Extremal edge polytopes

The"edge polytope"of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of k-neighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.


Introduction
The main object of our investigation is a special class of 0/1-polytopes (cf. [31]): The edge polytope P(G) of a graph G on the vertex set {1, 2, . . . , n} is the polytope generated by all vectors e i + e j such that i is adjacent to j, where e i and e j stand for the ith and jth unit vectors of R n . For example, the edge polytopes of trees are simplices, while the edge polytope of the complete graph K n is the second hypersimplex ∆ n−1 (2). Thus the edge polytopes are the subpolytopes of the second hypersimplex. A study of edge polytopes of general graphs was initiated by Ohsugi & Hibi [25] and Villarreal [30], who both provided the half-space description of these polytopes (cf. Theorem 19). Dupont & Villarreal [8] have recently connected this to the setting Rees Algebras in combinatorial commutative algebra. For further discussions of edge polytopes, see [12], [22], [24], and [26].
In this paper we demonstrate that edge polytopes form a rich family of 0/1polytopes with interesting random and extremal properties. In particular, we obtain edge polytopes with an exponential number of facets (see Theorem 22) and k-neighborly 0/1-polytopes with more than linearly many vertices for any k ≥ 2 (Corollary 17). On the other hand we will show that edge polytopes can be described and analyzed in terms of parameters of the graphs they are based on (and thus are not as intractable as the same problems for general 0/1-polytopes [16] [31]). Thus we obtain structural overview concerning three main topics: (1) a description of the low-dimensional faces of the polytope P(G); (2) non-linear relations between the components of the f -vector of P(G); (3) the asymptotics of the maximal number of facets of d-dimensional edge polytopes for large d. Here are some remarks connected to this. To describe all low-dimensional faces of P(G) we only need to consider "small" induced subgraphs of G. The second topic is closely related to the problem of finding minimal density of a fixed bipartite graph in a dense graph. Concerning the third topic Gatzouras et al. [10], improving on a breakthrough by Bárány & Pór [5], showed that there are random 0/1-polytopes in R d with as many as cd log 2 d d/2 facets (or more), where c > 0 is an absolute constant. The situation for d-dimensional random edge polytopes, however, where we have only a polynomial (quadratic) number of potential vertices, turns out to be quite different from that of general 0/1-polytopes.
The paper is divided into five sections. In the next section we introduce the object of our investigation and determine the dimension of an arbitrary edge polytope. A criterion for determining faces of edge polytopes is provided.
In Section 3 we compute the number of edges of P(G) in terms of the number of vertices of P(G), the number of 4-cycles and the number of 4-cliques in G. The function g(n) := max {f 1 (P(G)) : G has n vertices} is in Theorem 12 shown to be of order Θ(n 4 ). The lower bound for this is provided by random edge polytopes.
In Section 4 we characterize k-neighborly edge polytopes for k ≥ 2. We then obtain a tight upper bound on the number of vertices of these edge polytopes, by counting various types of walks in the graph. All edge polytopes which attain these bounds are pseudo-random in some sense.
In Section 5 we use results of Ohsugi & Hibi [25] to show that a d-dimensional edge polytope has at most 2 d + d facets. Inspired by Moon & Moser [23], we provide a construction for d-dimensional edge polytopes with roughly 4 d/3 facets.

Preliminaries
All graphs in this paper are finite, undirected, with no loops, no multiple edges and no isolated vertices. The vertex and edge sets of a graph G are denoted by V (G) and E(G). We write |G| for the number of vertices of G, and e(G) for the number of edges. We write G[S] for the subgraph of G induced by a set S ⊆ V (G). Given two sets S, T ⊆ V (G), not necessarily disjoint, we write e G (S, T ) for the number of ordered pairs (s, t) with s ∈ S, t ∈ T and st ∈ E(G). Given a non-empty subset The main object of our consideration is a special class of 0/1-polytopes.
Definition 1. Let G be a graph on the vertex set {1, 2, . . . , n} := [n]. The edge polytope P(G) of G is the convex hull of all vectors e i + e j such that i is adjacent to j, where e i and e j denote the ith and jth unit vectors of R n .
Thus the edge polytopes of n-vertex graphs correspond to the subpolytopes of the second hypersimplex of order n. Example 2. The second hypersimplex of order n is defined as It is the edge polytope of the complete graph K n . The second hypersimplex ∆ n−1 (2) has dimension n − 1 (it is contained in {x ∈ R n : x 1 + · · · + x n = 2}), n 2 vertices, and 2n facets if n ≥ 4. For example, ∆ 3 (2) ⊆ R 4 is affinely equivalent to the regular octahedron.
Example 3. Let C n be the cycle (1, 2, . . . , n−1, n). If n is odd, then e 1 + e 2 , e 2 + e 3 , . . . , e n−1 +e n , e n +e 1 are affinely independent, so the edge polytope P(C n ) is an (n−1)-simplex. If n is even, then the edge polytope P(C n ) has dimension n−2: It is a sum of two ( n 2 −1)-simplices in {x ∈ R n : The definition of edge polytope can be rephrased in the following way. Let G be a graph. The incidence matrix of G is the matrix The edge polytope P(G) of G is precisely the convex hull of the column vectors of the matrix A. Hence P(G) can be obtained by taking the intersection between the cone cone (A) and the hyperplane {x ∈ R n : x 1 + · · · + x n = 2}. Thus On the other hand, by [ This result enables us to obtain a quadratic upper bound on the number of vertices of the polytope P(G) in terms of its dimension. For further investigation, we need a criterion for determining faces of an edge polytope. From now on, the symbol e ij is used to denote the vector e i + e j .
We will use the following simple criterion. For edge polytopes, this criterion can be reformulated as follows. So P(G[U]) is a face of P(G), by Lemma 6.

Graphs of edge polytopes
The following simple result of Ohsugi and Hibi identifies the edges of P(G).  Using Lemma 8, we can compute the number of edges of P(G). For this, let c 4 (G) and k 4 (G) be the numbers of copies of C 4 and of K 4 in G, respectively. As usual in polytope theory, we denote by f k the number of k-dimensional faces of a polytope. Proposition 9. If P(G) be the edge polytope of a simple graph G, then f 0 (P(G)) = e(G) and Proof. Let r 4 (G) denote the number of pairs of disjoint edges of G which are contained in some 4-cycle of G. Lemma 8 shows that f 1 = f 0 2 − r 4 (G). Thus Proposition 9 will be established if r 4 (G) − 2c 4 (G) + 3k 4 (G) = 0 holds. Now so we can assume from the beginning that |G| = 4. The rest is left to the reader.
We next give a sharp lower bound for f 1 (P(G)) in terms of f 0 (P(G)).
Theorem 10. If f 0 and f 1 denote the number of vertices resp. edges of the edge polytope P(G), then f Equality holds if and only if G is a complete bipartite graph with equal size parts.
Proof. Without restriction we can assume that G is a connected graph on n vertices. Letd be the average degree of G. Since G is connected, it has at most one bipartite component. Thus we have d := dim (P(G)) = n−c 0 (G)−1 ≥ n−2. Now we count the following set S in two ways: S is the set of incidence pairs (v, e) where v is a vertex of P(G), and e is an edge of P(G). Here where u is adjacent to v and to w, with v = w. Since {u, v} and {u, w} are two edges of G which have one common node, they form an edge of P(G), by Lemma 8. Therefore we have Combining this inequality with the previous one, we get It is easy to check that the equality holds if and only if G ∼ = K n/2,n/2 .
Let P(G) be the edge polytope of a graph G and f 1 (P(G)) be its number of edges. How large can f 1 (P(G)) be if the number of vertices of G is fixed? Formally, we want to know the asymptotic behaviour of the number of edges of the polytope in terms of the number n of vertices of the graph, For that we use the following bound on the number of copies of a fixed complete bipartite subgraph of a graph of given density.
Lemma 11 (Alon [2, Corollary 2.1]). For every fixed ε > 0, any two fixed integers s ≥ t ≥ 1, and for any graph G with n vertices and at least εn 2 edges, the number of subgraphs of G isomorphic to K s,t is at least where the o(1) terms tend to 0 as n tends to infinity.
It is worth noting that the assertions of the above lemma are tight, as shown by the random graph G(n, 2ε) on n labeled vertices in which each pair of vertices is an edge with probability 2ε.
And now, as promised, we provide bounds for the function g(n).
Theorem 12. For every integer n ≥ 6, the function g(n) = max {f 1 (P(G)) : |G| = n} satisfies (i) Lower bound For simplicity of notation, we write G instead of G(n, p). Define  Figure 1). From this, we get p 2 = p 2 (1 − p 2 ) 2 . Thus by linearity of expectation From this it follows that g(n) ≥ n 4 /54.
(ii) Upper bound Let G be an arbitrary graph with n vertices and ρ n 2 2 edges (0 < ρ < 1). Since the cycle of length 4 is isomorphic to the complete bipartite graph K 2,2 , Lemma 11 shows that c 4 (G) ≥ ( 1 8 + o(1))ρ 4 n 4 . Furthermore, as each clique of size 4 contains exactly three cycles of length 4, we have c 4 (G) ≥ 3k 4 (G). Therefore, the number of edges of P(G) is at most Remark. The upper bound in Theorem 12 is not tight: For any pair of graphs F and G, let N(F, G) denote the number of labeled copies of F in G. A sequence According to Chung, Graham, and Wilson [7] this happens iff N( If the upper bound of Theorem 12 is tight, then we can find a sequence (G n ) of graphs such that e(G n ) = ( 1 It remains to be explored whether the upper bound can be improved by formalizing the subgraph count via flag algebras as in [14].

Neighborly edge polytopes
Here we provide a forbidden subgraph characterization for k-neighborly edge polytopes, and then determine the maximal number of vertices of such polytopes. For this we first prepare some notation.
Given a family F of graphs, a graph G is F -free if it contains no copy of a graph in F as a subgraph. The Turán number ex (n, F ) is the maximal number of edges in an F -free graph on n vertices. The Zarankiewicz number z(n, F ) is the maximal number of edges in an F -free bipartite graph on n vertices.
A polytope P is k-neighborly if every subset of at most k of its vertices defines a face of P . Thus every polytope is 1-neighborly, and a polytope is 2-neighborly if and only if its graph is complete. Except for simplices, no d-dimensional polytope is more than ⌊ d 2 ⌋-neighborly. For k ≥ 2, let F k be a family of graphs on at most 2k vertices consisting of • even cycles, • graphs obtained by joining two odd cycles by a path. For example, F 2 = {C 4 } and F 3 = {C 4 , F 2 , 2K 3 + e, C 6 } (see Figure 2).  Theorem 13. For k ≥ 2 the edge polytope P(G) of a graph G with at least k edges is k-neighborly if and only if G is F k -free.
Before proving Theorem 13 we state a consequence. Let C even 2k denotes the family of all even cycles of lengths at most 2k. As a bipartite graph is F k -free if and only if it is C even 2k -free, Theorem 13 implies the following result. Corollary 14. For k ≥ 2 the edge polytope P(G) of a bipartite graph G with at least k edges is k-neighborly if and only if G is C even 2k -free. For the proof of Theorem 13, we use the following lemma, which is a straight- Proof. The "if" part is obvious. For the "only if" part we observe that if a graph contains two odd cycles that intersect in more than one vertex, then it also has an even cycle.
We are now ready to prove Theorem 13.
Proof of Theorem 13. Assume that G is F k -free. Let {i 1 , j 1 }, . . . , {i k , j k } be k different edges of G. Set U = {i 1 , j 1 , . . . , i k , j k }, then |U| ≤ 2k. By Lemma 16 and Lemma 15, P(G[U]) is a simplex. Therefore, conv{e i 1 j 1 , . . . , e i k j k } is a face of P(G) by Lemma 7. From this it follows that P(G) is k-neighborly.
Assume that P(G) is k-neighborly.
Theorem 13 can be used to obtain the following upper bound on the number of vertices of k-neighborly edge polytopes.
Corollary 17. Let k ≥ 2 be a fixed integer. Then a k-neighborly edge polytope of an n-vertex graph has at most 1 2 n 1+1/k + k−1 2k + o(1) n vertices. Furthermore, for each k ∈ {2, 3, 5} there are infinitely many positive integers n for which there is an n-vertex graph G n whose edge polytope is k-neighborly with at least 1 2 n 1+1/k + k−1 2k n − n 1−1/k vertices.
Proof. In the following we will use results on Turán numbers and on pseudorandomness, Theorems 25 and 26, which are presented in the appendix. Since P(G) is k-neighborly, Theorem 13 implies that the graph G is F k -free. Under this condition we will show that, as n goes to infinity, e(G) ≤ 1 2 n 1+1/k + ( k−1 2k + o(1))n. We may assume that e(G) ≥ 1 2 n 1+1/k . Now let T be the set of all odd cycles of length at most k in G, and let U be the set of all vertices in G which is contained in some element of T . Because G is F k -free, elements in T are pairwise disjoint, and consequently |T | ≤ n/3. By removing one edge from each element of T we get a (C even 2k ∪ C k )-free graph H with e(H) ≥ 1 2 n 1+1/k − 1 3 n, where C k denotes the family of all cycles of lengths at most k. Since H is (C even 2k ∪ C k )free, Theorem 25 tells us that e(H) ≤ 1 2 n 1+1/k + k−1 2k n + n 1−1/k . Therefore, H has average degree d H ∼ n 1/k . Theorem 26 can be applied showing that Since H is obtained from G by deleting o(n 1+1/k ) edges, a similar formula holds for G, namely e G (S, T ) = n −1+1/k |S||T | + o(n 1+1/k ) for every S, T ⊆ V (G).
We also have the following lower bound on the maximal number of vertices of a k-neighborly edge polytope.
Corollary 18. Let k ≥ 2 be a fixed integer. Then the following holds.
(i) A k-neighborly edge polytope of an n-vertex graph has at most 1 2 n 1+1/k +O(n) vertices. Moreover, there are graphs G n on n vertices such that P(G n ) is a k-neighborly polytope with Ω(n 1+2/3k ) vertices.
(ii) A k-neighborly edge polytope of an n-vertex bipartite graph has at most ( 1 2 ) 1+1/k n 1+1/k + O(n) vertices. Moreover, for k ∈ {2, 3, 5} this bound is tight up to the linear term for infinitely many n.
(ii) If the edge polytope P(G) of a bipartite graph G is k-neighborly, then the graph is C even 2k -free by Theorem 13. It follows from [13] that the polytope P(G) has at most z(n, C even 2k ) ≤ 1 2 1+1/k n 1+1/k + O(n) vertices.

Edge polytopes with many facets
In this section we study the maximal number of facets of a d-dimensional edge polytope. Here we only deal with edge polytopes of connected graphs; all results can easily be extended to the general case.
We use some terminology of Ohsugi & Hibi [25], as follows. Let G be a connected graph on the vertex set [n].
If A is independent in G, then the bipartite graph induced by A in G is defined to be the graph having the vertex set A ∪ N(A) and consisting of all edges {i, j} of G with i ∈ A and j ∈ N(A). This graph will be denoted by G [A, N(A)]. When G is non-bipartite, we say that a subset and by H A the hyperplane If i ∈ [n], then we write H + i for the closed half-space , and H i for the hyperplane Let d ∈ N. We write f (d) for the maximal number of facets of P(G), where G ranges over all connected graph such that dim (P(G)) = d.
Proof. Let G be a connected graph on [n] with dim (P(G)) = d ≥ 3. Denote by f d−1 the number of facets of P(G). It is sufficient to prove that f d−1 ≤ 2 d + d.
We distinguish two cases. If G is bipartite, then d = n − 2. Let V 1 ∪ V 2 be the partition of V (G). We can assume that |V 1 | ≤ |V 2 |. Applying Theorem 19, we get If G is non-bipartite, then d = n−1. We denote by F the family of independent sets in G.  Proof. Without restriction we can assume that d = 3k. Let G be the windmill graph Wd (4, k) on the vertex set [3k + 1] with the edge set As G is a connected non-bipartite graph, we have dim (P(G)) = (3k + 1) − 1 = d.
We will now determine all fundamental sets in G. Observe that a non-empty subset A ⊆ [3k + 1] is independent in G if and only if (i) A = {3k + 1}, or (ii) 3k + 1 / ∈ A, and |A ∩ {3i − 2, 3i − 1, 3i}| ≤ 1 for all i = 1, . . . , k. We claim that such a set A is fundamental in G. There are two possible cases. We can see that this graph is connected and non-bipartite. Hence i is a regular vertex in G, as desired. Finally, applying Theorem 19 we conclude that the number of facets of P(G) is k ℓ=1 k ℓ 3 ℓ + 3k = 4 k + 3k − 1. Therefore, f (d) > 4 k = 4 ⌊d/3⌋ . As a consequence of these lemmas, we obtain the following bounds on f (d).
Theorem 22. For all d ≥ 3, the maximal number f (d) of facets of a d-dimensional edge polytope satisfies

Appendix: Turán numbers
Here we give a tight asymptotic upper bound on the Turán number ex(n, C even 2k ∪ C k ), where C even 2k = {C 4 , C 6 , . . . , C 2k } and C k = {C 3 , C 4 , . . . , C k }. We also show that every nearly extremal (C even 2k ∪ C k )-free graph is pseudorandom. The basic estimates for Turán numbers for even cycles are obtained by counting various types of walks in graphs: A non-returning walk of length k in G is a sequence v 0 e 0 v 1 e 1 . .
and e i = e i+1 for 0 ≤ i < k. Let ν k (G) denote the average number of nonreturning walks of length k in G. If G is a d-regular graph on n vertices then clearly ν k (G) = d(d − 1) k−1 . For irregular graphs, we have the following lower bound.
Proposition 23 (Alon, Hoory & Linial [3]). If G is a graph with minimum degree at least 2 and average degree d, The following simple result will be very useful for our investigation. It is probably well-known, but we couldn't find a reference for it.
Lemma 24. Suppose that P and Q are two different paths of length k ≥ 2 with the same endpoints. If P ∪ Q is C k -free, then P = αP ′ β and Q = αQ ′ β for some vertex-disjoint paths α, β, P ′ and Q ′ .
Proof. The lemma is obviously true for k ∈ {2, 3}. So let k > 3 and proceed by induction. Let x and y be endpoints of P and Q. We distinguish three cases. Case 1: N P (y) = N Q (y) := v. Let P 1 and Q 1 be the subpaths of P and Q from x to v, respectively. Then they are two different paths of length k − 1 from x to v, and their union is C k -free. By induction, P 1 = αP ′ β and Q 1 = αQ ′ β for some vertex-disjoint paths α, β, P ′ , Q ′ with x ∈ α and v ∈ β. Hence P = αP ′ βy and Q = αQ ′ βy. Case 2: N P (y) = N Q (y) and N P (y) ∈ Q. In this case, N P (y)Qy is a cycle of length at most k. This contradicts the assumption that P ∪ Q is C k -free. Case 3: N P (x), N P (y) / ∈ Q and N Q (x), N Q (y) / ∈ P . We identify x, N P (x) and N Q (y) (resp. y, N P (y) and N Q (y)) as a new vertex x ′ (resp. y ′ ). Let P 2 and Q 2 be the new paths corresponding to P and Q. Then they are two different paths of length k − 2 from x ′ to y ′ . Since P ∪ Q is C k -free, P 2 ∪Q 2 is C k−2 -free. By induction P 2 = α ′ P ′′ β ′ and Q 2 = α ′ Q ′′ β ′ for some disjoint paths α ′ , β ′ , P ′′ and Q ′′ . We claim that x ′ is the only vertex of α ′ . Otherwise, α ′ = x ′ uα ′′ for some vertex u and path α ′′ . In this case, the vertices x, N P (x), u and N Q (x) form a cycle of length 4 in P ∪ Q, which contradicts the fact that P ∪ Q is C k -free. Similarly, y ′ is the only vertex of β ′ . From this it follow that P and Q are two internally vertex-disjoint paths. Consequently, they have the desired structures.
Proof of Theorem 25. As discussed above, it is enough to prove the theorem for k ≥ 3. Let G be a minimal counterexample to the theorem. Then G has minimum degree at least 2 and average degree d > n 1/k + k−1 k + 2n −1/k . We denote by P k the family of paths of length k in G. Since G is C k -free, every non-returning walk of length k is nothing but a path of length k. It now follows from Proposition 23 that |P k | ≥ nd(d − 1) k−1 > n 2 . By Lemma 24, for any ordered pair of vertices u, v there is at most one path of length k from u to v. Thus, the number of paths of length k is at most n 2 , a contradiction.
Let α ∈ N. Set q = 2 2α+1 if k = 3, and q = 3 2α+1 if k = 5. Lazebnik et al. [21] constructed a (C even 2k ∪ C k )-free graph G on n = q k + . . . + q + 1 vertices with 1 2 (q + 1)(q k + . . . + q + 1) − (q ⌊ k+1 2 ⌋ + 1) edges. We can verify that e(G) ≥ 1 2 n 1+1/k + k−1 2k n − n 1−1/k for large n. Another key ingredient in the proof of Corollary 17 is the notion of pseudorandomness. We refer the reader to Krivelevich & Sudakov [18] for a survey. The following result expresses the pseudorandomness property of a nearly extremal (C even 2k ∪ C k )-free graph: For any two large sets the number of ordered edges between them is close to what one would expect in a random graph of the same edge density.
Theorem 26. Let k ≥ 2 be a fixed integer. Suppose G is a (C even 2k ∪ C k )-free graph on n vertices with average degree d ∼ n 1/k . Then e G (S, T ) = d n |S||T | + o(n 1+1/k ) for any S, T ⊆ V (G).
Sketch. Our proof follows the lines of a remark of Keevash et al. [15,Section 9]. We just sketch the argument, and refer to [15] for the omitted details. Suppose that G has eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n . Let w • 2k+2 (G) denote the number of closed walks of length 2k + 2 in G divided by n. Since G is (C even 2k ∪ C k )free, Lemma 24 shows that there is at most one path of length k between any pair of vertices of G. Using this property we control the maximum degree as ∆ < (1 + ε)d by deleting o(n 1+1/k ) edges. Then the argument of [15,Lemma 3.4] shows w • 2k+2 (G) < (1+o(1))n∆ 2 ; the only difference is that there are n−1 choices for u rather than n/2 + o(n). On the other hand, w • 2k+2 (G) = 1 n λ 2k+2 i has a contribution of d 2k+2 /n ∼ nd 2 from the first eigenvalue, so the other eigenvalues are o(d) as ε → 0. The pseudorandomness property now follows from the nonbipartite version of [15,Lemma 5.4], which is provided in [18,Section 2.4].
Verstraëte [15]. We would like to thank an anonymous referee for very instructive comments and suggestions.