h-vectors of PS ear-decomposable graphs

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1 h-vectors of PS ear-decomposable graphs Nima Imani 2, Lee Johnson 1, Mckenzie Keeling-Garcia 1, Steven Klee 1 and Casey Pinckney 1 1 Seattle University Department of Mathematics, th Avene, Seattle, WA University of Washington Department of Mathematics, Box , Seattle, WA Jne 29, 2013 Abstract In this paper we consider a family of simple graphs known as PS ear-decomposable graphs. These graphs are one-dimensional specializations of the more general class of PS ear-decomposable simplicial complexes, which were introdced by Chari as a means of nderstanding matroid simplicial complexes. In this paper we otline a shifting algorithm for PS ear-decomposable graphs that allows s to explicitly show that the h-vector of a PS ear-decomposable graph is a pre O-seqence. 1 Introdction This paper concerns the combinatorial strctre of a certain family of simple graphs known as PS ear-decomposable graphs. PS ear-decomposable graphs and more generally, PS eardecomposable simplicial complexes, were introdced by Chari [1] and provide a nified framework for proving a nmber of combinatorial reslts abot the combinatorial strctre of matroid simplicial complexes. imanima@.washington.ed johns193@seattle.ed keelingg@seattle.ed klees@seattle.ed pinckne1@seattle.ed 1

2 In the late 1970 s, Stanley [3] conjectred that the h-vector of a matroid simplicial complex is a pre O-seqence. Since Chari [1] was able to se the strctre of PS eardecomposable simplicial complexes to prove a nmber of reslts on h-vectors of matroid complexes, it is natral to conjectre [1, Conjectre 3] that the h-vector of a PS ear-decomposable simplicial complex is a pre O-seqence. In this paper, we focs or attention on the family of PS ear-decomposable graphs, which contains the family of all rank-two matroids. For any PS ear-decomposable graph Γ, we define a canonical PS ear-decomposable graph S(Γ) with the same nmber of vertices and edges as Γ, called a shifted PS ear-decomposable graph. Having defined this shifted PS ear-decomposable graph, it is easy to find a corresponding pre mlticomplex whose F -vector is the h-vector of S(Γ). This approach of defining a shifting algorithm as a means of preserving combinatorial data while simplifying the algebraic or geometric strctre of a simplicial complex is not new, and we refer to the work of Kalai [2] and the references therein for frther information. It is or hope that the shifting approach presented in this paper cold be generalized to higher-dimensional PS ear-decomposable simplicial complexes as an alternative approach to solving Stanley s conjectre. The remainder of the paper is strctred as follows. In Section 2, we provide the necessary backgrond on PS ear-decomposable graphs and pre mlticomplexes. In Section 3 we define or shifting algorithm on PS ear-decomposable graphs and se this constrction to prove that the h-vector of a PS ear-decomposable graph is a pre O-seqence. 2 Backgrond and definitions We will be interested in stdying two families of combinatorial objects in this paper. The first is the family of PS ear-decomposable graphs, and the second is the family of pre mlticomplexes. 2.1 Graphs and PS ear-decompositions In this paper we only consider finite, simple graphs, which we typically denote by Γ. The most natral combinatorial data that can be conted for a graph Γ are its nmber of vertices and edges, which we denote by f 0 (Γ) and f 1 (Γ) respectively. Here the sbscripts indicate that a vertex is zero-dimensional and an edge is one-dimensional when we draw a graph. We are interested in stdying an integer linear transformation of these nmbers called the 2

3 h-nmbers of Γ, which are defined by h 0 (Γ) = 1, h 1 (Γ) = f 0 (Γ) 2, and h 2 (Γ) = f 1 (Γ) f 0 (Γ) + 1. Notice that f 1 (Γ) = h 0 (Γ) + h 1 (Γ) + h 2 (Γ) and f 0 (Γ) = h 1 (Γ) + 2 so that knowing the h-nmbers of Γ is eqivalent to knowing the nmber of vertices and edges in Γ. We encode the h-nmbers of Γ in a vector called the h-vector, which is defined as h(γ) = (h 0 (Γ), h 1 (Γ), h 2 (Γ)). Following Chari [1], we will stdy a certain family of simple graphs known as PS eardecomposable graphs, which are defined indctively as follows. A PS cycle is a graph that is either a 3-cycle or a 4-cycle. A PS ear is a graph that is either a path of length two or a path of length one (a single edge). We call these PS ears of Type 1 and PS ears of Type 2 respectively. The bondary of a PS ear is defined as the set of vertices that are only incident to a single edge. It may seem conterintitive to define an ear of Type 1 as a path of length two and an ear of Type 2 as a path of length one, bt it will be more natral to consider ears of Type 1 first in or constrctions later in the paper. Table 2.1 illstrates all possible PS cycles and PS ears. The bondary vertices of the PS ears are colored white, while all other vertices are colored black. PS cycle h-vector PS ear Type 1: h-vector contribtion (1, 1, 1) (0, 1, 1) Type 2: (1, 2, 1) (0, 0, 1) Table 1: PS cycles and ears Definition 2.1 [1, Section 3.3] A graph Γ is PS ear-decomposable if it can be decomposed as a nion of the form Γ = Σ 0 Σ 1 Σ m, so that 3

4 1. Σ 0 is a PS cycle, 2. Σ j is a PS ear for all 0 < j m, and 3. the intersection Σ j i<j Σ i consists precisely of the bondary vertices of Σ j for all 0 < j m. One advantage to stdying PS ear-decomposable graphs is that their h-vectors can also be compted indctively in terms of the ears of the decomposition. Specifically, adding an ear of Type 1 adds one vertex and two new edges to the graph, so it contribtes (0, 1, 1) to the h-vector. Similarly, adding an ear of Type 2 adds one edge and zero vertices to the graph, so it contribtes (0, 0, 1) to the h-vector. Example 2.2 Consider the following graph Γ. v 3 v 6 v 1 v 4 v 2 v 5 We exhibit the following PS ear-decomposition of Γ. v 3 v 3 v 6 v 4 v 3 v 6 v 1 v 4 v 2 v 5 v 4 v1 v 4 v 2 v 2 v 5 Since Γ has 6 vertices and 11 edges, we can directly compte h(γ) = (1, 4, 6). We can also compte h(γ) in terms of the given PS ear-decomposition as h(γ) = (1, 1, 1) + (0, 1, 1) + (0, 1, 1) + (0, 1, 1) + (0, 0, 1) + (0, 0, 1) = (1, 4, 6). Remark 2.3 Not all graphs are PS ear-decomposable (e.g. a tree), and some graphs may admit several combinatorially distinct PS ear-decompositions. 4

5 2.2 Mlticomplexes A collection of monomials M in the variables {x 0, x 1,..., x m } is called a mlticomplex if, whenever µ M and ν divides µ, then ν M as well. We say that M is a mlticomplex of rank d if d is the maximal degree of any monomial in M. A mlticomplex M is pre of rank d if each monomial in M divides into some monomial of degree d in M. For a given mlticomplex M of rank d, we gather combinatorial data on M in the form of the F -vector, written F (M) = (F 0 (M), F 1 (M),..., F d (M)), where F j (M) conts the nmber of monomials of degree j in M. An integer vector F = (F 0, F 1,..., F d ) is a (pre) O-seqence if there is a (pre) mlticomplex M sch that F = F (M). Example 2.4 The vector F = (1, 3, 1) is an O-seqence, bt not a pre O-seqence. The mlticomplex M = {1, x 0, x 1, x 2, x 0 x 1 } has F -vector F (M) = (1, 3, 1); bt F is not a pre O-seqence since a pre mlticomplex with one monomial of degree two spports at most two monomials of degree one. Example 2.5 The vector (1, 4, 6) is a pre O-seqence. The following table exhibits a pre mlticomplex whose F -vector is (1, 4, 6). degree monomials 2 x 2 0 x 2 1 x 2 2 x 2 3 x 0 x 1 x 0 x 2 1 x 0 x 1 x 2 x h-vectors of PS ear-decomposable graphs Stanley [3] conjectred that the h-vector of any matroid simplicial complex is a pre O- seqence. We will not define matroid simplicial complexes or their h-vectors here, bt we refer to Stanley s book [4] for frther details. Chari [1] proved that any matroid simplicial complex is PS ear-decomposable, a definition that specializes to the given Definition 2.1 for graphs. The family of graphs that are matroid simplicial complexes are somewhat ninteresting as they correspond to the family of complete mltipartite graphs, while the family of PS eardecomposable graphs is larger, as is exhibited in Example 2.2. Or main contribtion in this paper is to show that Stanley s conjectre contines to hold for PS ear-decomposable graphs. Theorem 3.1 Let Γ be a PS ear-decomposable graph on n + 3 vertices. Then there is a pre mlticomplex M sch that h(γ) = F (M). Moreover, there is a canonical PS eardecomposable graph S(Γ) sch that 5

6 1. h(γ) = h(s(γ)), 2. the vertices of S(Γ) are labeled as {, v, x 0, x 1,..., x n }, and 3. the mlticomplex M arises natrally from the PS ear-decomposition of S(Γ) as a pre mlticomplex on {x 0, x 1,..., x n }. Proof: We will prove Theorem 3.1 in two main steps. The first step is motivated by the observation that the h-vector of a PS ear-decomposable graph Γ depends only on the types of ears that are sed in the PS ear-decomposition of Γ and is independent of the how these ears are attached. We begin by defining the graph S(Γ), which we call a shifted PS ear-decomposable graph. Let Γ be a PS ear-decomposable graph on n + 3 vertices with PS ear-decomposition Γ = Σ 0 Σ 1 Σ m. For any 0 < j < m, let Γ j := Σ 0 Σ 1 Σ j. We define a new PS ear-decomposable graph S(Γ) satisfying conditions 1 and 2 of Theorem 3.1 by indction on the nmber of ears in the PS ear-decomposition of Γ. If Σ 0 is a 3-cycle, we define S(Γ) 0 to be a 3-cycle whose vertices are labeled, v, and x 0. On the other hand, if Σ 0 is a 4-cycle, we define S(Γ) 0 to be 4-cycle whose vertices are cyclically labeled, v, x 0, and x 1 as shown below. x 0 v x 1 For 0 < j m, sppose we have indctively constrcted a PS ear-decomposable graph S(Γ) j 1 that satisfies conditions 1 and 2 of Theorem 3.1. Sppose the vertices of S(Γ) j 1 are labeled as {, v, x 0, x 1,..., x i }. If Σ j is a PS ear of Type 1, we obtain S(Γ) j from S(Γ) j 1 by adding a new vertex labeled x i+1 that is adjacent to vertices and v. Otherwise, if Σ j is a PS ear of Type 2, observe that there is a missing edge in S(Γ) j 1 becase (1) S(Γ) j 1 has the same nmber of vertices and edges as Γ j 1 and (2) Γ j is obtained from Γ j 1 by adding a single edge. To form S(Γ) j, we add the lexicographically smallest missing edge to S(Γ) j 1 according to the alphabet order < v < x 0 < x 1 < < x n. Recall that an edge {a, b} with a < b precedes an edge {c, d} with c < d lexicographically if either a < c, or a = c and b < d. By or constrction it is clear that h(γ j ) = h(s(γ) j ). In order to complete the proof of Theorem 3.1, we need to show that h(s(γ)) is a pre O-seqence. Again, this will follow by indction on the nmber of ears in the PS eardecomposition of Γ. For each 0 j m, we will constrct a pre mlticomplex M j sch that F (M j ) = h(s(γ) j ). 6

7 We begin with the PS cycle Σ 0. If Σ 0 is a 3-cycle, then h(σ 0 ) = (1, 1, 1), which is the F - vector of the pre mlticomplex M 0 = {1, x 0, x 2 0}. On the other hand, if Σ 0 is a 4-cycle, then h(σ 0 ) = (1, 2, 1), which is the F -vector of the pre mlticomplex M 0 = {1, x 0, x 1, x 0 x 1 }. Indctively, for 0 < j m, sppose we have constrcted a pre mlticomplex M j 1 on variables {x 0,..., x i } sch that F (M j 1 ) = h(s(γ) j 1 ). We define a pre mlticomplex M j sch that F (M j ) = h(s(γ) j ) as follows: 1. If Σ j is a PS ear of Type 1, define M j := M j 1 {x i+1, x 2 i+1}. Clearly F (M j ) = F (M j 1 ) + (0, 1, 1), and hence h(s(γ) j ) = F (M j ). Moreover, it is clear that M j is a pre mlticomplex since M j 1 was a pre mlticomplex, and we have added a new monomial of degree one and its sqare. 2. If Σ j is a PS ear of Type 2, define M j := M j 1 X, where we define X according to the following rle. (a) If the missing edge added to S(Γ) j 1 has the form {x k, x l }, then X := {x k x l }. In this case, M j is a mlticomplex becase the monomials of degree one that divide x k x l, which are x k and x l, belong to M j 1 by constrction; and M j is pre becase we have simply added another monomial of maximal degree. (b) If the missing edge added to S(Γ) j 1 is {, x 0 } then X := {x 2 0}; if the missing edge is {v, x 1 }, then X := {x 2 1}. This only arises in the case that Σ 0 is a 4-cycle. The monomials x 2 0 and x 2 1 do not belong to M 0 in this case; bt their divisors, x 0 and x 1 respectively, do. Ths M j is a mlticomplex, and it is pre becase we have only added a monomial of maximal degree to M j 1. In either case, it is again clear that F (M j ) = F (M j 1 ) + (0, 0, 1) so that h(s(γ) j ) = F (M j ). This constrction of the reslting pre mlticomplex M is well-defined becase we do not allow mltiple edges in or graphs. In the case that Σ 0 is a 3-cycle, a monomial x 2 k is introdced when the corresponding vertex labeled x k is introdced, and this only happens when an ear of Type 1 is attached. Otherwise, all other monomials that are introdced have the form x k x l with k l and correspond to an edge {x k, x l } being introdced to the graph. The same argment applies when Σ 0 is a 4-cycle except that x 2 0 and x 2 1 are introdced to the mlticomplex when the edges {v, x 0 } and {, x 1 } are introdced. Here, we say that the graph S(Γ) is shifted for the following reason. At each step in the PS ear-decomposition, an ear is attached in sch a way that its bondary vertices are the lexicographically smallest pair of vertices that spport the reqired type of ear when we order the vertices < v < x 0 < < x n. 7

8 Example 3.2 Let Γ be the PS ear-decomposable graph presented in Example 2.2. The shifted PS ear-decomposable graph S(Γ) is shown in Figre 3.2. We exhibit the PS ear-decomposition otlined in Theorem 3.1, as well as the corresponding pre mlticomplex encoded by S(Γ) in Figre 3.2. v x 0 x 1 x 2 x 3 Figre 1: The shifted graph S(Γ) x 1 x 2 Ears x 0 x 1 x 2 x 3 v v v v x 0 x 0 Monomials {1, x 0, x 2 0} {x 1, x 2 1} {x 2, x 2 2} {x 3, x 2 3} {x 0 x 1 } {x 0 x 2 } Figre 2: Decomposing the shifted graph S(Γ) References [1] M. Chari. Two decompositions in topological combinatorics with applications to matroid complexes. Trans. Amer. Math. Soc., 349(10): , [2] Gil Kalai. Algebraic shifting. In Comptational commtative algebra and combinatorics (Osaka, 1999), volme 33 of Adv. Std. Pre Math., pages Math. Soc. Japan, Tokyo, [3] R. P. Stanley. Cohen-Macalay complexes. In Higher combinatorics (Proc. NATO Advanced Stdy Inst., Berlin, 1976), pages NATO Adv. Stdy Inst. Ser., Ser. C: Math. and Phys. Sci., 31. Reidel, Dordrecht, [4] R.P. Stanley. Combinatorics and commtative algebra, volme 41 of Progress in Mathematics. Birkhäser Boston Inc., Boston, MA, second edition,