Extendable shellability for $d$-dimensional complexes on $d+3$ vertices

We prove that for all $d \geq 1$ a shellable $d$-dimensional simplicial complex with at most $d+3$ vertices is extendably shellable. The proof involves considering the structure of `exposed' edges in chordal graphs as well as a connection to linear quotients of quadratic monomial ideals.


INTRODUCTION
A pure d-dimensional simplicial complex ∆ is said to be shellable if there exists an ordering of the facets F 1 , F 2 , . . . , F s such that for all k = 2, 3, . . . , n the simplicial complex induced by Shellability is an important combinatorial tool that has consequences for the topology of ∆ as well as algebraic properties of its Stanley-Reisner (face) ring K [∆]. Examples of shellable simplicial complexes include the independence complexes of matroids [13], boundary complexes of simplicial polytopes [5], as well as the skeleta of shellable complexes [4]. In particular for any k = 1, 2, . . . , n − 1 the k-skeleton of a simplex on vertex set [n] is shellable. It was recently shown [10] that for every d ≥ 2 deciding if a given pure d-dimensional simplicial complex is shellable is NP-hard.
Given a shellable complex a natural question to ask is whether one can get 'stuck' in the process of building a shelling order. A shellable complex ∆ is said to be extendably shellable if any shelling of a subcomplex of ∆ can be extended to a shelling of ∆. Here a subcomplex of ∆ is a simplicial complex Γ whose set of facets consists of a subset of the facets of ∆. Any 2-dimensional triangulated sphere (which is necessarily polytopal) is extendably shellable [8], and Kleinschmidt [11] has shown that any d-dimensional sphere with d + 3 vertices is extendably shellable. Björner and Eriksson [3] proved that independence complexes of rank 3 matroids are extendably shellable. On the other hand Ziegler [16] has shown that there exist simplicial 4-polytopes that are not extendably shellable.
Simon [15] has conjectured that every k-skeleton of a simplex is extendably shellable. The Björner-Eriksson result establishes the k = 2 case of Simon's conjecture by considering U 3 n , the uniform matroid of rank 3. In [1] Simon's conjecture was established for k ≥ n − 3, with the case k = n − 3 the new contribution (a simpler proof was provided independently in [9] based on results from [6]). Here we prove that much more is true. Theorem 1.1. Suppose X is a shellable d-dimensional simplicial complex on at most d + 3 vertices. Then X is extendably shellable.
This of course implies the k = n − 3 case of Simon's conjecture and also provides a generalization of Kleinschmidt's results. Our result is also best possible in the sense that there are 2-dimensional complexes on 6 vertices that are not extendably shellable (see [12], [2]).
To prove Theorem 1.1 we use the correspondence between shellings of d-dimensional simplicial complexes on d + 3 vertices and linear quotients of monomial ideals generated by quadratics. In previous work of the authors ( [6], [9]) it is shown that such constructions are equivalent to removing exposed edges from chordal graphs. Using these ideas we will see that Theorem 1.1 follows from the following graph-theoretic result. Proposition 1.2. Suppose G is a chordal graph and suppose H ⊂ G is a subgraph of G that is also chordal. Then H can be obtained from G via a sequence of removing exposed edges.
In the next section we recall the relevant definitions and set notation. In Section 3 we prove Proposition 1.2 and show how it leads to a proof of Theorem 1.1.

NOTATIONS
For a finite set E we let E k denote the family of k-subsets of E. A subset C ⊂ E k will be called a k-clutter on vertex set E. For a k-clutter C we use C to denote the (pure) (k − 1)-dimensional simplicial complex generated by C, that is the collection of all subsets of elements of C (including the empty set ∅). Given a k-clutter C a shelling step is the addition of a some e ∈ E k − C to C such that for some d ⊂ e. In other words the intersection of e with C is a pure simplicial complex of dimension k − 2.
If ∆ is pure d-dimensional simplicial complex ∆ with shelling order of its facets (F 1 , F 2 , . . . , F s ), the restricted set of the facet F i is the set of (d − 1) dimensional faces in the intersection of the facet We next recall some basic notions from graph theory. For us a graph G consists of a finite set of vertices V(G) along with a set E(G) of unordered pairs of elements of V(G). In particular our graphs are undirected and simple, with no loops or multiple edges. An element of E(G) will be written vw, with set brackets and comma suppressed. If G is a graph on vertex set V(G) then a subgraph H ⊂ G is a graph on a vertex set V(H) ⊂ V(G) with the property that if e ∈ E(H) then e ∈ E(G). An induced subgraph H ⊂ G has the property that whenever v, w ∈ V(H) and vw ∈ E(G) we have vw ∈ E(H). If S ⊂ V(G) is a subset of vertices of G we let G[S] denote the subgraph induced by S. A clique in a graph G is an induced subgraph K ⊂ G with the property that all pairs of vertices in V(K) form edges in K (so that K is a complete graph).
A vertex v is simplicial if the subgraph induced on N(V) is a complete graph. We recall the notion of exposed edges introduced in [6]. Definition 2.1. Suppose G is a graph. An edge e ∈ E(G) is said to be exposed if it is uniquely contained in a maximal clique. If in addition the edge is properly contained in the clique (i.e. e is contained in some triangle) then we say that e is properly exposed. We refer to the operation of removing a (properly) exposed edge from G as an (proper) erasure. A sequence of edges (e 1 , . . . , e k ) is an (proper) erasure sequence, if e i is a (properly) exposed edge of the graph G − {e s | s < i} for all i = 1, . . . , k.
An edge which is exposed but not properly exposed will also be called a facet edge. A chordal graph is a graph with no induced cycles of length four or more. In [6] it is shown that a graph G on vertex set [n] = {1, 2, . . . , n} is chordal if and only if G can be obtained from the complete graph K n via a sequence of erasures. Furthermore, this G is connected (and chordal) if and only if each edge in that sequence is properly exposed.
We note that if we fix V(G) = [n] = {1, 2, . . . , n} to be the vertex set of the graph G, the collection of edges E(G) = {e 1 , e 2 , . . . e m } corresponds to a clutter X(G) ⊂ [n] n−2 with n − 2-subsets given by F i = [n]\e i . This correspondence can of course be reversed, so that an (n − 2)-clutter X corresponds to a graph G(X). We will use the following lemma from [9]. . . , f k is a sequence of shelling steps (resulting in the (n − 3)-dimensional simplicial complex X(G)). An edge e i is properly exposed if and only if the restricted set of f i consists of less than n − 2 elements.

PROOFS
In this section we provide the proofs of Proposition 1.2 and Theorem 1.1, which will follow from a number of graph-theoretic lemmas. Having obtained Proposition 1.2, we discovered that a slightly weaker result had already been proved as Lemma 2 of [14], though that result does not characterize the class of edges allowed for removal. The notion of exposed edges allows for a new and simplified treatment of chordality that is independent of any particular structural descriptions of a chordal graph (such as the one obtained from a fixed vertex elimination ordering), resulting in a much simpler proof of Proposition 1.2 than that found in [14]. Believing this to be of some independent interest, we include a self-contained discussion here, in the form of Lemmas 3.1-3.4.
We first collect some simple observations regarding exposed edges which follow straight from the definitions: Lemma 3.1. For any graph G and any edge xy ∈ E(G) the following are equivalent: (1) xy ∈ G is exposed (respectively, properly exposed).
(3) y is a simplicial (resp. simplicial and non-isolated) vertex of N G (x). (4) x is a simplicial (resp. simplicial and non-isolated) vertex of N G (y).
The basic relationship between exposed edges and chordality is as follows:

Lemma 3.2. Suppose G is a chordal graph. Then:
(1) An edge e ∈ E(G) is exposed if and only if G − e is chordal; (2) if v ∈ V(G) is simplicial, every edge xv ∈ E(G) is exposed.
Proof. For the implication ⇒ of (1) we follow the proof of Theorem 8 in [6]. Suppose G is a chordal graph and xy ∈ ∂G is an exposed edge. If C is an induced cycle in H := G − e such that {x, y} ⊂ C (i.e., C possibly contains x or y, but not both), then C is also an induced cycle of G and so of length 3. Otherwise, suppose {x, y} ⊂ C and |C| > 3. Note that if |C| > 4, then the induced subgraph C = C ∪ xy of G has an induced cycle of length greater than 3, a contradiction. This leaves us with the case where Since xy is exposed in G, we must have v 1 v 2 ∈ E(G), since otherwise xy would lie in two distinct maximal cliques so that xy would not be exposed. However, v 1 v 2 ∈ E(G) (hence in E(H)) means that C would not be an induced cycle in H, a contradiction.
Conversely, note that if e = xy is not exposed, then there are two non adjacent vertices z, w ∈ For (2), notice that if v ∈ G is simplicial then every vertex x ∈ N G (v) is simplicial in N G (v). Thus either N G (v) = {x} so that xv is a facet edge, or else Lemma 3.1 gives the result.
The following result for chordal graphs is well-known, but is usually derived in the literature from the characterization of chordality via vertex elimination orderings. Here we derive it directly from the hereditary property of chordality and the preceding characterization of exposed edges. (1) every facet edge in G is a cut edge; (2) either G is complete or G has at least two nonadjacent simplicial vertices.
Proof. To prove (1), consider a facet edge wz in G. If H := G − wz is connected, then there is a path from z to w in G − wz. Since G is chordal, we conclude there is a vertex u ∈ N G (w) ∩ N G (z). But this contradicts wz being a facet edge.
To prove (2), suppose G is a counterexample with f(G) := |E(G)| + |V(G)| as small as possible. Then G is chordal, incomplete (obviously), and connected: otherwise, each connected component is a chordal graph that is not a counterexample and we can take one simplicial vertex from each component to obtain an independent set of simplicial vertices of cardinality at least 2.
Also, G may not contain a facet edge wz since otherwise G−wz would be chordal with f(G−wz) < f(G), contradicting minimality: indeed, since G − wz is disconnected by (1), each connected component K of G − wz would contain at least one simplicial vertex u K that is neither w nor z, and u K is necessarily simplicial in G.
Next, we claim that G has a properly exposed edge. To see this let v ∈ V(G) be any vertex of G. Thus f(N G (v)) < f(G) and G[N G (v)] is chordal (all induced subgraphs of a chordal graph are chordal). Therefore either (i) G[N G (v)] is complete, meaning v is a simplicial vertex and every edge incident to v is exposed (and there must be one such edge because G is connected and not complete); or (ii) N G (v) contains two simplicial vertices, each of which gives rise to a properly exposed edge in G that is incident to v. Now, let e = xy be a properly exposed edge in G. Then G − e is not complete and is chordal, and f(G − e) < f(G). Hence there exist v 1 , v 2 nonadjacent simplicial vertices in G − e. Since Thus v 1 , v 2 are also simplicial in G.
The following lemma is the promised (mild) strengthening of Lemma 2 in [14], with a new and simplified proof using exposed edges rather than perfect elimination orderings.  H) is minimal, we can find an exposed edge e = xy in G − z such that e / ∈ E(H − z). To see that e is also exposed in G, notice that since e / ∈ E(H − z) and hence e / ∈ E(H) and z is simplicial in H, we must have that {x, y} ⊂ N H (z) = N G (z), or in other words z / , which is complete and e is exposed in G.
is chordal we can apply Lemma 3.3 to see that either (i) G[N G (z)] is complete, in which case z is simplicial in G and any edge zx with x ∈ N G (z)\N H (z) is exposed (Lemma 3.2); or (ii) G[N G (z)] has two non-adjacent simplicial vertices v 1 , v 2 . Since z is simplicial in H, we must have {v 1 , v 2 } ⊂ N H (z) and so one of xv 1 , xv 2 / ∈ E(H), but both are exposed in G.
We can now prove the results stated in the introduction.
Proof of Proposition 1.2. Suppose G is a chordal graph and suppose H ⊂ G is a subgraph of G that is also chordal. By Lemma 3.4 there exists and edge e of G such that G − e is chordal and contains H. Continue removing edges in this way until we obtain the graph H.
Proof of Theorem 1.1. Suppose X is a shellable d-dimensional complex on vertex set V = V(X). We first note that if |V| = d + 1 then X is a simplex, and if |V| = d + 2 it is not hard to see that any ordering on the facets of X is a shelling: after we add the first facet F 1 , every subsequent facet F i intersects the previous collection in i − 1 faces of dimension d − 1.
Hence we can assume that |V| = d + 3. Let (F 1 , F 2 , . . . , F ) be some shelling order for X, where each F i is a facet of X (a subset of V of cardinality d + 1). Let H = G(X) be the corresponding graph on the same vertex set, with edges given by e i = V\F i . By Lemma 2.2 the graph H is chordal. Now suppose that we have a partial shelling (F i 1 , F i 2 , . . . F i k ) of X resulting in a subcomplex Y ⊂ X. Again by Lemma 2.2 we have that e i 1 , e i 2 , . . . , e i k is an erasure sequence in the complete graph K d+3 , resulting in the graph G = G(Y) which by Lemma 3.2 is chordal. Hence we see that H is a chordal subgraph of the chordal graph G, both on the same vertex set V. By Proposition 1.2 we can obtain H by removing exposed edges in G, which (again by Lemma 2.2) corresponds to completing the partial shelling into a shelling of X. The result follows.