n (log2 logbn)" ] < logn
Combining these two results, we see that:
E[ log 2
(G)]
(nlog 2 s)(1 + o(1)), 28
H(G)
in ( log 2 a )
=

log2
n/ ai)
\

log 2 !oc(G) n
=
log 2
(
and moreover, that log 2 (G) ~ E[ log 2 e(G)] a.s. holds. The second necessary inequality comes, firstly, from using Observation 1.4.1, so that e(G) < n!Vol (STAB (G)), and then from a direct application of Theorem 1.4.10. log 2 e(G). Now, we further observe that for We obtain that n(log 2 n  H(G)) 1 y(G), and then: a E STAB (G), we have Ej a.
s + c holds A classic result of Grimmett and McDiarmid {1975} states that X(G) a.s., where c =2 log b+1. Hence, a.s., H(G) (log 2 g)= log 2 nlog 2 (s+0(1)), and then nlog 2 (s + 0(1)) : log 2 E(G). From here, we directly obtain: log 2
F(G)
nlog2 s+O
(n)
(1.4.3)
a.s..
Therefore, from inequalities 1.4.2 and 1.4.3: log 2 F(G) = nlog2 s+O(n) a.s..
Calculating inequality 1.4.3 more precisely by dropping the 0notation and using Grimmett and McDiarmid's constant, we obtain:
n!
e
Sns"
2/(0og2 b)
n Vol(STAB(G)) <  . Cnj* a.s., where c = 2 n!2
Corollary 1.4.13. Let G ~ GnP with 0 < p < 1, b = n 2 logo logo n. Then, for large enough n: log 2
1.5
+ +
H(G) 5 log 2 (
+
and s = 2log n

(1n
logn
.
Sn
2

Corollary 1.4.12. Let G  G,, with 0 < p < 1, b = 11P and s = 2 logo logo n. Then, for large enough n:
a.s..
Further techniques.
In this section, we will see how the main problem of the chapter has two more presentations as selecting a region in the graphical arrangement with maximal fractional volume, or as selecting a vertex of the graphical zonotope that is farthest from the origin in Euclidean distance.
29
Definition 1.5.1. Consider a simple undirected graph G = G([n], E). The graphical arrangement of G is the central hyperplane arrangement in IRH1 given by: AG = Ix
C Rn] : xi  xj = 0 , for some {i, j}
E}.
The regions (see Definition 2.2.1 and the comments thereafter) of the graphical arrangement AG with G = G([n], E) are in onetoone correspondence with the acyclic orientations of G (Proposition 2.2.2). Moreover, the complete fan in R[n] given by AG is combinatorially dual to the graphicalzonotope of G: Za"'''
=
[e

e3 , e

e1,
{i,j}EE
and there is a clear correspondence between the regions of AG and the vertices of Zntral.
Following Klivans and Swartz {2011}, we define the fractional volume of a region
R of AG to be: Vol' (R)

Vol (B n nR)
Vol (Bn)
where B" is the unit ndimensional ball in R [n. With little work, using Proposition 2.2.2 and the symmetry of the permutohedron, it is possible to say the following about these volumes: Proposition 1.5.2. Let G = G([n], E) be an undirected simple graph, and let AG be its graphical arrangement. If R is a region of AG and P is its corresponding partial order on [n] under the map of Proposition 2.2.2, then:
e(P) n!
Volf (R)= n!
The problem of finding the regions of AG with maximal fractional volume is, intuitively, closely related to the problem of finding the vertices of Z* t ral that are farthest from the origin under some appropriate choice of metric. It turns out that, with Euclidean metric, a precise statement can be formulated when G is a comparability graph: Theorem 1.5.3. Let G = G(V, E) be a comparability graph. Then, the vertices of the graphical zonotope of Zcntral that have maximal Euclidean distance to the origin are precisely those that correspond to the transitive orientations of E, which in turn have maximal number (G) of linear extensions. To prove Theorem 1.5.3, we first note that for a simple (undirected) graph G = G(V, E), the vertex of Zn t a corresponding to a given acyclic orientation of E is precisely the point:
(outdeg (v)  indeg(v))vEV, where outdeg (.) and indeg (.) are calculated using the given orientation. We need to establish a preliminary lemma.
30
Lemma 1.5.4. Let Go = Go(V, E) be an oriented graph. Then, (indeg (v)  outdeg (v)) 2 = E + tri (Go) + incom (Go)  com (Go), vEV
where: 1. tri (Go) is the number of directedtriangles(u, v), (v, w), (u, w) E E. 2. incom(Go) is the number of triples u,v,w C V such that (v,w), (w,v) 0 E but
either (u, v), (u, w) E E or (v, u), (w, u) E E.
E E such that (u, w)
$
3. com (Go) is the number of directed2paths (u, v), (v, w)
E. Proof. For v E V, outdeg (v) 2 is equal to outdeg (v) plus two times the number of pairs u / w such that (v, u), (v, w) E E, indeg (v) 2 is equal to indeg (v) plus two times the number of pairs u, 74 w such that (u, v), (w, v) E E, and outdeg (v)  indeg (v) is equal to the number of pairs u 74 w such that (u, v), (v, w) E E. If we add up these terms and cancel out terms in the case of directedtriangles, we obtain the desired equality.
An important consequence of Lemma 1.5.4 is the following: If G = G(V, E) is a simple graph, all the in their values of tri() and of E, which incom (.). Moreover, com () + incom () G of the form {u, v}, {v, w} E E with u choice of orientation for E.
acyclic orientations of E will not vary depend on G, but only in com (.) and is equal to the number of 2paths in 74 w, so it is also independent of the
Proof of Theorem 1.5.3. We apply Lemma 1.5.4 directly. Since G is a comparability graph, from Theorem 1.3.7, we know that the value of incom (.)  com () will be maximized precisely on the transitive orientations of G, since all transitive orientations force com () = 0.
Remark 1.5.5 (To Theorem 1.5.3). In fact, per the Second proof of Theorem 1.2.8 in Subsection 1.4.1, the analogous result to Theorem 1.5.3 also holds true for odd cycles. In joint work with Kyle Gettig, we have again made use of Observation 1.4.2 to prove that odd anticycles (i.e. the complement graphs to odd cycles) also have this property.
31
32
Chapter 2 Largest Eigenvalue of the Laplacian Matrix. 2.1
Introduction.
Let G = G([n], E) be a simple (undirected) graph, where [n] = {1, 2,..., n}, n The adjacency matrix of G is the n x n matrix A = A(G) such that:
(
=
1
C P.
if {i,j} E E,
A The Laplacian matrix of G is the n x n matrix L = L(G) such that: (L)=
di
if i = j,
:
where di = dG(i)= (dG)i is the degree of vertex i in G. The spectral theory of these matrices, i.e. the theory about their eigenvalues and eigenspaces, has been an object of much study for the last 40 years. The roots of this beautiful theory, however, can arguably be traced back to Kirchhoff's matrixtree theorem, whose first proof is often attributed to Borchardt {1860} even though at least one proof was already known by Sylvester {1857}. A recollection of some of the interesting applications of the theory can be found in Spielman {2009}, and more complete accounts of the mathematical backbone are Brouwer and Haemers {2011} and Chung {1997}. Still, it would be largely inconvenient and prone to unfair omissions to attempt here a brief account of the many past and present contributors and contributing papers to the modern spectral theory of graphs, and we refer the reader to our references for further inquiries of the literature. This chapter aims to fill one (of the many) gap (s) in our present knowledge of the spectrum of the Laplacian matrix, namely, the lack of results about its eigenvectors with largest eigenvalue. We will answer the question: What information about the structure of a graph is carried in these eigenvectors? This question will be studied while remaining loyal to the central theme of this thesis: The mathematics of acyclic
33
axAmax(G),
be the (real) eigenvalues of L, and note that A 2 > 0 if G is a connected graph; we have effectively dropped G from the notation for convenience but remark that eigenvalues and eigenvectors depend on the particular graph in question, which will be clear from the context. We will also let EA, be the eigenspace corresponding to A. In its most primitive form, Fiedler's nodal domain's theorem {Fiedler, 1975} states that when G is connected and for all x e EA 2 , the induced subgraph G [{i E [n] : xi > 0}] is connected. Related ideas and results might be found in Merris {1998}. On this chapter, we will go even further in the way in which eigenvectors of the Laplacian may be used to learn properties of G. To explain this, let us firstly call a map,
0 :E + ([n] x [n]) U E
=
[n] 2 U E,
such that O(e) c {e, (i, j), (j, i)} for all e := {i, j} E E, an (partial) orientation of E (or G), and say that, furthermore, 0 is acyclic if O(e) $ e for all e and the directedgraph on vertexset [n] and edgeset O[E] has no directedcycles. Along the exposition of the chapter, we will oftentimes also identify the object 0 with the set
0[E]. During this second part of the thesis, eigenvectors of the Laplacian and more precisely, elements of Ekmax, will be used to obtain orientations of certain (not necessarily induced) subgraphs of G. Henceforth, given G and for all x c R[n, the reader should always automatically consider the orientation (map) Ox = Ox(G) associated to x, Ox: E + [n] 2 U E, such that for e := {ij} c E: if Xi = X3
e
,
 "n =
orientations of graphs. Our work follows the spirit of Fiedler {2011}, who pioneered the use of eigenvectors of the Laplacian matrix to learn about a graph's structure. One of the first observations that can be made about L is that it is positivesemidefinite, a consequence of it being a product of incidence matrices. We will thus let, 0 = A A2 < .
Ox(e) =
(i, j) if xi < xj, (j, i)
if Xi > Xj.
The orientation Ox will be said to be induced by x (e.g. Figure 21c). Implicit above is another subtle perspective that we will adopt, explicitly, that vectors x E RMn] are real functions from the vertexset of the graph in question (all our graphs will be on vertexset [n]). In our case, this graph is G, and even though accustomed to do so otherwise, entries of x should be really thought of as being indexed by vertices of G and not simply by positive integers. Later on in Section 2.3, for example, we will regularly state (combinatorial) results about the fibers of x when x belongs to a certain subset of RM"I (e.g. Emax), thereby regarding these fibers as vertexsubsets of the particular graph being discussed at that moment. Using this perspective, we will learn that the eigenspace ENx. is closely related to the theory of modular decomposition of Gallai {1967}; orientations induced by elements of EXax lead naturally to the discovery of modules. This connection will
34
most concretely be exemplified when G is a comparability graph, in which case these orientations iteratively correspond to and exhaust the transitive orientationsof G. It will be instructive to see Figure 21 at this point. In Section 2.2, we will introduce the background and definitions necessary to state the precise main contributions of this chapter. These punch line results will then be presented in Section 2.3. The central theme of Section 2.3 will be a stepwise proof of Theorem 2.3.1, our main result for comparability graphs, which summarily states that when G is a comparability graph, elements of Ex... induce transitive orientations of the copartition subgraph of G. It will be along the natural course of this proof that we present our three main results that apply to arbitrary simple
graphs: Propositions 2.3.10 and 2.3.11, and Corollary 2.3.12. Finally, in Section 2.4, we will present a curious novel characterization of comparability graphs that results from the theory of Section 2.3.
2.2
Background and definitions.
Let us first review the mathematical background relevant to this chapter. A few of the concepts that we will present here have already been introduced in Chapter 1, but we will restate them to free the reader from having to skip repeatedly over a large number of pages. In any case, we will also add small further clarifications to the original setting of Chapter 1, adapting the narrative of this chapter to fit our current needs.
2.2.1
The graphical arrangement.
Definition 2.2.1. Let G = G([n], E) be a simple (undirected) graph. The graphical arrangement of G is the union of hyperplanes in RlR: AG := {x G RI'l : xi  xj = 0 , for some {i,j} E E}.
Basic properties of graphical arrangements and, more generally, of hyperplane arrangements, are presented in Chapter 2 of Stanley {2004}. For G as in Definition 2.2.1, let R(AG) be the collection of all (open) connected components of the set RHl\Ac. An element of R(AG) is called a region of AG, and every region of AG is therefore an ndimensional open convex cone in RI"1. Furthermore, the following is true about regions of the graphical arrangement: Proposition 2.2.2. Let G be as in Definition 2.2.1. Then, for all R x, y E R, we have that:
C R((AG)
and
OR:= OX  O. Moreover, the map R '+ OR from the set of regions of AG to the set of orientations of E is a bijection between R(AG) and the set of acyclic orientations of G. Motivated by Proposition 2.2.2 and the comments before, we will introduce special notation for certain subsets of R "I obtained from AG
35
Notation 2.2.3. Let G be as in Definition 2.2.1. For an acyclic orientation0 of E, we will let CO denote the ndimensional closed convex cone in R 'n that is equal to the topological closure of the region of AG corresponding to 0 in Proposition2.2.2.
2.2.2
Modular decomposition.
We need to concur on some standard terminology and notation from graph theory, so let G = G([n], E) be a simple (undirected) graph and X a subset of [n]. As customary, C denotes the complement graph of G. The notation N(X) denotes the open neighborhood of X in G: N(X)
{j E [n]\X: there exists some i E X such that {i,j} E E}.
The induced subgraph of G on X is denoted by G[X], and the binary operation of graph disjoint union is represented by the plus sign +. Lastly, for Y C [n], X and Y are said to be completely adjacent in G if: X n Y = 0, and for all i E X and
j
E Y, we have that {i, j} E E.
The concepts of module and modular decomposition in graph theory were introduced by Gallai {1967} as a means to understand the structure of comparability graphs. The same work would eventually present a remarkable characterization of these graphs in terms of forbidden subgraphs. Section 2.3 of the present work will present an alternate and surprising route to modules. Definition 2.2.4. Let G = G([n], E) be a simple (undirected) graph. A module of G is a set A C [n] such that for all i, j G A: N(i)\A = N(j)\A
=
N(A).
Furthermore, A is said to be proper if A C [n], nontrivial if if G[A] is connected.
Al > 1, and connected
Corollary 2.2.5. In Definition 2.2.4, two disjoint modules of G are either completely adjacent or no edges exist between them. Let us now present some basic results about modules that we will need.
Lemma 2.2.6 (Gallai {1967}). Let G = ([n], E) be a connected graph such that Z is connected. If A and B are maximal (by inclusion) proper modules of G with A then A n B = 0.
$
B,
Corollary 2.2.7 (Gallai {1967}). Let G = ([n], E) be a connected graph such that C is connected. Then, there exists a unique partitionof [n] into maximal propermodules of G, and this partition contains more than two blocks.
36
From Corollary 2.2.7, it is therefore natural to consider the partition of the vertexset of a graph into its maximal modules; the appropriate framework for doing this is presented in Definition 2.2.8. Hereafter, however, we will assume that our graphs are connected unless otherwise stated since (1) the results for disconnected graphs will follow immediately from the results for connected graphs, and (2) this will allow us to focus on the interesting parts of the theory.
Definition 2.2.8 (RamirezAlfonsin and Reed {2001}). Let G = G([n], E) be a connected graph. such that:
We will let the canonical partition of G be the set P
=
P(G)
a. If G is connected, P is the unique partition of [n] into the maximal proper modules of G. b. If C is disconnected, P is the partitionof [n] into the vertexsets of the connected components of C. Hence, in Definition 2.2.8, every element of the canonical partition is a module of the graph. Elements of the canonical partition of a graph on vertexset [8] are shown
in Figure 21b. Definition 2.2.9. In Definition 2.2.8, we will let the copartition subgraph of G be the graph GP on vertexset [n] and edgeset equal to:
E\ {{i, j} E E: i, j E A for some A E P}.
2.2.3
Comparability graphs.
We had anticipated the importance of comparability graphs for the results of this chapter, so let us now recall what they are. Definition 2.2.10. A comparability graph is a simple (undirected) graph G = G(V, E) such that there exists a partial order on V under which two different vertices u, v E V are comparable if and only if {u, v} G E. A comparability graph on vertexset [8] = {1, 2, . . , 8} is shown in Figure 21b. Comparability graphs are perfectly orderable graphs and more generally, perfect graphs. These three families of graphs are all large hereditary classes of graphs. Note that, given a comparability graph G = G(V, E), we can find at least two partial orders on V whose comparability graphs (obtained as discussed in Definition 2.2.10) agree precisely with G, and the number of such partial orders depends on the modular decomposition of G. Let us record this idea in a definition. Definition 2.2.11. Let G = G(V, E) be a comparabilitygraph, and let 0 be an acyclic orientation of E. Consider the partial order induced by 0 under which, for u, v E V, u is less than v iff there is a directedpath in 0 that begins in u and ends in v. If the comparability graph of this partial order on V (obtained as in Definition 2.2.10) agrees precisely with G, then we will say that 0 is a transitive orientation of G.
37
(c)
(b)
(a)
1 2 3 4 5 6
1
2
a
0.1515... b = 0.2587... a c =0.1021.. d = 0.1866... d
e0=.8855...
2
8
a
Figure 21: (21a) Hasse diagram of a poset P on ground set [8]. (21b) Comparability graph G = G([8], E) of the poset P, where the closed regions depict the maximal proper modules of G. (21c) Unit eigenvector x E Exmax of G fully calculated, where dim (EAniax) = 1. Arrows represent the induced orientation Ox of G. Notice the relation between Ox, the modules of G, and the poset P.
2.2.4
Linear algebra.
Some standard terminology of linear algebra and other related conventions that we adopt are presented here. Firstly, we will always be working in Euclidean space RI'J, and all (Euclideannormed real) vector spaces considered are assumed to live therein. Euclidean norm is denoted by 11 I1. The standard basis of R[n] will be {ei}[], as customary. Generalizing this notation, for all I C [n], we will also let:
Zj:=Eei. iEI
The orthogonal complement in R["I to spanR (e[n]) will be of importance to us, so we will use special notation to denote it: R*[n] := (spanR
e[l]))
For an arbitrary vector space V and a linear transformation T : V + V, we will say that a set U C V is invariant under T, or that T is Uinvariant, if T(U) C U. Lastly, a key concept of this chapter: C[n],
For a vector x E Rc
and a set
C
we will say that E is a fiber of x if there exists oc E R such that xi = oc if and only if i E
4.
The notion of being a generic vector in a certain vector space, to be understood from the point of view of Lebesgue measure theory, is a central ingredient in many of our results. We now make this notion precise.
Definition 2.2.12. Let V be a linear subspace of R[n] with dim (V) > 0. We will say that a vector x E V is a uniformly chosen at random unit vector or u.c.u.v.
38
if x is
uniformly chosen at random from the set {y E V : IIyI = 1}. For x E V a u.c.u.v., a certain event or statement about x is said to occur or hold true almost surely if it is true with probability one.
2.2.5
Spectral theory of the Laplacian.
We will need only a few background results on the spectral theory of the Laplacian matrix of a graph. We present these below in a single statement, but refer the reader to Brouwer and Haemers {2011} for additional background and history. Lemma 2.2.13. Let G = G([n], E) be a simple (undirected) graph. Let L = L(G)  0, z + e(x p)
Q.
On the contrary, (x, z) 9 int ([P, and F > 0, such that z + (x  p) E
Q])
(
i y E int ([P, Q]) if and only if there exist p* E relint (P) and q* E Q such that y E (p*, q*), if and only if there exist p** C relint (P) and q** C relint (Q) such that y E (p**,q**).
if and only if there exists p E relint (P)
Q.
iii Let 7 arff(P) : R n) 4 R[n] be the projection operator onto the affine hyperplane containing P. If taff(P)[(x, z)]n relint (P)n relint (aff(P) [ 0, then (x, z) G
int ([P, QI). Proof. We will obtain these results in order. i (See also Figure 32b) We prove the "only if" direction for both statements. Suppose that y E int ([P, Q]) and let p C P and q c Q be such that y E (p, q). Let us assume that p E a (P). Take an open (n  1)dimensional ball By C
int ([P, Q]) centered at y such that aff(By) is parallel to aff(P) and aff(Q). Let 60
C be the positive open cone generated by By  q, and consider the affine open cone q + C. Then, B,, : (q + C) n aff(P) is an open (n  1)dimensional ball in aff(P) such that p E relint (B.). Hence, since P is also (n  1)dimensional, there exists some pi E relint (P) n B.. Now, let yi = (p1, q) n By E int ([P, Q]). Since Y2 := y + (y  yi) E B, 9 int ([P, Q]), there exist P2 E P and q2 E Q such that Y2 = (p2 , q2) n BY. But then, there exist p* E (P1, P2) C relint (P) and q* E (q, q2 ) g Q such that y = (*, q*) nBy, as we wanted. If q* E a (Q), we can now repeat an analogous construction starting from q* and p* to find p** E relint (P) and q** E relint (Q) such that y E (p**, q**). ii This is a consequence of i, and not easy to prove without it. We prove the second statement, which is equivalent to the first. For the "if" direction, suppose that for some p E relint (P) and e > 0, z + F(x  p) E Q. Take some y E (x, z) and consider the line containing both z + e(x  p) and y. For a sufficiently small e, this line intersects aff(P) in some pi E relint (P). But then, for a small open ball Bp, C relint (P) centered at p, and with aff(Bp 1 ) = aff(P), the open set (Bp 1 , z) contains y and lies completely inside int ([P, Q]), so y E int ([P, Q]). For the "only if" direction, suppose that (x, z) C int ([P, Q]) and take y E (x, z). If x E relint (P), then we are done since Q is also (n  1)dimensional. If x c a (P), from i, take p c relint (P), p # x, and q E Q with y E (p, q). But then, z + E(x  p) = q E Q for some e > 0. iii Take p E flafP) [(x, z)] n relint (P) n relint (7rtaffLp) [Q]) and let p = 0 be a normal to aff(P). Then, for some y E (x, z) and real number Oc : 0, y E (p, p + Ocp) and p + Ocp E relint (Q), so i shows that y E int ([P, Q]). Clearly then (x, z) C
int ([P, Q]).
,
Definition 3.2.21. Let G = G([n], E) be a simple graph, and let 0 be a p.a.o. of G with connected partition Y and acyclic orientation OF of G'. Let us write Y0,in for the set of elements of E that are minimal in (E, o5 ), and for i E [n] with i E p E let:
IG(i, O) ={ E Z: a or p}. With this notation, we now define certain functions associated to 0 and G, called
height and depth: height , depthG : height(i)
[n]
*
Q, 1

depthG(2) =
height(j).
jEIV(i,O) Example 3.2.22. Figure 32a exemplifies Definition 3.2.21 on a particular graph G on vertexset [14] = {1, 2,... ,14}, with given p.a.o. 0. Since both heights
61
aff(P
0:
I
a 614414
'
5"
b)aff
1 3
E ={0,U2,
3},/p=
heightg constant on each
{117, 1} ci. i E
depthG constant on each oa, i
E
161, e.g. height (6) [61, e.g. depthG(13)
=
1
=
(a) (b)
Figure 32: Visual aids/guides to the proofs of (32a) Proposition 3.2.23 and (32b) Proposition 3.2.20.i. (32a) also offers an example for Definition 3.2.21. are constant within each element/block of the connected partition E = {oa = {7}, 2 = {1, 2}, 0 3 = {6,10,14}, .., o6 = {3, 4, 8}} associated to 0, we present only that common value for each block in the figure. and
depth'
1 >
depthG(i) >
.
Proposition 3.2.23. In Definition 3.2.21, let 9 E JG(O) and let p 9 [n] intersect every element of 5i~ni in exactly one point and contain only minimal elements of 0. Then: iEpn(T
if
and only if
= [n], and
.
Moreover, if G is connected, then iE np ePtho(i) > n depthGi)  E> whenever this holds,
Remark 3.2.24. Figure 32a shows one such choice of a set p in Proposition 3.2.23 that works for Example 3.2.22 (in red). Proof. The verification is actually a simple doublecounting argument using the fact that o is an order ideal, so we omit it. When G is connected, if o 5 [n], then there must exist i E [n]\ that is strictly greater in 0 than some element of u (and hence strictly greater than some element of p), again since u is an order ideal. Clearly, we must have heightg(i) >

Theorem 3.2.25. Let G = G([n], E) be a connected simple graph with abstract cell complex G as in Definition 3.2.18. For N > 0, N / n + E, consider the (n 1)dimensional simplex NA = cony (Nei, Ne 2 ,.... Nen) in R "n.If we let YG be
62
the polytopal complex obtained from the join [ZG, NA] after removing the (open) ndimensional cell and the (relatively open) (n  1)dimensional cell corresponding to NA, then YG is a polytopal complex realization of g(G. Proof. Let the faces of 3G obtained from 2H'1\{1 n1,o correspond to the faces of 0 (NA) in the natural way. Also, let the faces of XG of the form ([n], 0) correspond to the faces of ZG as in Theorem 3.2.15. The result is clearly true for the restriction to this two subcomplexes, so we will concentrate our efforts on the remaining cases. First, for the sake of having a lighter notation during the proof, we will let ' = [n]\p for any set p g [n]. A (relatively open) cell of YG\(ZG U a (NA)) can only be obtained as the strict join of a cell of 0 (ZG) and a cell of a (NA), so let us adopt some conventions to refer to this objects. Convention 3.2.26. During the course of the proof, we will let S (or So) denote a generic nonempty relatively open cell of NA obtained from p 9 [n] (resp. po), and F (or Fo) a generic relatively open cell of ZG with p. a. o. 0 of G, associated connected partition Y of G, and acyclic orientation 0 ' of G' yielding 0 (resp. 00, Eo, 0o1o). We argue that we will be done if we can prove the following claim: Claim i a) (F, S) is a cell of YG if and only if b) p = [n] and p is a nonempty union of elements from the set {u E E: u is maximal in (1, oz)}. When this equivalence is established, then we will let (F, S) correspond to the pair (', O3) E XG, where O1i denotes the restriction of 0 to E(G[]). Indeed, assume that Claim i holds. Then, under the stated correspondence of ground sets of cells, all elements of XG are uniquely accounted for as cells of YG. This is true for ZG clearly, and for the remaining cases since for any choice of a1 G [n], 5 0, and of p.a.o. 01 of G[u1], we can always extend uniquely 01 to a p.a.o. of G $in which all the elements of 'ij are maximal. Secondly, we verify that < corresponds to face containment in 9 G. Suppose that
(Fo, So) and (F, S) are cells of YG. Then, (Fo, So) 9 (F, S) if and only if Fo g F and So g 5, if and only if JG(O) C JG(OO) and po 9 p. Now, assuming Claim i, the last statement is true if and only if JG[p](OI) 9 JG[ ](0j): The difficult part here is the "if" direction. Clearly, po g p. Since p is a union of elements of Y that are maximal in (1, Icl + IE(G o])I= E= xI and x  p must have a negative entry in a. Therefore, z + (x  p) all E > 0 and Proposition 3.2.20.ii shows that (x, z) C 0 ([ZG, NA]).
'
NA for
Claim i.2 Let F0 , 00, 1o, 0", So, po be as in Convention 3.2.26. Then, there exist F, 0, T, O', S, p also as in Convention 3.2.26, such that p is a union of elements of the set {0 E Y : ux is maximal in (T, oY)} and (Fo, So) 9 (F, S). (See Figures 33a and 33b for a particular example of the objects and setting considered during this proof) Let: o'po :={xE Y0 : If g E Eo and g
Q:o
u, then g n po =0
Then, define: c 0 :=
U (.
+ G[Uk] is the decomposition of G[6'O] into its connected components, we will let E = Xo,p, U {C, ... , Uk}. We will use the acyclic orientation O of G' obtained from the two conditions 1) OEroK = Oo and 2) 71,   I, l are maximal in (E, oz). The p.a.o. 0 is now obtained from OE, and let F be associated to 0 and S be obtained from ' =1 U ... U UkWe now prove that (Fo, So) C (F, S). if G[& 0'] = Gla1] + 
Since (Fo, So) C (F, S), it is enough to find x E F and z E So such that (x. z) C (F, S), so this is precisely what we will do. To begin, we note that for i E [k], the restriction Oi := Ool, is a p.a.o. of Gi := G[x], so we will let i be the connected partition of Gi and Of' the acyclic orientation of G' associated to Oi; moreover, we note that po intersects every element of i minimal in (Is, 5z). Hence, let us select go C po so that for every i E [k], go intersects every element of i minimal in (Yi, i) in exactly one point and so that go n i contains only minimal elements in Oi. Now, take any x c F and let:
z
= E
E
depth
(j) 
ej
S.
iE[k] iEeoni
We will make use of the technique of Proposition 3.2.20.ii to prove that (x, z) E
64
00: Eo = {{1, 15}, {2, 18}, {3, 6, 7, 16}, {4, 13, 17}, {5,9,12}, {8}, {1
0: E
11},{14}}
{{1, 15}, {2, 3, 6, 7, 8, 16, 18}, {4, 13, 17}, {5,9, 12}, {10, 11}, {14}}
15
po = {1, 2 ,3 ,16 } w Eo,p, = {{4, 13, 17}, {5, 9, 12}, {14}, {10, 11}} 0o = {4, 5, 9, 10, 11, 12, 13, 14, 17}
E= 01
{{1, 15}}, {1, 15}, 02
{{2,18}, {3,6,7,16}, {8}} {2, 3,6,7,8, 16, 18}
2 = =
(b)
(a)
Figure 33: An example to the proof of Claim i.2 in Theorem 3.2.25. (F, S), so for that we need to consider a point in S, which we select as:
S=s=LN
X1~
E
iE[k]
S
ej (E S.
jEi
For i E [k], if we consider a pi E JG,(O) with pi , a, Proposition 3.2.23 gives us:
)
depthQ d
j Qi
>N (eoi k jai

k
ji
/
\jEeingo
1
Ii12
lid
Jai I
_
N
k.IcrI 2
> 0. Hence, for a sufficiently small e > 0, x + E(z  s) E F, so for each y E (x, z) we can find x' E F and s' E S such that y E (X', z'). That implies (FO, So) ; (F, S). Claim i.3 Let both F, S and FO, So satisfy the conditions of Claim i.b). Then, (F, S) n
#
0 if and only if F = FO and S = So. Moreover, (F, S) is a face of YG and dimff ((F, S)) = IpI + dimG[I(01) (similarly for (F,So)). Let O E (0, 1) and consider the polytope P, = {x E R[n] : i[]xi = El) + (1  cx)N} n [ZG, NA]. Every x E P, satisfies the inequalities E
(n
+
(Fo, So)
Xi =
(1  a)N + oc(n + JEJ) and Eje, xi ;> oc(lu+ E(G[u])I) for all u C [n], a =' 0. Per Claim i.1 and Claim i.2, the set (F, s) n P, can be characterized by the condition that it contains all the points x E P, which, among those inequalities,
65
satisfy the and only the following equalities:
Z xi = (1 
(3.2.3)
c)N + cx(n + El) and
iE[n]
(3.2.4)
Exi = OC(jj + E(G[u)), iE9
for all U E JG[p](Olp), o
$
0.
This observation proves the first statement. For the second statement, we assume without loss of generality that N > n+ El and select generic coefficients 0, E R+ with U E JG[p](Ojp)\{0}, such that:
B:,(ju +
IE(G[u])I)
=
N  (n + El).
E JG[p](O1 )\{O}
The linear functional,
f:=
Ze iE[n]
+
E UEJG[O](Oji)\{0}
f.Ze,
(3.2.5)
jEU
satisfies that, for x E P,
f(x) ;> (1  oc)N + cv(n + El) + c (N  (n + El))
=
N.
By the proof of the first claim, this inequality is tight if and only if x C (F, S) n P, = (F, S) n Po. Moreover, since this minimum is independent of c, the linear functional f is minimized in [ZG, NA] exactly at (F, S). If N < n + IEl, we must select negative coefficients and consider instead the maximum of the linear functional in question, analogously. For the third statement, we simply note that an open ball in the affine space determined by all x E R "n satisfying Equalities 3.2.33.2.4 can be easily (but tediously) found inside (F, S). Hence, dim,,. ((F, S)) = II + dimG[ ](OI).
Definition 3.2.27. Let G = G([n], E) be a connected simple graph. Let X% = (XG,  0 if i E 9} : F E F(AG)} is, as suggested by our choice of notation, equal to the totally positive part of the fan of the graphical arrangement of G[o], regarding here R' as a subspace of RHn1. Per Theorem 3.2.15, since F(AG[,]) is precisely the normal fan of ZG], and F(AG[])+ the totally positive part of this fan, we know that the relatively open cones of F(AG[,)) correspond to the p.a.o.'s of G[a]. From the description of the cells of ZG[,], the cone C+, is exactly the cone in F(AG[,])+ normal to the cell of ZG[Y] corresponding to 0. This establishes the correspondence between cells of XG and elements of XG, since we can go both ways in this discussion. Using this same lens to regard cells of XG, the correctness of Definition 3.2.27.1 now follows from the analogous verification done in Theorem 3.2.25, by a standard result on normal fans of polytopes, namely, the duality of face containment.
3.3
Two ideals for acyclic orientations.
Definition 3.3.1. Let G = G([n, E) be a simple graph.
67
1. For an orientation 0 of G and for every i E [n], let:
j E [n]}I
Outdeg(G,1O)
11ij E
nod(G,O)(i)
,
indeg(G,O)(i) := {(j, i) E O[E] []jE[
7I
I{e E O[E] : either e = (j, i) or e = {i, j}, j E [nG},
where we denote the respective associated vectors in R["I as indeg(0 0 and nod(c,o). 2. For a C [n] with a $ 0, define 1 Let now, for every i E [n]:
=
inof(G,,) (i)
:=
0
:= 1[n].
if i C o, otherwise,
I{{i,j} E E :
Outof(Ga)()
outdeg(G,O),
ei,e E RNI, further writing 1
E E :j E a}
f{{ij}
),
[n]\}
0
if i E a, otherwise,
and denote the respective associated vectors of R [n] as inof(G,,) and outof(0,0). Remark 3.3.2. During this section, we will follow the notation and definitions of Miller and Sturmfels {2005}, Chapters 1,4,5,6 and 8, in particular, those pertaining to labelled polytopal cell complexes. We refer the reader to this standard reference on the subject for further details. Some key conventions worth mentioning here are: 1. The letter k will denote an infinite field.
2. For a := (ai, a2 ,.
. . , a,)
E NHn, ma := (x
: i
C [n]) is the ideal of k[x1,.
..
, xn]
associated to a. Definition 3.3.3. Let G = G([n], E) be a connected simple graph. The ideal AG of acyclic orientations of G is the monomial ideal of k[x1,... ,xn] minimally generated as: AG :
Xindei(GO)
J
: 0 is an acyclic orientation of G).
indeg(Go)(i)+l
iE[nI Definition 3.3.4. Let G = G([n], E) be a connected simple graph. The tree ideal TG of G is the monomial ideal of k[x1,..., x.] minimally generated as:
TG :=
Xoutof(G,a)1c
x
utof(G,)(0+1
:9 E
2 [nI\{o}
and G[31 is connected)
t 3iEoc
Definition 3.3.5. Given two vectors a, b E
68
N En]
with b  a (bi < ai
for all i E [n]),
let a\b be the vector whose ith coordinate is: ai+ 1  bi 0
ai\bi ar~i =
1, if bi if bi = 0.
If I is a monomial ideal whose minimal generators all divide x", then the Alexander dual of I with respect to a is: Ial := fl{ma\b
Xb
is a minimal generator of I}.
Theorem 3.3.6. Let G = G([n], E) be a simple connected graph. Then, the ideals AG and TG of Definitions 3.3.33.3.4 are Alexander dual to each other with respect to dG + 1, So AdG+1] = TG and T [dG+l
=
AG
Proof. It is enough to prove one of these two equalities, so we will prove that A[dG 11  TG. Take some o E 2["l\{o} such that G[o] is connected and consider Utof(G,fJ)M1. We will the minimal generator of TG given by XOUtOf(G,)+ l = verify that x"Utf(G,g)1' E m(dG+I)\b for every minimal generator Xb of AG. Se(G,) HE[] i lect an acyclic orientation 0 of G and let xindg(G,o) the minimal generator of AG associated to 0. If we take m E u to be maximal in ([n], 0 m and i E a imply i = m, then ( dG(m) outof(GM)(m)
+
)
\
outdeg(GO)(m) + 1 < ING(m)\oi + 1
(indeg(GO)(M) + )
+ 1. Hence,
Xoutof(G,+1,
E
OtOf(GU)(m)+1) C KdG(m)+1)\(inde9(GO)(m)+1) (dG+1A(indeg(G,O)
1)
This proves that TG C A[dG+l] Now, consider a monomial x' V TG with 0 < b (so bi > 0 for some i E [n]). Then, for every U E 2" \{0 } there exists i E  such that bi < outof(G,cr)(i) + 1, noting here that the condition on G[o] being connected can be dropped. Hence, consider a bijective labeling f : [n] + [n] of the vertices of G such that bf1(j) < outof(G,fIl{,)(f(i) + 1 for all i E [n]. If we let 0 be the acyclic orientation of G such that for every e = {i,j} E E, O(e) = (ij) if and only if f(i) < f(j), then for all i E [n], bf1(j) < outof(G'f(1,i])(@) + 1 = outdeg(GO)(f(i)) + 1 =
(dG(f'(i) + 1) \ (indeg(G,O)(fr(i)) + I therefore TG A [dG+1I that xb O A [dG+l, GG
,
or xb
'
M(dG+1)\(indeg(G,O)+1).
Corollary 3.3.7. Let G = G([n], E) be a simple connected graph. Then: AG
{minof(Ga)+1
a
E 2 [n]\{0} and G[] is connected},
69
This shows
is the irreducible decomposition of AG. Also:
TG =
{moutdeg(0,G)+1 : 0 is an acyclic orientation of G},
is the irreducible decomposition of TG. Definition 3.3.8. For a simple connected graph G = G([n], E), consider the polytopal complexes ZG, YG and XG, which respectively realize the abstract cell complexes _TG, G and X of Definitions 3.2.11, 3.2.18 and 3.2.27. We will let ZG = (ZGz (YG, fy) and XG = (XG, x) be the Nin] labelled cell complexes with underlying YG polytopal complexes given by ZG, YG and XG, respectively, and face labelling functions Ze 1 , EX, defined according to: 1. ZG: For a face F of ZG correspondingto 0 E 2G:
[n].
fz(F)i = nod(G,O)(i) + 1, i
2. YG: (a) For a face F of YG correspondingto (u0,O) E
f(F)

fy(F~i
G:
+ 1 if i E u,
nod(G[a],o)() dG(i) + 2
(b) For a face F of YG corresponding to u
G
otherwise.
C2
dG(i) + 2 0
{]\{[n],0} g 3G:
if i Co, otherwise.
3. XG: For a face F of XG corresponding to (,,O)
Ec
:
( outdeg(G[U], 0 )(i + outof(G,,(i) + 1 fF.0 otherwise.
if
E,
Lemma 3.3.9. Let G = G([n], E) be a simple connected graph. Then, for any face F of ZG with vertices v 1,... , V, we have that: Xez(F) = LCM {x z(vi)}iE[k]
where LCM stands for "least common multiple". Proof. Let F be a face of ZG with corresponding p. a. o. 0 of G and connected partition Y. Every acyclic orientation of G that corresponds to a vertex of F is obtained by 1) selecting an acyclic orientation for each of the G[o] with u CE, and then by 2) combining those III acyclic orientations with 0[E] n V 2 . For a fixed vertex i c o with u c 5, it is possible to select an acyclic orientation of G[] in which i
70
is maximal and then to extend this to an acyclic orientation of G that refines 0, so if vertex v, of F corresponds to one such orientation, then e,(vj)j = nod(G,o)(i) + 1. On the other hand, clearly Ez(vj)i 5 nod(G,o)(i) + 1 for all vertices vj of F. Hence, xez(F) = L CM{xfz (v)}Elk]
Corollary 3.3.10. Similarly, for G as in Lemma 3.3.9 and for any face F of YG with vertices v 1 ,... ,V, we have that: X y(F) = LCM {x 4 (vi)}iE[k] where LCM stands for "least common multiple". Proof. If F is a face of YG inside the simplex NA, then this is immediate. If F corresponds to some (u, 0), then this is a consequence of the proof of Lemma 3.3.9, since the vertices of F are all the N  ej with i E [n]\c, and all the vertices of YG that correspond to acyclic orientations of G whose restrictions to G[u] refine 0 and in which all edges of G connecting u with [n]\u are directed out of u.
Proposition 3.3.11. Let G = G([nl, E) be a simple connected graph. The cellular free complex FyG supported on YG is a minimal free resolution of the artinianquotient k~xi, . . . , x.]/ (AG + mdG+2). Proof. Without loss of generality, we assume here that N > n + IEj. From standard results in topological combinatorics it is easy to see that for b E N1"I, the closed b} form a faces of YG that are contained in the closed cone Cs = {v E R in) :v contractible polytopal complex, whenever this cone contains at least one face of YG. Now, suppose that b satisfies that bi 5 dG(i) + 1 for all i E [n]. Then, the complex of faces of YG in the cone Csb coincides with YG, dG(i) + 2, and let Db = [n]\Ub. Consider the vector a e R["] such that: 
N
{
ai= bi
if i E Ub,
if i E Db.
Then, the set of faces of YG in the cone C 1 of the recursion to define fb(T), N < n, the edge that will be added to the set E (Tb(T)) is an edge of T. Initially, when N = 0, we have fb(T) (0) = p1(0) = r and E (Tb(T)) = 0. Suppose that the result is true for all N < i, i E [n], and let us consider the ith step of the recursion, so that K = {0,1,..., i  1}. By induction, if j E [n]\fb(T)[K] and since 1 fb(T)(k) = p (k) for all k E K, we have that {lo
so when
3, and select an arbitrary bijecttvemap fo : V + [n] (regarded as a labelling of V). Let us consider a sequence (ft)t=o,1,2,... of bijective maps V + [n] such that for t > 1, ft is obtained from ft1 through the following random process: Let vt E V be chosen uniformly at random, and let, n if v = vt,
ft (v)=
ft i(v)  1
if ft i(v) > ft i(v),
fti(v)
otherwise.
Consider now the sequence of acyclic orientations (Ot)t=o.1,2,... of G induced by the labellings (ft)t=0,1,2,..., so that for all e = {u, v} E E and t > 0, we have that Ot(e) = (u,v) if and only if ft(u) < ft(v). The sequence (Ot)t=o,1,2... is called the Cardshuffling (CS) Markov chain on the set of acyclic orientations of G. Equivalently, we can define this Markov chain by selecting an arbitrary acyclic orientation Oo of G, and then for each t > 1, letting Ot be obtained from Ot_1 by selecting vt e V uniformly at random and taking, for all e c E: Ot(e) =
Oti(e)
(v,vt)
if vt g e,
if e= {v,vt}.
Theorem 3.5.4. The CardShuffling Markov chain of G in Definition 3.5.3 is a welldefined, irreducible and aperiodic Markov chain on state space equal to the set of all acyclic orientations of G; its unique stationary distribution 7rC is given by: S=
,e(O) for all acyclic orientations 0 of G,
where e(O) denotes the number of linear extensions of the induced poset (V, 0 and Pr[Ot = Ot1Ot1] > 0 for all t > 1), and has a unique stationary distribution 7rCs, necessarily then given by for every acyclic orientation 0 of G, from Equations 3.5.1. i(! 710 = Definition 3.5.5 (EdgeLabelReversal Stochastic Process). Let G = G(V, E) be a connected simple graph with VI = n, and select an arbitrary bijective map fo : V + [n] (regarded as a labelling of V). Let us consider a sequence (ft)t=0,1,2,... of bijective maps V + [n] such that for t > 1, ft is obtained from ft1 through the following random process: Let et = {ut, vt} E E be chosen uniformly at random from this set, and let, if V = vt, ifv=ut,
fti(ut)
ft_1(Vt) ft_1(v)
ft(v)=
otherwise.
Consider now the sequence of acyclic orientations (0t)t=0,1,2,... of G induced by the labellings (ft)t=0,1,2,..., so that for all e = {u, v} E E and t > 0, we have that Ot(e) = (u, v) if and only if ft(u) < ft(v). The sequence (Ot)t=,1,2,... is called the EdgeLabelReversal (ELR) stochastic process on the set of acyclic orientations of G. Theorem 3.5.6. The EdgeLabelReversal stochastic process of G in Definition 3.5.5 satisfies that, for every acyclic orientation 0 of G: N (irELR)
=ELR
:.
Jim
Lt
_L
=0]
e(Q)
t=1
where e(O) denotes the number of linear extensions of the induced poset (V, this result holds independently of the initial choice of 00.
O), and
Proof. Consider the simple graph H on vertexset equal to the set of all bijective maps V + [n], and where two maps f and g are connected by an edge if and only if there exists {u, v} E E such that f(u) = g(v), f(v) = g(u), and f(w) = g(w) for all w E V\{u, v}. Since G is connected, a standard result in the algebraic theory of the symmetric group shows that H is connected, e.g. consider a spanning tree T of G; then, any permutation in 6v can be written as a product of transpositions of the form (u v) with {u, v} E E(T). Moreover, by considering the parity of permutations in 6V, we observe that H is bipartite. Now, the sequence (ft)t=,1,2,... of Definition 3.5.5 87
is precisely a simple random walk on H, and the degree of each bijective map f : V * [n] in H is clearly JE, so the stationary distribution for this Markov chain is uniform. Necessarily then, the result follows from the construction of (Ot)t=o,1,2,... and Equations 3.5.1.
Definition 3.5.7 (Sliding(n + 1) Stochastic Process). Let G = G(V, E) be a connected simple graph withV = n, and consider the graphGr. Let us select an arbitrary bijective map fo : V LI {r} + [n + 1], which we regard as a labelling of the vertices of Gr, and define a sequence (ft)t=o,1,2... of bijective maps V Li {r} * [n + 1] such that for t > 1, ft is obtained from ft1 through the following random process: Let Vt_1 E V U {r} be such that ft_1(vt_) = n + 1, and select vt G NGr(vt1) uniformly at random from this set. Then,
ft(v)=
n n+1
ifv=vt,
ft1(vt) ft_1(v)
if V t=v_, otherwise.
Consider now the sequence of acyclic orientations (Ot)t=0,1,2,... of Gr induced by the labellings (ft)t=0,1,2,.., so that for all e = {u, v} E E(Gr) and t > 0, we have that Ot(e) = (u, v) if and only if ft(u) < ft(v). The sequence (Ot)t=0,1,2,... is called the Sliding(n + 1) (SL) stochastic process on the set of acyclic orientations of Gr. Theorem 3.5.8. The Sliding(n + 1) stochastic process of Gr of Definition 3.5.7 satisfies that, if Sr is the set of all acyclic orientations of Gr whose unique maximal element is r, then Z0 Pr[Ot Sr3= oc and for every 0 E Sr:
(nSL )O = 7TSL:0
liM
N+o
LN
Nt= Nt=
Pr[Ot =0]
Pr[ Ot
Sr ]
e(O1)
n!
where Ov is the restriction of 0 to E (hence an acyclic orientation of G) and e(Ov) denotes the number of linear extensions of the induced poset (V, oi ). These results hold independently of the initial choice of 00. Proof. Consider the simple graph H on vertexset equal to the set of all bijective maps V U {r} + [n + 1], and where two maps f and g are connected by an edge if and only if there exists {u, v} E E(Gr) such that f(u) = g(v) = n + 1, f(v)  g(u),
and f(w) = g(w) for all w E V\{u, v}. If two bijective maps
f, g
: V U {r}
+
[n + 1]
differ only in one edge of Gr, so that f(u) = g(v) 7 n + 1 and f(v) = g(u) $ n + 1 for some {u,.v} c E(Gr), but f(w) = g(w) for all w E V\{u, v}, then we can easily but somewhat tediously show that f and g belong to the same connected component of H, making use of the facts that vertex r is adjacent to all other vertices of Gr and that G is connected. But then, the proof of Theorem 3.5.6 shows that H is a connected graph. Now, the sequence (ft)t=o,1,2,... of Definition 3.5.7 is a simple random walk on H, and the degree of a bijective map f : V + [n] in H is clearly dc,(vi), where vf c V depends on f and is such that f(vf) = n + 1, so the stationary distribution
88
7r for this Markov chain satisfies that 7Tf = c  dG (vf), for some fixed normalization constant c E R+. The vertices of H that induce acyclic orientations of Gr from the set Sr are exactly the bijective maps f : V Li {r} + [n + 1] such that f(r) = n + 1, and for these we have that 7Tf = c  n. The result then follows from the construction of (Ot)t=0,1,2,... and from Equations 3.5.1.
Cov(0t_ 1 ) :=
{e E Oti[E] : e represents a cover relation in (V,
os)}
,
Definition 3.5.9 (CoverReversal Random Walk). Let G = G(V, E) be a simple graph with Vj = n, and select an arbitrary acyclic orientation Oo of.G. Let us consider a sequence (Ot)t=0,1,2,... of acyclic orientations of G such that for t > 1, Ot is obtained from Ot_1 through the following random process: Let (u, v) be selected uniformly at random from the set,
and for all e E E, let,
if e = {u, v},
(v,u)
Ot(e) =
Oti(e) otherwzise.
The sequence (Ot)t=,1,2,... is called the CoverReversal (CR) random walk on the set of acyclic orientations of G. Theorem 3.5.10. The CoverReversal random walk in G of Definition 3.5.9 is a simple 2period random walk on the 1skeleton of the clean graphical zonotope ZG of Theorem 3.2.15 (hence, on a particularsimple connected bipartite graph on vertexset equal to the set of all acyclic orientations of G), and its stationary distribution 7rcR satisfies that, for every acyclic orientation 0 of G:
7r
R =
c  ICOv(0),
where c E R+ is a normalization constant independent of 0. Proof. From the proof of Theorem 3.2.15, the edges 'of ZG are in bijection with the set of all p.a.o.'s 0 of G such that if 2 is the connected partition associated to 0, then II = n  1. Hence, the edges of ZG are in bijection with the set of all pairs of the form (e, 0), where e E E and 0 is an acyclic orientation of the graph G/e, obtained from G by contraction of the edge e. The two vertices of ZG adjacent to an edge corresponding to a (e, 0) with e = {u, v} are, respectively, obtained from the acyclic orientations 01 and 02 of G such that 0 1 (e) = (u, v), 0 2 (e) = (v, u), and such that 021E\, are naturally induced by 0 (e.g. see Definition 3.2.4). Necessarily then, both (u, v) and (v, u) correspond respectively to cover relations in the posets (V, o,) and (V, 02), since otherwise the orientation 0 of G/e would not be acyclic. On the other hand, given an acyclic orientation 01 of G and an edge (u, v) E 01 [E] such that v covers u in (V, Oi), then, reversing the orientation of (only) that edge in 01 yields a new acyclic orientation 02 of G, so (v, u) E 02 [E]. Otherwise, using a directedcycle formed by edges from 02 [E], which must then include the edge (v, u), 011E\e =
89
we observe that the relation u 1, Ot is obtained from Ot_1 through the following random process: Let {u, v} c E be selected uniformly at random from this set, with (u, v) E Ot_1[E], and for all e = {x, y} c E with (x, y) G Ot 1 [E], let, (e) OtOt e) 
{
,'X) (xY)
=
Ot (e)
if u o, X