The Topology of Intersections of Coloring Complexes

The Topology of Intersections of Coloring Complexes Jakob Jonsson October 19, 2005 Abstract In a construction due to Steingrímsson, a simplicial complex is associated to each simple graph; the complex is the coloring complex of the graph. Some of the nonfaces of the coloring complex correspond in a natural manner to proper colorings of the graph. Indeed, the h-vector of the complex is a certain affine transformation of the chromatic polynomial. In an earlier paper we showed that the coloring complex is constructible and hence Cohen-Macaulay. In this paper, we consider some other variants of the coloring complex corresponding to colorings that are proper for a certain number of graphs in a given sequence of graphs. Again, these complexes have attractive topological properties as soon as certain technical conditions are satisfied. 1 Introduction The primary goal of this paper is to analyze the topology of simplicial complexes related to a certain simplicial complex G defined in terms of a (simple) graph G = (V, E); see Section 1.1 for basic concepts and Section 2 for a formal definition of G. The complex G is the coloring complex of G and was introduced by Steingrímsson [8]. The faces in G can be interpreted as chains φ X 1 X 2... X k V of vertex sets with the property that the component X i \X i 1 contains an edge from G for some i {1,..., k +1} (X 0 = φ and X k+1 = V ). Certain minimal nonfaces of G correpond in a natural manner to proper colorings of G. In particular, the h-vector of G is an affine transformation of the chromatic polynomial χ G of G; see Steingrímsson [8] for details and for a ring-theoretic interpretation. For example, Figure 1 illustrates the coloring complex C4 of the square graph C 4. C4 contains the 1-cell {1, 134}, because the component 134 \ 1 = 34 is an edge in C 4. However, C4 does not contain the 1-cell {1, 124} as the components 1, 124 \ 1 = 24, and 1234 \ 124 = 3 contain no edges from C 4. Research financed by EC s IHRP Programme, within the Research Training Network Algebraic Combinatorics in Europe, grant HPRN-CT-2001-00272. 1

1 1 2 134 14 4 124 2 12 123 34 234 23 4 3 3 C 4 C4 Figure 1: The coloring complex of the square graph C 4. In an earlier paper [7], we showed that G is constructible and hence Cohen- Macaulay. The purpose of this paper is to prove similar results for a class of related complexes. Specifically, we consider a sequence G = (G 1,..., G m ) of m graphs on a fixed vertex set. We show how to associate a simplicial complex G,r corresponding to colorings that are proper for at least r out of the m graphs; 1 r m. G,m coincides with the coloring complex of the union of all graphs in the sequence, while G,1 has an h-vector determined by an alternating sum of chromatic polynomials related to the graphs in the sequence. For m = 2, G,r turns out to be constructible as soon as the underlying graphs have disjoint and nonempty edge sets. For larger values of m, we have been able to establish that the complexes are Cohen-Macaulay as soon as the graph sequence satisfies a technical condition. While we do not know whether these Cohen-Macaulay complexes are also constructible, they are what we refer to as pseudo-constructible or P C. The class of P C complexes is defined just as the class of constructible complexes, except that we are also allowed to go backwards in the definition: Given two P C complexes 1 and 2 of the same dimension d such that the union 1 2 is P C, their intersection 1 2 is P C if dim 1 2 is pure of dimension d 1. Related work Hultman [6] has demonstrated that the coloring complex of any graph is not only constructible but also shellable. Phil Hanlon (personal communication) obtained a refined result about the homology by imposing a Hodge structure on the complex. The dimensions of the resulting Hodge pieces are exactly the absolute values of the coefficients of the chromatic polynomial. It is plausible that some of the above results would extend to the constructions in this paper. 2

1.1 Basic concepts Let G = (V, E) be a simple graph; V is the set of vertices and E ( V 2) is the set of edges in G. The edge between a and b is denoted as ab. (More generally, a set {a 1, a 2,..., a d } is sometimes denoted as a 1 a 2... a d.) Two vertices a and b are adjacent in G if ab is an edge in G. A singleton vertex is a vertex not adjacent to any other vertex in G. Whenever the underlying vertex set V is fixed, we identify G with its edge set E; e G means that e E. G is empty if E = φ and nonempty otherwise. Let G \ e = (V, E \ e) and G e = (V, E e). For W V, let G(W ) = (W, E ( ) W 2 ); G(W ) is the induced subgraph of G on the vertex set W. For a graph G and an edge e = xy, G/e = (V/e, E/e) is the contraction along the edge e. Here, V/e = V \y and E/e = (E {ax : a x and ay E}) ( ) V \y 2. We refer to y as the removed vertex. For r 1, [r] denotes the set {1,..., r}. An r-coloring of G is a function γ : V [r]. A coloring γ is proper if γ(v) γ(w) whenever vw E. The chromatic polynomial of G is the function χ G : C C with the property that χ G (r) is equal to the number of proper r-colorings for r 0. It is well-known that χ G is indeed a polynomial. A simplicial complex on a finite set V is a nonempty family of subsets of V closed under deletion of elements. Members of a simplicial complex Σ are denoted as faces; maximal faces in Σ are denoted as facets. The dimension of a face σ is defined as σ 1. The dimension of a complex Σ is the maximal dimension of any face in Σ. A complex is pure if all facets have the same dimension. For d 1, the d-simplex is the simplicial complex of all subsets of a set V of size d+1. The boundary of the d-simplex is obtained by removing the full set V. Note that the ( 1)-simplex is the complex containing only the empty set. For two simplicial complexes 1 and 2, 1 2 means that 1 and 2 are isomorphic (i.e., identical up to the names of the vertices). Whenever we discuss the homology of a simplicial complex, we are referring to the reduced Z-homology. 2 The coloring complex For a given nonempty graph G on a vertex set V of size n, the coloring complex G introduced in [8] is defined as follows. Let F V be the family of all nonempty and proper subsets of V (thus φ and V are not contained in F V ). A (not necessarily nonempty) family {X 1, X 2,..., X k } = X 1 X 2... X k of sets from F V, ordered from the smallest to the largest, is contained in G if and only if φ = X 0 X 1 X 2... X k X k+1 = V (1) and at least one of the sets Y i = X i \ X i 1 (2) 3

(1 i k + 1) is not a stable set in G; a set S is stable in G if no edge in G is contained in S. We refer to X 1 X 2... X k as a chain and to the sets Y i in (2) as the components of the chain X 1 X 2... X k. Note that the set of 0-cells (vertices) in G is a proper subset of F V if G is bipartite. For example, the complex in Figure 1 does not contain the 0-cells 13 and 24. The coloring complex of the graph with the single edge e will be denoted as e. While this notation is ambiguous, the underlying vertex set will always be clear from context. A chain X 1 X 2... X k satisfying (1) can be interpreted as a coloring where the vertices in the i-th component Y i is given color i for 1 i k + 1. We will refer to this coloring as the coloring induced by X 1 X 2... X k. If X 1 X 2... X k G, then some component is non-stable, which means that the induced coloring is not proper. G is well-known to be pure of dimension n 3; see [8]. 2.1 Properties of the coloring complex It was demonstrated in [7] that G is a constructible complex. Definition 2.1 The class of constructible simplicial complexes is defined recursively as follows. 1. Every simplex (including the ( 1)-simplex {φ}) is constructible. 2. If 1 and 2 are constructible complexes of dimension d and 1 2 is a constructible complex of dimension d 1, then 1 2 is constructible. The concept of constructible complexes was introduced by Hochster [5]. Every shellable complex is constructible, but the converse is not always true; see [2]. Definition 2.2 A simplicial complex is homotopy-cohen-macaulay (abbreviated homotopy-cm) if every link link (σ) of (including itself) is homotopy equivalent to a wedge of spheres of maximal dimension (i.e., link (σ) is (dim link (σ) 1)-connected; see [2]). Let R be Z or a field. is Cohen- Macaulay over R (denoted CM/R) if every link link (σ) of (including itself) is (dim link (σ) 1)-acyclic over R. A constructible complex is also homotopy-cm and CM/R, but the converse is not always true; see [2]. If a complex is CM/Z, then it is CM over any field. For the purposes of this paper, say that a CM/Z complex is homology-cm. Theorem 2.3 ([7]) For any nonempty graph G on n vertices, G is a constructible complex. As a consequence, G is homotopy-cm. In particular, G is homotopy equivalent to a wedge of spheres of dimension n 3. Remark. The homotopy type of G has been determined earlier in [4]. A crucial step in the proof of Theorem 2.3 was the observation for any graph G with at least two edges and any edge e in G that G = G\e e ; (3) G\e,e := G\e e G/e. (4) 4

Theorem 2.4 ([7]) If G is a graph with at least two edges, then the reduced Euler characteristic χ( G ) of G satisfies In particular, any graph G satisfies χ( G ) = χ( G\e ) χ( G/e ) + χ( e ). (5) χ( G ) = χ G ( 1) + ( 1) n, (6) where χ G is the chromatic polynomial of G. Thus G is homotopy equivalent to a wedge of χ G ( 1) 1 spheres of dimension n 3. Remark. (6) was first proved in [8] by Steingrímsson. For a simplicial complex of dimension d 1, the f-vector (f 0,..., f d 1 ) of is defined by letting f i be the number of faces of dimension i in. The h-vector (h 0,..., h d ) of is defined by the formula f 1 = 1. Let d f i 1 (t 1) d i = i=0 h(, t) = d h i t d i ; (7) i=0 d h i t i and f(, u) = i=0 With u = t/(1 t), it is easily seen that d f i 1 u i. i=0 h(, t) = f (, u). (8) (1 t) d As is shown in [8], (6) is a consequence of the following result. Theorem 2.5 (Steingrímsson [8]) Let u = t/(1 t). For any nonempty graph G, h( G, t) (1 t) n = f( G, u) (1 + u) 2 = T G (r + 1)t r, (9) r 0 where n is the number of vertices in G and T G (r) = r n χ G (r). Remark. To be precise, Theorem 2.5 is proved in [8] for the double cone of the complex G (meaning that chains are allowed to contain the empty set φ and the full set V ). Note that the polynomial f( G, u) (1 + u) 2 in (9) corresponds to the f-vector of this double cone. 3 Intersections of coloring complexes We have noted that the intersection G\e,e G/e of the two complexes G\e and e inherits the property of being pure and constructible, being a coloring 5

complex. A natural question to ask is whether this is true for a more general class of intersections. In the subsequent sections, we will show that this is indeed true for any intersection of two coloring complexes such that the two underlying graphs are edge-disjoint. For intersections of more than two complexes, we have succeeded in proving that an intersection is pure and homology-cm (see Section 2.1) as soon as the underlying graphs satisfy a certain technical condition diagonal cycle-freeness to be explained later in this section. However, we do not know whether such an intersection is also constructible and homotopy- CM in general. Let G = (G 1, G 2,..., G m ) be an arbitrary sequence of graphs on a fixed vertex set V such that all graphs have nonempty edge sets. Define G = G1,G 2,...,G m = Let G = (G 1,..., G m 2 ). It is clear that m Gi. i=1 G,G m 1 G m = ( G,G m 1 \ G ) G,G m ;. (10) the union is disjoint. This implies that χ( G ) = χ( G,G m 1 ) χ( G,G m 1 G m ) + χ( G,G m ). (11) Define the chromatic polynomial χ G of G in the same spirit; χ G (j) counts the number of colorings of the vertex set V with j colors such that the coloring is proper for at least one graph G i. The function χ G being a polynomial is a consequence of the recursive identity χ G (j) = χ G,G m 1 (j) χ G,G m 1 G m (j) + χ G,G m (j) (12) for all positive integers j. As an interesting special case, we have χ G1,G 2 = χ G1 χ G1 G 2 + χ G2. Proposition 3.1 Let m 1 be an integer and let G = (G 1,..., G m ) be a sequence of nonempty graphs on a vertex set of size n. Let G I = i I G i. The reduced Euler characteristic χ( G ) of G is (χ Gφ (t) = t n ). χ( G ) = ( 1) I χ GI ( 1) = χ G ( 1) + ( 1) n (13) I [m] Proof. Use induction over m. The case m = 1 is exactly Theorem 2.4. Consider a sequence G = (G, G m 1, G m ) = (G 1,..., G m 2, G m 1, G m ) 6

of length at least 2. By the induction hypothesis, we have χ( G,G m 1 ) = ( 1) I χ GI ( 1); χ( G,G m 1 G m ) = χ( G,G m ) = I [m 1] I [m 2] I [m]\{m 1} Consider a set I. There are four cases: ( 1) I (χ GI ( 1) χ GI {m 1,m} ( 1)); ( 1) I χ GI. I is contained in [m 2]. Then the term χ GI ( 1) appears in each of the three sums, each time with the same sign ( 1) I. I contains m 1 and m. Then χ GI ( 1) appears only in the second sum with sign ( 1) I 2 = ( 1) I. I contains m 1 but not m. Then χ GI ( 1) appears only in the first sum with sign ( 1) I. I contains m but not m 1. Then χ GI ( 1) appears only in the third sum with sign ( 1) I. To form the reduced Euler characteristic of G, use (11); thus we take the sum of the first and the third sum and subtract the second sum. It is clear that the first equality in (13) is satisfied. The second equality is proved in a similar manner. Now, as we have determined the Euler characteristic of G, we would like to proceed with a proof that G is homotopy equivalent to a wedge of spheres of maximal dimension. Unfortunately however, this is not true in general. Moreover, G is not a pure complex in general: Example 1. If V = [4], E 1 = {12, 13, 23}, and E 2 = {23, 24, 34}, then {23, 123} and {12} are both maximal in G1,G 2, where G i = ([4], E i ). Furthermore, dim H 1 ( G1,G 2 ) = 1 and dim H 0 ( G1,G 2 ) = 8. Example 2. For the complex to be pure, it is not even sufficient that the edge sets are disjoint. For example, if V = [5], E 1 = {12}, E 2 = {13, 34}, and E 3 = {23, 35}, then {123, 1234} and {12} are both maximal in G1,G 2,G 3, where G i = ([5], E i ). This time, dim H 1 ( G1,G 2,G 3 ) = 1 and dim H 0 ( G1,G 2,G 3 ) = 2. Yet, while there is no straightforward correspondence to Theorem 2.3 in general, we have already indicated that a certain class of complexes turns out to have attractive properties. A diagonal of a graph sequence G = (G 1, G 2,..., G m ) on a vertex set V is an edge set E = {e 1, e 2,..., e m } such that e i G i for 1 i m. The sequence G is diagonally cycle-free if all graphs have nonempty 7

and mutually disjoint edge sets and if the graph (V, E) is cycle-free for any diagonal E. (These two conditions can be merged into one condition if we interpret E = {e 1, e 2,..., e m } as a multi-set; if e i = e j for some i j, then E contains the cycle (e i, e j ).) For instance, the sequence in Example 2 is not diagonally cycle-free; the three edges 12 G 1, 13 G 2, and 23 G 3 form a triangle. However, the sequence obtained from (G 1, G 2, G 3 ) by removing 23 from G 3 is diagonally cycle-free. Proposition 3.2 Let G = (G 1, G 2,..., G m ) be a diagonally cycle-free sequence of graphs on a vertex set of size n. Then G is pure of dimension n m 2. Proof Consider a face σ in G and let Y 1,..., Y r be the components of σ. By definition of G, for each G i there is an edge e i contained in some component Y ji. Consider the diagonal E = {e 1,..., e m }. Being cycle-free, this edge set forms a forest with n m connected vertex components Y 1, Y 2,..., Y n m. This forest is a subgraph of the graph consisting of all edges contained in some Y i. In particular, {Y 1, Y 2,..., Y n m} is a subdivision of {Y 1,..., Y r }. As a consequence, σ is contained in a facet with components Y 1, Y 2,..., Y n m reordered in a manner consistent with the order of Y 1,..., Y r. Clearly, the dimension of this facet is the number n m of components minus 2. The generalization of Theorem 2.5 reads as follows. Theorem 3.3 Let G = (G 1, G 2,..., G m ) be a sequence of nonempty graphs on a vertex set of size n and let u = t/(1 t). Then h( G, t) (1 t) dim G+3 = f( G, u) (1 + u) 2 = T G (r)t r 1, r 1 where T G (r) = r n χ G (r). In particular, if G is diagonally cycle-free, then h( G, t) (1 t) n m+1 = r 1 T G (r)t r 1. (14) As a consequence, deg T G = n m if G is diagonally cycle-free. Proof. Use induction over m. m = 1 is Theorem 2.5. Consider m 2. Let Via induction, we obtain that G 1 = (G 1,..., G m 2, G m 1 ); G 2 = (G 1,..., G m 2, G m ); G 12 = (G 1,..., G m 2, G m 1 G m ). (15) f( Gi, u) (1 + u) 2 = r 1 T Gi (r)t r 1 8

for i = 1, 2, 12. (10) yields that f( G, u) (1 + u) 2 = (f( G1, u) + f( G2, u) f( G12, u)) (1 + u) 2 = T G1 (r)t r 1 + T G2 (r)t r 1 T G12 (r)t r 1 r 1 r 1 r 1 = [(12)] = r 1 T G (r)t r 1. The last claim (14) follows from the fact that h( G, t) is nonzero (h( G, 1) is the number of facets in G ) and Proposition 3.2; dim G = n m 2. 3.1 The homotopy type of the intersection of two coloring complexes While the previous section was devoted to enumerative properties of intersections of coloring complexes, we now proceed with a more detailed topological analysis. Our first object is to present a proof that G is constructible if G is a diagonally cycle-free graph sequence of length two, meaning that G = (G 1, G 2 ), where G 1 and G 2 are edge-disjoint. The proof is similar in nature to the proof of Theorem 2.3 but slightly more complicated. The reason is that the most obvious approach does not work straight away. Namely, consider a complex G1,G 2 such that G 1 and G 2 are edge-disjoint and such that G 1 contains at least two edges. Pick an edge e G 1. Then we have G1,G 2 = G1\e,G 2 e,g2. Via the appropriate induction hypothesis, the complexes in the right-hand side may be assumed to be constructible. Unfortunately however, their intersection G1\e,e,G 2 may be non-pure; the underlying sequence (G 1 \ e, e, G 2 ) is not necessarily diagonally cycle-free and is hence not necessarily constructible. Even worse, there are situations where all edges e in both graphs have this bad property; for example, consider G 1 = {12, 23, 34} and G 2 = {13, 24, 14}. Yet, as it turns out, this concern can be addressed if G1,e is replaced with a certain subcomplex obtained by replacing G 1 with a certain subgraph. Before we proceed with the main result, we state and prove the counterpart of (4) for sequences of arbitrary length. Lemma 3.4 Let m 2 and let G = (G 1,..., G m ) be a sequence of mutually edge-disjoint graphs on a fixed vertex set such that G m consists of a single edge e m. Then G = G/em, (16) where G/e m = (G 1 /e m,..., G m 1 /e m ). In addition, if G is diagonally cyclefree, then so is G/e m. Proof. First, we prove that if G = (G 1,..., G m ) and G m consists of a single edge e m, then G/e m = (G 1 /e m,..., G m 1 /e m ) is diagonally cycle-free if G is 9

diagonally cycle-free. Suppose that G/e m is not diagonally cycle-free. Then there are edges e i G i/e m for 1 i m 1 such that (e i 1,..., e i r ) forms a cycle for some i 1,..., i r [m 1]. With appropriate orientations of the edges, this can be described as the chain group identity r (e i r ) = 0. Let e i G i be edges such that e i maps to e i in G i/e m under the contraction operation. Then r (e i r ) = λ (w v) for some constant λ, where w v = (e m ) (thus e m = vw). In particular, j (e i j ) λ (e m ) = 0, which is a contradiction to the fact that G is diagonally cycle-free. Next, we want to settle (16); our proof is almost identical to the proof of (4) in [7] and is included for completeness. Let e m = xy and let y be the removed vertex in G/e m (of course, we remove the same vertex in each G i ). For a chain X e with components Y 1,..., Y k+1, let Y j be the component containing x and y (they must be in the same component). Define ϕ(x ) to be the chain with components Y 1,..., Y j \ y,..., Y k+1 (all components but Y j remain unchanged). This clearly gives an isomorphism from em to the complex Σ V \y of all possible chains of the form (1) on the vertex set V \ y. Namely, we may easily reconstruct a chain X = ϕ 1 (X ) em from a chain X Σ V \y by adding y to the component containing x. Hence we need only prove for each X em that X G1,...,G m 1 = i G i if and only if ϕ(x ) G/em = i G i/e m. It is clear that it suffices to consider each G i separately. Let X em and let Y j be the component containing x and y. Clearly, each of the other components is stable in G i if and only if it is stable in G i /e m. Moreover, the same is true for Y j. Namely, for each z Y j \ {x, y}, xz is an edge in G i /e m if and only if at least one of xz and yz is an edge in G i. It follows that X Gi if and only if ϕ(x ) Gi/e m, and we are done. Theorem 3.5 Assume that G = (G 1, G 2 ) is diagonally cycle-free. Then G is constructible. As a consequence, G is homotopy-cm. In particular, G is homotopy equivalent to a wedge of χ( G ) = χ G1 G 2 ( 1) χ G1 ( 1) χ G2 ( 1) + ( 1) n spheres of dimension n 4. Proof. We use induction on the size of the underlying vertex set and the number of edges in the graphs. Consider a diagonally cycle-free sequence G = (G 1, G 2 ). If one of the graphs, say G 2, consists of a single edge e 2, then by (4) we have that G = G1/e2 ; hence we obtain that G is constructible. Thus assume that the two graphs in G have at least two edges. For an edge e 1 G 1, let G 2,e1 be the graph obtained by removing all edges e 2 G 2 such that (G 1 \ e 1, e 1, e 2 ) is not diagonally cycle-free. This means that e 1 = ab and e 2 = ac for some vertices a, b, c and that bc G 1. For e 2 G 2, define G 1,e2 in the analogous manner. We claim that G1,G 2 = G1\e 1,G 2 e1,g 2,e1 for any e 1 = ab G 1. Namely, suppose that a chain X is a face in e1,g 2 but not in e1,g 2,e1. This implies that some edge e 2 say e 2 = ac is contained 10

in G 2 and in a component of X but not in G 2,e. Since e 2 shares an endpoint a with e 1, the set e 1 e 2 = abc must be contained in a component of X. Yet, by construction of G 2,e1, the edge bc is contained in G 1. As a consequence, X is a face of G1\e 1,G 2, and the claim follows by contradiction. The reason why we introduce G 2,e1 is that the two complexes G1\e 1,G 2 and e1,g 2,e1 have no facets in common (this is easily proven in the same manner as above), which implies that their intersection has dimension smaller than n 4. Note that the intersection satisfies G1\e 1,G 2 e1,g 2,e1 = G1\e 1,e 1,G 2,e1. The sequence (G 1 \ e 1, e 1, G 2,e1 ) is easily seen to be diagonally cycle-free; we have removed all edges from G 2 that may cause troubles. However, G 2,e1 is not necessarily nonempty. G 2,e1 being empty would be a bad property as it would imply that G1\e 1,e 1,G 2,e1 is the empty complex and hence not of the desired dimension n 5. Yet, as it turns out, we can always find some e 1 G 1 such that G 2,e1 is nonempty or some e 2 G 2 such that G 1,e2 is nonempty. Namely, pick an arbitrary e 1 = ab G 1 and assume that G 2,e is empty. Consider an edge in G 2, say e 2 = ac. Since e 2 is not in G 2,e1, the edge bc must be contained in G 1. In particular, bc is not contained in G 2, which implies that G 1,e2 = G 1,ac is nonempty. Without loss of generality, we may thus assume that we have an edge e 1 G 1 such that G 2,e1 is nonempty. By Lemma 3.4, G1\e 1,e 1,G 2,e1 = G1/e 1,G 2,e1 /e 1. Since G 1 /e 1 and G 2,e1 /e 1 are both nonempty, we obtain via induction that G1/e 1,G 2,e1 /e 1 is constructible of dimension (n 1) 4 = n 5. This concludes the proof. 3.2 Pseudo-constructible complexes and the homology of the intersection of coloring complexes Our next object is to analyze the homology of G = G1,...,G m for a diagonally cycle-free graph sequence G = (G 1,..., G m ). While we have been unable to prove that G belongs to the family of constructible complexes for m 3, we will demonstrate that G belongs to a larger family with some nice properties. We do not know whether all complexes in this larger family are homotopy-cm, but they are at least homology-cm (i.e., CM/Z). The family is defined as follows. Definition 3.6 Let W be any set (finite or infinite). We define the class of pseudo-constructible simplicial complexes on W, denoted P C(W ), recursively as follows. 1. Every finite simplex (including the ( 1)-simplex {φ}) with vertex set contained in W is P C(W ). 11

2. If 1 and 2 are P C(W ) complexes of dimension d 0 and 1 2 is a P C(W ) complex of dimension d 1, then 1 2 is P C(W ). 3. If 1, 2, and 1 2 are P C(W ) complexes of dimension d 0 and 1 2 is pure of dimension d 1, then 1 2 is P C(W ). A complex is P C if it is P C(W ) for some set W. It is clear that each P C complex is pure and that each P C(W ) complex has as vertex set a subset of the set W. By the following elementary but useful result, the class of P C complexes inherits some attractive properties of the subclass of constructible complexes. Theorem 3.7 For any set W, a P C(W ) complex is homology-cm. Proof. We need to prove that the property of being CM over Z is inherited for complexes derived as in steps 1, 2, or 3 of Definition 3.6. 1. Clearly, every simplex is CM. 2. Assume that 1 and 2 satisfy the conditions in Step 2 of Definition 3.6 and let 12 = 1 2 and = 1 2. We have to prove that 12 is CM/Z if 1, 2, and are CM/Z. Let σ be a face in 12. If σ i \ 3 i for i = 1 or 2, then link 12 (σ) = link i (σ), which is hence (dim link 12 (σ) 1)-acyclic by assumption. If σ 1 2, consider the Mayer-Vietoris long exact sequence H k (link (σ)) H k (link 1 (σ)) H k (link 2 (σ)) H k (link 12 (σ)) H k 1 (link (σ)) H k 1 (link 1 (σ)) H k 1 (link 2 (σ)) By assumption, this sequence vanishes for k < d 1 = dim link (σ), which implies that the only nontrivial part of the sequence is H d (link 1 (σ)) H d (link 2 (σ)) H d (link 12 (σ)) H d 1 (link (σ)). As a consequence, link 12 (σ) is (d 1)-acyclic. CM/Z. It follows that 12 is 3. Assume that 1 and 2 satisfy the conditions in Step 3 of Definition 3.6; again, let 12 = 1 2 and = 1 2. Consider a face σ 1 2. We obtain the same long exact sequence as in Step 2; again this sequence vanishes except for the short exact sequence H d (link 1 (σ)) H d (link 2 (σ)) H d (link 12 (σ)) H d 1 (link (σ)), which implies that link (σ) is (d 2)-acyclic. Hence is CM/Z. For a set V, recall that F V is the family of nonempty and proper subsets of V. Theorem 3.8 Let V be a vertex set of size n. If G = (G 1,..., G m ) is diagonally cycle-free on V, then G is P C(F V ) of dimension n m 3 and hence homology- CM. 12

Proof. Use induction on m to show that G is P C(F V ) if G is diagonally cyclefree. By Theorem 2.3, this is true for m = 1 (any constructible complex with vertex set X is P C(X)). Consider m 2. By Proposition 3.2, G is a pure complex of dimension n m 2. The property of being diagonally cycle-free is clearly preserved under union and deletion in the sense that (G 1,..., G i G i+1,..., G m ) and (G 1,..., Ĝi,..., G m ) (the hat denotes deletion) are diagonally cycle-free if (G 1,..., G m ) is diagonally cycle-free and m 2. Define G 1, G 2, and G 12 as in (15). Via induction, we obtain that the complexes G1, G2, and G12 = G1 G2 are P C(F V ) of dimension n m 1; G 1, G 2, and G 12 are clearly diagonally cycle-free. Since G = G1 G2 and since the dimension of G is n m 2, we conclude that G is P C(F V ). 3.3 Unions of intersections of coloring complexes An intersection of m coloring complexes Gi corresponds to colorings of vertices that are proper for at least one G i. We will now consider complexes corresponding to colorings that are proper for at least r of the m graphs, where 1 r m. For a graph sequence G = (G 1,..., G m ) and a set I = {i 1,..., i s }, let G I = (G i1,..., G is ). For 1 r m, define ( ) G,r = GI = Gi I [m]: I =m r+1 I [m]: I =m r+1 i I (17) It is easy to see that a chain is a face in G,r if and only if the corresponding components contain edges from at least m r + 1 of the graphs G 1,..., G m. In particular, non-faces correspond to colorings that are proper for at least m (m r + 1) + 1 = r graphs. Note that G,1 = G and G,m = m i=1 G i. (17) can be rewritten as G,r = = G[m]\J. I m r+1 GI J <r The set {J [m] : J < r} is a simplicial complex, which suggests the following generalization. For a simplicial complex Σ on the vertex set [m], define G,Σ = G[m]\σ. σ Σ We want to prove that G,Σ is P C(F V ) if Σ is P C([m]) and G is diagonally cycle-free. This is a consequence of the following more general result. Theorem 3.9 For an integer m 1 and a set W, let { σ : σ [m]} be a family of P C(W ) complexes such that the following hold: 13

dim σ = σ + c 1 for some fixed constant c 0; σ1 σ2 = σ1 σ 2 for all σ 1, σ 2 [m]. Let Σ be a P C([m]) complex. Then Σ = σ Σ is P C(W ) of dimension dim Σ + c. If Σ is P C([m]) and σ is CM/R for each σ Σ, then Σ is CM/R. If Σ is constructible and σ is constructible for each σ Σ, then Σ is constructible. Proof. It is clear that Σ is pure, being the union of pure complexes of dimension dim Σ + c. We need to show that Σ inherits the property of being P C(W ) when Σ is derived as in steps 1, 2, or 3 of Definition 3.6. 1. When Σ is a simplex, Σ = σ, where σ is the maximal facet of Σ. By assumption, this complex is P C(W ). 2. Suppose that Σ = Σ 1 Σ 2, where Σ 1 and Σ 2 are P C([m]) of dimension d and Σ 1 Σ 2 is P C([m]) of dimension d 1. We need to show that Σ is P C(W ) of dimension d + c if Σ1, Σ2, and Σ1 Σ 2 are P C(W ). Note that Σ = ( ) ( ) σ = = Σ1 Σ2. σ Σ σ Σ 1 σ σ σ Σ 2 σ and Σ1 Σ 2 = σ = σ Σ 1 Σ 2 σ 1 Σ 1 σ 2 Σ 2 σ1 σ 2 = Σ1 Σ2. To prove that Σ1 Σ2 is P C(W ), we need hence only show that the dimensions of the subcomplexes are correct. Yet, since Σi is a union of complexes σ such that dim σ = d, we have that dim Σi = d + c for i = 1, 2. Also, Σ1 Σ 2 is a union of complexes σ such that dim σ = d 1, which implies that dim Σ1 Σ 2 = d + c 1. It follows that Σ1 Σ2 is P C(W ). 3. Suppose that Σ = Σ 1 Σ 2 is pure of dimension d 1 and that Σ 1, Σ 2, and Σ 1 Σ 2 are P C([m]) of dimension d. We need to show that Σ is P C(W ) of dimension d + c 1 if Σ1, Σ2, and Σ1 Σ 2 are P C(W ). This is done in exactly the same manner as in Step 2. To prove the last two claims in the theorem, replace P C(W ) with CM/R in steps 1-3 in the first case and replace P C(W ) and P C([m]) with constructible in the second case. 14

Corollary 3.10 If G = (G 1,..., G m ) is diagonally cycle-free on a vertex set V of size n and Σ is a P C([m]) complex, then G,Σ is P C(F V ) of dimension dim Σ + n m 1 and hence homology-cm. Proof. Write σ = G[m]\σ. By Theorem 3.9, we need only prove the following. σ is P C(F V ) for each σ [m]. This is Theorem 3.8. dim σ = σ + c 1 for some fixed constant c 0 for each σ [m]. This is Proposition 3.2; c = n m 1. σ1 σ2 = σ1 σ 2 for all σ 1, σ 2 [m]. This is immediate from the definition of σ. Corollary 3.11 If G = (G 1,..., G m ) is diagonally cycle-free on a vertex set V of size n and 1 r m, then G,r is P C(F V ) of dimension n m + r 3 and hence homology-cm. Proof. This is an immediate consequence of the fact that the complex {σ [m] : σ < r} is shellable; a shelling order is given by the obvious lexicographic order. Remark. While we have discovered a huge class of P C complexes related to diagonally cycle-free graph sequences, only a few of them are so far known to be constructible. Even worse, we have not been able to prove that the complexes are simply connected (if the dimension is different from 1); this would imply that the complexes are homotopy equivalent to wedges of spheres. Another simple application of Theorem 3.9 is as follows. A poset is defined to be CM/R if the order complex (P ) is CM/R. Let P be a CM/R poset of rank r and let P i be the set of elements in P of rank i. For σ [r], let P σ = i σ P i (with the induced partial order). Theorem 3.12 (Rank Selection Theorem [1]) Let P be a CM/R poset of rank r. For any subset σ [r], the poset P σ is CM/R. For a proof from The Book of Theorem 3.12, see Sundaram [9]. Corollary 3.13 Let P be a CM/R poset of rank r and let Σ be a P C([r]) complex. Then Σ = (P σ ) is CM/R. σ Σ Proof. By Theorem 3.12, each (P σ ) is CM/R. Also, (P σ1 ) (P σ2 ) = (P σ1 P σ2 ) = (P σ1 σ 2 ) and dim (P σ ) = dim σ. Thus, by Theorem 3.9, Σ is CM/R. We conclude with a problem that might be of some interest. 15

Problem 3.14 With notation as in Theorem 3.9, is there a family { σ : σ [m]} of CM/R complexes and a CM/R complex Σ on [m] such that Σ is not CM/R? References [1] K. Baclawski, Cohen-Macaulay ordered sets. J. Algebra 63 (1980), 226 258. [2] A. Björner, Topological methods, Handbook of Combinatorics, R. Graham, M. Grötschel and L. Lovasz (eds), North-Holland/Elsevier, Amsterdam, 1995, 1819 1872. [3] R. Forman, Morse theory for cell complexes, Adv. Math. 134 (1998), 90 145. [4] J. Herzog, V. Reiner and V. Welker, The Koszul property in affine semigroup rings, Pacific J. Math. 186 (1998), 39 65. [5] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. Math. 96 (1972), 318 337. [6] A. Hultman, Link complexes of subspace arrangements, Preprint, 2005. [7] J. Jonsson, The topology of the coloring complex, J. Algebraic Combin. 21 (2005), 311-329. [8] E. Steingrímsson, The coloring ideal and coloring complex of a graph, J. Algebraic Combin. 14 (2001), 73 84. [9] S. Sundaram, Homotopy of non-modular partitions and the Whitehouse module, J. Algebraic Combin. 9 (1999), 251 269. 16