Graph Orientations and Linear Extensions.
Abstract.
Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph and, therefore, we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem will be solved essentially for comparability graphs and odd cycles, presenting several proofs. The corresponding enumeration problem for arbitrary simple graphs will be studied, including the case of random graphs; this will culminate in 1) new bounds for the volume of the stable polytope and 2) strong concentration results for our main statistic and for the graph entropy, which hold true for random graphs. We will then argue that our problem springs up naturally in the theory of graphical arrangements and graphical zonotopes.
Key words and phrases:
acyclic orientation, linear extension, poset, comparability graph, stable polytope.1. Introduction.
Linear extensions of partially ordered sets have been the object of much attention and their uses and applications remain increasing. Their number is a fundamental statistic of posets, and they relate to everrecurring problems in computer science due to their role in sorting problems. Still, many fundamental questions about linear extensions are unsolved, including the wellknown  Conjecture. Efficiently enumerating linear extensions of certain posets is difficult, and the general problem has been found to be Pcomplete in Brightwell and Winkler (1991).
Directed acyclic graphs, and similarly, acyclic orientations of simple undirected graphs, are closely related to posets, and their problemmodeling values in several disciplines, including the biological sciences, needs no introduction. We propose the following problem:
Problem 1.1.
Suppose that there are individuals with a known contagious disease, and suppose that we know which pairs of these individuals were in the same location at the same time. Assume that at some initial points, some of the individuals fell ill, and then they started infecting other people and so forth, spreading the disease until all of them were infected. Then, assuming no other knowledge of the situation, what is the most likely way in which the disease spread out?
Suppose that we have an underlying connected undirected simple graph with vertices. If we first pick uniformly at random a bijection , and then orient the edges of so that for every we select (read directed to ) whenever , we obtain an acyclic orientation of . In turn, each acyclic orientation induces a partial order on in which if and only if there is a directed path in the orientation. In general, several choices of above will result in the same acyclic orientation. However, the most likely acyclic orientations so obtained will be the ones whose induced posets have the maximal number of linear extensions, among all posets arising from acyclic orientations of . Our problem then becomes that of deciding which acyclic orientations of attain this optimality property of maximizing the number of linear extensions of induced posets. This problem, referred to throughout this article as the main problem for , was raised by Saito (2007) for the case of trees, yet, a solution for the case of bipartite graphs had been obtained already by Stachowiak (1988). The main problem brings up the natural associated enumerative question: For a graph , what is the maximal number of linear extensions of a poset induced by an acyclic orientation of ? This statistic for simple graphs will be herein referred to as the main statistic (Definition 3.12).
The central goal of this initial article on the subject will be to begin a rigorous study of the main problem from the points of view of structural and enumerative combinatorics. We will introduce 1) techniques to find optimal orientations of graphs that are provably correct for certain families of graphs, and 2) techniques to estimate the main statistic for more general classes of graphs and to further understand aspects of its distribution across all graphs.
In Section 2, we will present an elementary approach to the main problem for both bipartite graphs and odd cycles. This will serve as motivation and preamble for the remaining sections. In particular, in Section 2.1, a new solution to the main problem for bipartite graphs will be obtained, different to that of Stachowiak (1988) in that we explicitly construct a function that maps injectively linear extensions of nonoptimal acyclic orientations to linear extensions of an optimal orientation. As we will observe, optimal orientations of bipartite graphs are precisely the bipartite orientations (Definition 2.1). Then, in Section 2.2, we will extend our solution for bipartite graphs to odd cycles, proving that optimal orientations of odd cycles are precisely the almost bipartite orientations (Definition 2.7).
In Section 3, we will introduce two new techniques, one geometrical and the other posettheoretical, that lead to different solutions for the case of comparability graphs. Optimal orientations of comparability graphs are precisely the transitive orientations (Definition 3.2), a result that generalizes the solution for bipartite graphs. The techniques developed on Section 3 will allow us to rediscover the solution for odd cycles and to state inequalities for the general enumeration problem in Section 4. The recurrences for the number of linear extensions of posets presented in Corollary 3.11 had been previously established in Edelman et al. (1989) using promotion and evacuation theory, but we will obtain them independently as byproducts of certain network flows in Hasse diagrams. Notably, Stachowiak (1988) had used some instances of these recurrences to solve the main problem for bipartite graphs.
Further on, in Section 4, we will also consider the enumeration problem for the case of random graphs with distribution , and obtain tight concentration results for our main statistic, across all graphs. Incidentally, this will lead to new inequalities for the volume of the stable polytope and to a very strong concentration result for the graph entropy (as defined in Csiszár et al. (1990)), which hold for random graphs.
Lastly, in Section 5, we will show that the main problem for a graph arises naturally from the corresponding graphical arrangement by asking for the regions with maximal fractional volume (Proposition 5.2). More surprisingly, we will also observe that the solutions to the main problem for comparability graphs and odd cycles correspond to certain vertices of the corresponding graphical zonotopes (Theorem 5.3).
Convention 1.2.
Let be a simple undirected graph. Formally, an orientation of (or ) is a map such that for all , we have . Furthermore, is said to be acyclic if the directed graph on vertex set and directededge set is acyclic. On numerous occasions, we will somewhat abusively also identify an acyclic orientation of with the set , or with the poset that it induces on , doing this with the aim to reduce extensive wording.
When defining posets herein, we will also try to make clear the distinction between the ground set of the poset and its order relations.
2. Introductory results.
2.1. The case of bipartite graphs.
The goal of this section is to present a combinatorial proof that the number of linear extensions of a bipartite graph is maximized when we choose a bipartite orientation for . Our method is to find an injective function from the set of linear extensions of any fixed acyclic orientation to the set of linear extensions of a bipartite orientation, and then to show that this function is not surjective whenever the initial orientation is not bipartite. Throughout the section, let be bipartite with vertices.
Definition 2.1.
Suppose that has a bipartition . Then, the orientations that either choose for all with and , or for all with and , are called bipartite orientations of .
Definition 2.2.
For a graph on vertex set with , we will denote by Bij the set of bijections from to .
As a training example, we consider the case when we transform linear extensions of one of the bipartite orientations into linear extensions of the other bipartite orientation. We expect to obtain a bijection for this case.
Proposition 2.3.
Let be a simple connected undirected bipartite graph, with . Let and be the two bipartite orientations of . Then, there exists a bijection between the set of linear extensions of and the set of linear extensions of .

Consider the automorphism rev of the set Bij given by for all and . It is clear that . However, since implies , then rev reverses all directed paths in any induced acyclic orientation of , and in particular the restriction of rev to the set of linear extensions of has image , and viceversa.
∎
We now proceed to study the case of general acyclic orientations of the edges of . Even though similar in flavour to Proposition 2.3, our new function will not in general correspond to the function presented in the proposition when restricted to the case of bipartite orientations.
To begin, we define the main automorphisms of Bij that will serve as building blocks for constructing the new function.
Definition 2.4.
Consider a simple graph with . For different vertices , let be the automorphism of Bij given by the following rule: For all , let
It is clear that for all . Moreover, we will need the following technical observation about .
Observation 2.5.
Let be a simple graph with and consider a bijection . Then, if for some with we have that but , then and furthermore, at least one of or must be equal to one of or .
Let us present the main result of this section, obtained based on the interplay between acyclic orientations and bijections in Bij.
Theorem 2.6.
Let be a connected bipartite simple graph with , and with bipartite orientations and . Let also be an acyclic orientation of . Then, there exists an injective function from the set of linear extensions of to the set of linear extensions of and furthermore, is surjective if and only if or .

Let be a linear extension of , and without loss of generality assume that . We seek to find a function that transforms into a linear extension of injectively. The idea will be to describe how acts on as a composition of automorphisms of the kind presented in Definition 2.4. Now, we will find the terms of the composition in an inductive way, where at each step we consider the underlying configuration obtained after the previous steps. In particular, the choice of terms in the composition will depend on . The inductive steps will be indexed using a positive integer variable , starting from , and at each step we will know an acyclic orientation of , a set and a function . The set will always be defined as the set of all vertices incident to an edge whose orientation in and differs, and will be a particular linear extension of that we will define.
Initially, we set and , and calculate . Now, suppose that for some fixed we know and , and we want to compute and . If , then and is a linear extension of , so we stop our recursive process. If not, then contains elements and such that and are respectively minimum and maximum elements of . Moreover, . We will then let , be the acyclic orientation of induced by , and calculate from .
If we let be the minimal positive integer for which , then . The existence of follows from observing that whenever . In particular, if , then and so . It follows that the pairs are pairwise disjoint, and for all , and . As a consequence, the automorphisms in the composition description of commute. Lastly, will be a linear extension of and we stop the inductive process by defining .
To prove that is injective, note that given and as above, we can recover uniquely by imitating our procedure to find . Firstly, set and , and compute as the set of vertices incident to an edge whose orientation differs in and . Assuming prior knowledge of and , and whenever for some positive integer , find the elements of whose images under are maximal and minimal in . By the discussion above and Observation 2.5, we check that these are respectively and precisely and . Resembling the previous case, we will then let , be the acyclic orienation of induced by , and compute accordingly as the set of vertices incident to an edge with different orientation in and . Clearly , and the procedure shows that is invertible in its image.
To establish that is not surjective whenever , note that then contains a directed path and . Without loss of generality, we may assume that the orientation of these edges in is given by and . But then, a linear extension of in which and is not in since otherwise, using the notation and framework discussed above, there would exist different such that and , which then contradicts the choice of and . This completes the proof.
∎
2.2. Odd cycles.
In this section will be a cycle on vertices with . The case of odd cycles follows as an immediate extension of the case of bipartite graphs, but it will also be covered under a different guise in Section 4. As expected, the acyclic orientations of the edges of odd cycles that maximize the number of linear extensions resemble as much as possible bipartite orientations. This is now made precise.
Definition 2.7.
For an odd cycle , we say that an ayclic orientation of its edges is almost bipartite if under the orientation there exists exactly one directed path, i.e. only one instance of and with .
Theorem 2.8.
Let be an odd cycle on vertices with . Then, the acyclic orientations of that maximize the number of linear extensions are the almost bipartite orientations.

Since the case when is straightforward let us assume that , and consider an arbitrary acyclic orientation of . Again, our method will be to construct an injective function that transforms every linear extension of into a linear extension of some fixed almost bipartite orientation of , where the specific choice of almost bipartite orientation will not matter by the symmetry of .
To begin, note that there must exist a directed path in , say and for some . Our goal will be to construct so that it maps into the set of linear extensions of the almost bipartite orientation in which our directed path is the unique directed path. To find , first consider the bipartite graph with vertex set and edge set , along with the orientation of its edges that agrees on common edges with and contains . Clearly is acyclic. If is a linear extension of , we regard the restriction of to as a strict orderpreserving map on , and analogously to the proof of Theorem 2.6, we can transform injectively into a strict orderpreseving map with of the bipartite orientation of that contains . Now, if we define via for all and , we see that is a linear extension of . We let .
The technical work for proving the general injectiveness of , and its nonsurjectiveness when is not almost bipartite, has already been presented in the proof of Theorem 2.6: That is injective follows from the injectiveness of the map transforming into , and then by noticing that . Nonsurjectiveness follows from noting that if is not almost bipartite, then contains a directed path with and , so we cannot have simultaneously and .
∎
3. Comparability graphs.
In this section, we will study our main problem using more general tehniques. As a consequence, we will be able to understand the case of comparability graphs, which includes bipartite graphs as a special case. Let us first recall the main object of this section:
Definition 3.1.
A comparability graph is a simple undirected graph for which there exists a partial order on under which two different vertices are comparable if and only if .
The acyclic orientations of the edges of a comparability graph that maximize the number of linear extensions are precisely the orientations that induce posets whose comparability graph agrees with .
Comparability graphs have been largely discussed in the literature, mainly due to their connection with partial orders and because they are perfectly orderable graphs and more generally, perfect graphs. Comparability graphs, perfectly orderable graphs and perfect graphs are all large hereditary classes of graphs. In Gallai’s fundamental work in Gallai et al. (2001), a characterization of comparability graphs in terms of forbidden subgraphs was given and the concept of modular decomposition of a graph was introduced.
Note that, given a comparability graph , we can find at least two partial orders on induced by acyclic orientations of whose comparability graphs (obtained as discussed above) agree precisely with , and the number of such posets depends on the modular structure of . Let us record this idea in a definition.
Definition 3.2.
Let be a comparability graph, and let be an acyclic orientation of such that the comparability graph of the partial order of induced by agrees precisely with . Then, we will say that is a transitive orientation of .
We will present two methods for proving our main result. The first one (Subsection 3.1) relies on Stanley’s transfer map between the order polytope and the chain polytope of a poset, and the second one (Subsection 3.2) is made possible by relating our problem to network flows.
3.1. Geometry.
To begin, let us recall the main definitions and notation related to the first method.
Definition 3.3.
We will consider with euclidean topology, and let be the standard basis of . For , we will define and ; furthermore, for we will let and .
Definition 3.4.
Given a partial order on , the order polytope of is defined as:
The chain polytope of is defined as:
Stanley’s transfer map is the function given by:
Let be a partial order on . It is easy to see from the definitions that the vertices of are given by all the with an order filter of , and those of are given by all the with an antichain of .
Now, a wellknown result of Stanley (1986) states that where is the number of linear extensions of . This result can be proved by considering the unimodular triangulation of whose maximal (closed) simplices have the form with a linear extension of . However, the volume of is not so direct to compute. To find Stanley made use of the transfer map , a pivotal idea that we now wish to describe in detail since it will provide a geometrical point of view on our main problem.
It is easy to see that is invertible and its inverse can be described by:
As a consequence, we see that for all antichains of , where is the order filter of induced by . It is also straightforward to notice that is linear on each of the with a linear extension of , by staring at the definition of . Hence, for fixed and for each , we can consider the order filters along with their respective minimal elements in , and notice that and also that . From there, is now easily seen to be a unimodular linear map on , and so . Since is invertible, without unreasonable effort we have obtained the following central result:
Theorem 3.5 (Stanley (1986)).
Let be a partial order on . Then, , where is the number of linear extensions of .
Definition 3.6.
Given a simple undirected graph , the stable polytope of is the full dimensional polytope in obtained as the convex hull of all the vectors , where is a stable (a.k.a. independent) set of .
Now, the chain polytope of a partial order on is clearly the same as the stable polytope of its comparability graph since antichains of correspond to stable sets of . In combination with Theorem 3.5, this shows that the number of linear extensions is a comparability invariant, i.e. two posets with isomorphic comparability graphs have the same number of linear extensions.
We are now ready to present the first proof of the main result for comparability graphs. We will assume connectedness of for convenience in the presentation of the second proof.
Theorem 3.7.
Let be a connected comparability graph. Then, the acyclic orientations of that maximize the number of linear extensions are exactly the transitive orientations of .

Without loss of generality, assume that . Let be an acyclic orientation of inducing a partial order on . If two vertices are incomparable in , then . This implies that all antichains of are stable sets of , and so .
On the other hand, if is not transitive, then there exists two vertices such that , but such that and are comparable in , i.e. the transitive closure of induces comparability of and . Then, is a vertex of the stable polytope of , but since is a subpolytope of the dimensional cube, . We obtain that if is not transitive, and so .
If is transitive, then . This completes the proof.
∎
3.2. Poset theory.
Let us now introduce the background necessary to present our second method. This will eventually lead to a different proof of Theorem 3.7.
Definition 3.8.
If we consider a simple connected undirected graph and endow it with an acyclic orientation of its edges, we will say that our graph is an oriented graph and consider it a directed graph, so that every member of is regarded as an ordered pair. We will use the notation to denote an oriented graph defined in such a way, coming from a simple graph .
Definition 3.9.
Let be an oriented graph. We will denote by the oriented graph with vertex set and set of directed edges equal to the union of and all edges of the form:
A natural flow on will be a function such that for all , we have:
In other words, a natural flow on is a nonnegative network flow on with unique source , unique sink , and infinite edge capacities.
First, let us relate natural flows on oriented graphs with linear extensions of induced posets.
Lemma 3.10.
Let be an oriented graph with induced partial order on , and with . Then, the function defined by
is a natural flow on . Moreover, the net flow from to is equal to .

Assume without loss of generality that , and consider the directed graph on vertex set whose set of directed edges consists of all:
As directed graphs, we check that is a subgraph of . We will define a network flow on with unique source and unique sink , expressing it as a sum of simpler network flows.
First, extend each linear extension of to by further defining and . Then, let be given by
Clearly, defines a network flow on with source , sink , and total net flow , and then defines a network flow on with total net flow . Moreover, for each we have that . It remains now to check that the restriction of to is still a network flow on with total flow .
We have to verify two conditions. First, for and if , then
so , i.e. the net flow between and is . Second, again for , if but , then . These two observations imply that defines a network flow on with total flow .
∎
The next result was obtained in Edelman et al. (1989) using the theory of promotion and evacuation for posets, and their proof bears no resemblance to ours.
Corollary 3.11.
Let be a partial order on , with . If is an antichain of , then , where denotes the induced poset on . Similarly, if is a cutset of , then . Moreover, if is a subset of that is either a cutset or an antichain of , then if and only if is both a cutset and an antichain of .

Let be any graph that contains as a subgraph the Hasse diagram of , and orient the edges of so that it induces exactly to obtain an oriented graph . Let be as in Lemma 3.10. Since edges representing cover relations of are in and are oriented accordingly in , the net flow is . Moreover, by the standard chain decomposition of network flows of Ford Jr and Fulkerson (2010) (essentially Stanley’s transfer map), which expresses as a sum of positive flows through each maximal directed path of , it is clear that for an antichain of , we have that , since antichains intersect maximal directed paths of at most once. Similarly, for a cutset of , we have that since every maximal directed path of intersects . Furthermore, equality will only hold in either case if the other case holds as well. But then, for each , the map Trans that transforms linear extensions of into linear extensions of and defined via: For a linear extension of and ,
is a bijection onto its image, and the number is precisely .
∎
Getting ready for the second proof of Theorem 3.7, it will be useful to have a notation for the main object of study in this paper:
Definition 3.12.
Let be an undirected simple graph. The maximal number of linear extensions of a partial order on induced by an acyclic orientation of will be denoted by .

Assume without loss of generality that . We will do induction on . The case is immediate, so assume the result holds for . Note that every induced subgraph of is also a comparability graph and moreover, every transitive orientation of induces a transitive orientation on the edges of every induced graph of . Now, let be a nontransitive orientation of with induced poset , so that there exists a comparable pair in that is stable in . Let be an antichain cutset of . Then, is a stable set of . Letting be the induced subgraph of on vertex set , we obtain that , where the first inequality is an application of Corollary 3.11 on a transitive orientation of , along with Definition 3.12 and the inductive hypothesis, the second inequality is obtained after recognizing that the poset induced by on each is a subposet of and by Definition 3.12, and the last equality follows because is a cutset of . If or , then by induction the second inequality will be strict. On the other hand, if or , then the first inequality will be strict since is stable in .
Lastly, the different posets arising from transitive orientations of have in common that their antichains are exactly the stable sets of , and their cutsets are exactly the sets that meet every maximal clique of at least once, so by the corollary, the inductive hypothesis and our choice of above, these posets have the same number of linear extensions and this number is in general at least , and strictly greater if or .
∎
4. Beyond comparability and enumerative results.
In this section, we will illustrate a short application of the ideas developed in Section 3 to the case of odd cycles, reestablishing Theorem 2.8 using a more elegant technique (Subsection 4.1) that applies to other families of graphs. Then, in Subsection 4.2, we will obtain basic enumerative results for . Finally, in Subsection 4.3, we will study the random variable when is a random graph with distribution , . As it will be seen, if , then concentrates tightly around its mean, and this mean is asymptotically equal to , where . This will permit us to obtain, for the case of random graphs, new bounds for the volumes of stable polytopes, and a very strong concentration result for the entropy of a graph, both of which will hold .
4.1. A useful technique.
We start with two simple observations that remained from the theory of Section 3.
Firstly, note that for a general graph , finding is equivalent to finding the chain polytope of maximal volume contained in , hence:
Observation 4.1.
For a simple graph , we have:
Also, directly from Theorem 3.7 we can say the following:
Observation 4.2.
Let and be partial orders on the same ground set, and suppose that the comparability graph of contains as a subgraph the comparability graph of . Then, and moreover, if the containment of graphs is proper, then .

Note that every acyclic orientation of induces a partial order on whose comparability graph contains (as a subgraph) the comparability graph of a poset given by an almost bipartite orientation, and this containment is proper if is not almost bipartite. By the symmetry of , then all of the almost bipartite orientations are equivalent.
Note to proof: The same technique allows us to obtain results for other restrictive families of graphs, like odd cycles with isomorphic trees similarly attached to every element of the cycle or, perhaps more importantly, oddanticycles, but we do not pursue this here.
∎
4.2. General bounds for the main statistic.
Let us now turn our attention to the general enumeration problem. Firstly, we need to dwell on the case of comparability graphs, from where we will jump easily to general graphs.
Theorem 4.3.
Let be a comparability graph, and further let . For , let be the induced subgraph of on vertex set . Then,
where denotes the symmetric group on and denotes the chromatic number of the graph.

Let us first fix a perfect order of the vertices of , i.g. can be a linear extension of a partial order on whose comparability graph is . Let be an induced subgraph of with vertex set and edge set , let be the restriction of to , and let be the partial order of given by labeling every with and orienting accordingly. Using the colors of the optimal coloring of given by , we can find mutually disjoint antichains of that cover , so by Corollary 3.11 we obtain that
(4.1) Now, we note that each with is also induced by the respective restriction of to , and that the comparability of is , and then each of the terms on the right hand side can be expanded similarly. Starting from above and noting the fact that for this case, we can expand the terms of 4.1 exhaustively to obtain the desired expression.
∎
Corollary 4.4.
Let be any graph on vertices with chromatic number . Then .

We can follow the proof of Theorem 4.3. This time, starting from , will be a poset on given by a minimal coloring of , i.e. we color using a minimal number of totally ordered colors and orient accordingly. Then, and we can expand the right hand side of 4.1, but noting that can only be guaranteed to be partitioned into at most antichains, and that the chromatic number of a graph is at most the number of vertices of that graph.
∎
Noting that the number of cutsets is a least in most cases, a similar argument to that of Theorem 4.3 implies:
Observation 4.5.
Let be a connected graph. Then:
Example 4.6.
If is the odd cycle on vertices, then for each we have , the th Euler number, and , so . As goes to infinity, then .
Other upper bounds can be obtained from rather different considerations.
Proposition 4.7.
Let be a simple graph on vertices. Then, is at most equal to the number of acyclic orientations of the edges of , the complement of . Equality is attained if and only if is a complete partite graph, .

Let be the set of edges of , so that .
The inequality holds since two different linear extensions (understood as labelings of with the totally ordered set ) of the same acyclic orientation of induce different acyclic orientations of : As both induce the same orientation of , they must induce different orientations of .
To prove the equality statement, first note that if is not a complete partite graph, then there exist edges such that . Suppose that is a directed edge in an optimal orientation of . Then, if we label the vertices of with the (totally ordered) set in such a way that comparing vertices according to their labels, our labeling induces an acyclic orientation of which cannot be obtained from a linear extension of . Hence, is strictly less than the number of acyclic orientations of .
If is a complete partite graph, then suppose that there exists an acyclic orientation of that cannot be obtained from a linear extension of , where is any optimal orientation of . Then, in the union of the (directed) edges in both and , we can find a directed cycle that uses at least one (directed) edge from both and . Take one such directed cycle with minimal number of (directed) edges. As is a comparability graph, then is transitive, and so the directed cycle has the form , where is a directed edge in , is a directed path in , and . Let , and let be the first directed edge in along the directed cycle. Since is complete partite, then because . Since is transitive, must be a directed edge in . However, this contradicts the minimality of the directed cycle.
∎
4.3. Random graphs.
Changing the scope towards probabilistic models of graphs, specifically to , we will obtain a tight concentration result for the random variable with . The central idea of the argument will be to choose an acyclic orientation of a graph from a minimal proper coloring of its vertices. We expect this orientation to be nearly optimal.
Let us first recall two remarkable results that will be essential in our proof. The first one is a wellknown result of Bollobás, later improved on by McDiarmid:
Theorem 4.8 (Bollobás (1988),McDiarmid (1990)).
Let with , and define . Then:
where is the chromatic number of .
To state the second result, we first need to introduce the concept of entropy of a convex corner, originally defined in Csiszár et al. (1990). We only present here the statement for the case of stable polytopes of graphs.
Definition 4.9.
Let be a simple graph, and let be the stable polytope of . Then, the entropy of is the quantity:
In 1995, Kahn and Kim proved certain bounds for the volumes of convex corners in terms of their entropies. One of them, when applied to stable polytopes, reads as follows:
Theorem 4.10 (Kahn and Kim (1995)).
Let be a simple graph, and let be the stable polytope of . Then:
Equipped now with these background results, the following is true:
Theorem 4.11.
Let with , , and write . Then:
Also, .

Let tend to infinity. Consider the chromatic number of the graph , and color properly using colors, say with color partition . Then . By Theorem 4.8, we know that a.s., so:
(4.2) We remark here that inequality 4.2 gives a slightly better bound than the one obtained directly from Corollary 4.4.
Now, the function satisfies the edge Lipschitz condition in the edge exposure martingale since addition of a single edge to can alter by a factor of at most , so we can apply Azuma’s inequality to obtain:
Combining these two results, we see that:
and moreover, that a.s. holds.
The second necessary inequality comes, firstly, from using Observation 4.1, so that , and then from a direct application of Theorem 4.10. We obtain that . Now, we further observe that for , we have , and then:
A classic result of Grimmett and McDiarmid (1975) states that holds a.s., where . Hence, a.s., , and then