a combinatoric property to an algebraic property. Accordingly, if xi , . . . , xi is a minimal vertex cover of G, then, by Lemma 2.7, { 1 s } 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)= x ,x ,x x ,x ,x x ,x ,x x ,x ,x x ,x ,x . ! 1 2 4" ∩ ! 1 3 4" ∩ ! 1 3 5" ∩ ! 2 3 4" ∩ ! 2 3 5" 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.Acomplete graph is a graph where each vertex is connected to every other vertex. A complete graph on n vertices is denoted Kn.Aclique 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 com- c plement G . The largest clique in G is a K3, and the largest clique in c G is a five-way tie among the edges, all of which are K2’s. x3 x3

x2 x4 x2 x4

x1 x5 x1 x5 GGc

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)= x ,x ,x x ,x ,x x ,x ,x x ,x ,x x ,x ,x ! 1 2 4" ∩ ! 1 3 4" ∩ ! 1 3 5" ∩ ! 2 3 4" ∩ ! 2 3 5" x3 x3

x2 x4 x2 x4

x1 x5 x1 x5 GGc 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. c Let Mc denote the adjacency matrix of G . It is well known that k th (Mc )i,j is the number of walks of length k that start at the i vertex and and end at the jth vertex. (The proof of this is a straightforward induction argument.) In particular, the ith entry on the diagonal of k th Mc is the number of walks of length k that start and end at the i 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 2 and only if every entry on the diagonal of Mc 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

3 entry on the diagonal of Mc is zero. Therefore, it is possible to read 2 3 oﬀfrom Mc and Mc 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 x2 x4

x1 x5 x1 x5 GGc In this case, we have: 0 0 0 1 1 0 0 0 1 1   Mc = 0 0 0 0 1 1 1 0 0 0   1 1 1 0 0   and   2 2 1 0 0 0 0 0 4 5 2 2 1 0 0 0 0 0 4 5 2   3   Mc = 1 1 1 0 0 Mc = 0 0 0 2 3 0 0 0 2 2 4 4 2 0 0     0 0 0 2 3 5 5 3 0 0      2    Since the diagonal of Mc contains only non-zero entries and the diag- 3 onal of Mc 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 prod- uct 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)= x ,x ,x x ,x ,x x ,x ,x x ,x ,x x ,x ,x . ! 1 2 4" ∩ ! 1 3 4" ∩ ! 1 3 5" ∩ ! 2 3 4" ∩ ! 2 3 5" Therefore,

I(G)∨ = x x x ,x x x ,x x x ,x x x ,x x x . ! 1 2 4 1 3 4 1 3 5 2 3 4 2 3 5" 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 6 G 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 Cohen- Macaulay 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 equiv- alent 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. Addison- Wesley, 1969. [2] D. Cox, J. Little, D. O’Shea. Ideals, Algorithms, and Varieties. Springer- Verlag, 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.