Detecting cliques in CONGEST networks

The problem of detecting network structures plays a central role in distributed computing. One of the fundamental problems studied in this area is to determine whether for a given graph $H$, the input network contains a subgraph isomorphic to $H$ or not. We investigate this problem for $H$ being a clique $K_{l}$ in the classical distributed CONGEST model, where the communication topology is the same as the topology of the underlying network, and with limited communication bandwidth on the links. Our first and main result is a lower bound, showing that detecting $K_{l}$ requires $\Omega(\sqrt{n} / b)$ communication rounds, for every $4 \le l \le \sqrt{n}$, and $\Omega(n / (l b))$ rounds for every $l \ge \sqrt{n}$, where $b$ is the bandwidth of the communication links. This result is obtained by using a reduction to the set disjointness problem in the framework of two-party communication complexity. We complement our lower bound with a two-party communication protocol for listing all cliques in the input graph, which up to constant factors communicates the same number of bits as our lower bound for $K_4$ detection. This demonstrates that our lower bound cannot be improved using the two-party communication framework.


Introduction
We study the problem of detecting network structures in a distributed environment, which is a fundamental problem in modern computing.Our focus is on the subgraph detection problem, in which for a given graph H, one wants to determine whether the network graph G contains a subgraph isomorphic to H or not.We investigate this problem for H being a clique K for ≥ 4.
The nowadays classical distributed CONGEST model (see, e.g., [17]) is a variant of the classical LOCAL model of distributed computation (where in each round network nodes can send through all incident links messages of unrestricted size) with limited communication bandwidth.The distributed system is represented as a network (undirected graph) G = (V, E) with n = |V | nodes, where each node v ∈ V executes the same algorithm in synchronous rounds, and the nodes collaborate to solve a graph problem with input G.Each node is assumed to have a unique identifier from {0, . . ., poly(n)}.In any single round, all nodes can: (i) perform an unlimited amount of local computation, (ii) send a possibly different b-bit message to each of their neighbors, and (iii) receive all messages sent to them.
We measure the complexity of an algorithms by the number of synchronous rounds required.
In accordance with the standard terminology in the literature, we assume b = O(log n); we note though that our analysis generalizes to other settings of b in a straightforward manner.(We note that in our lower bound for detecting K 4 and K in Section 2, to ensure full generality of presentation, we will make the analysis parametrized by the message size b, in which case we will refer to such model of distributed computation as CONGEST b , the CONGEST model with messages of size b.) Our goal is, for a given network G = (V, E) and ≥ 4, to solve the subgraph detection problem for a clique K , that is, to design an algorithm in the CONGEST model such that (i) if G contains a copy of K , then with probability at least 2  3 at least one node outputs 1, and (ii) if G does not contain any copy of K , then with probability at least 2  3 no node outputs 1.The subgraph detection problem is a local problem: it can be solved efficiently solely on the basis of local information.In particular, in the CONGEST model, the problem of finding K in a graph can be trivially solved in O(n) rounds, or in fact, in O(max u∈V deg G (u)) rounds, where deg G (u) denotes the degree of node u in G. Indeed, if each node sends its entire neighborhood to all its neighbors, then afterwards, each node will be aware of all its neighbors and of their neighbors.Therefore, in particular, each node will be able to detect all cliques it belongs to.Since for each node u, the task of sending its entire neighborhood to all its neighbors can be performed in O(deg G (u)) rounds in the CONGEST model, the total number of rounds for the entire network is O(max u∈V deg G (u)) = O(n) rounds.In view of this simple observation, the main challenge in the clique K detection problem is whether this task can be performed in a sublinear number of rounds.

Our results
In this paper, we give the first non-trivial lower bound for the complexity of detecting a clique K in the CON-GEST b model, for ≥ 4. In Theorem 5, we prove that every algorithm in the CONGEST b model that with probability at least 2 3 detects K , for ≥ 4 and = O( then Ω(n/( b)) rounds are required.We are not aware of any other non-trivial (super-constant) lower bound for this problem in the CONGEST b model.
We complement our lower bound with a two-party communication protocol for listing all cliques in the input graph (see Theorem 10), which up to constant factors communicates the same number of bits as our lower bound for K 4 detection.This demonstrates that our lower bound is essentially tight in this framework, and cannot be improved using the two-party communication approach.

Techniques: Framework of two-party communication complexity
Our main results, the lower bound of clique detection in Theorem 5 and the upper bound in Theorem 10, rely on the two-party communication complexity framework and the use of a tight lower bound for the set disjointness problem in this framework.
We consider the classical two-party communication complexity setting (cf.[15]) in which two players, Alice and Bob, each have some private input X and Y .The players' goal is to compute a joint function f(X, Y ), and the complexity measure used is the number of bits Alice and Bob must exchange to compute f(X, Y ).In the two-party communication problem of set disjointness, Alice's input is X ∈ {0, 1} n and Bob holds Y ∈ {0, 1} n , and their goal is to compute DISJ n (X, Y ) := n i=1 X i ∧ Y i .In a seminal work, Kalyanasundaram and Schnitger [13] showed that in any randomized communication protocol, the players must exchange Ω(n) bits to solve the set disjointness problem with constant success probability.

Theorem 1 ([13]
).The randomized two-party communication complexity of set disjointness is Ω(n).That is, for any constant p > 0, any randomized two-party communication protocol that computes DISJ n (X, Y ) with probability at least p, has two-party communication complexity Ω(n).
Our main result, the lower bound for detecting K in the CONGEST model, relies on a reduction from the two-party communication problem set disjointness.The two-party communication framework, and, in particular, the two-party set disjointness problem, have been frequently used in the past to construct lower bounds for the CONGEST model, see, e.g., [4,6,8,10,14].A typical approach relies on a construction of a special graph G = (V, E) with some fixed edges and some edges depending on the input of Alice and Bob.One partitions the nodes of G into two disjoint sets V A and V B .Let C be the (V A , V B )-cut, that is, the set of edges in G with one endpoint in V A and one endpoint in V B .Let E A be the edge set of G[V A ] (subset of E on vertex set V A ) and E B be the edge set of G[V B ].We consider a scenario where Alice's input is represented by the subgraph (We denote this way of distributing the vertex and edge sets as the vertex partition model.)In order to learn any information about the structure of G[A]\C and G[B]\C, and hence about the input of the other player, Alice and Bob must communicate through the edges of the cut C. Therefore, in order to obtain a lower bound for a problem in the CONGEST b model, one wants to construct G to ensure that it has some property (in our case, contains a copy of K ) if and only if the corresponding instance of set disjointness is such that DISJ n (X, Y ) = 1, and in order to determine the required property, one has to communicate a large part of (essentially the entire graph) G[A] through C. With this approach, if the cut C has size |C|, and the private inputs of Alice and Bob (edges in G[A] \ C or G[B] \ C) are of size s, one can apply Theorem 1 to argue that the round complexity of any distributed algorithm in the CONGEST b model for a given problem is Ω( s |C|•b ).The central challenge is to ensure that for the encoded set disjointness instance of size s and the cut of size |C|, the ratio s |C| is as large as possible.For example, Drucker et al. [6] incorporated a similar approach to obtain a lower bound for the subgraph detection problem in a broadcast variant of the CONGEST b model (in fact, even for a (stronger) broadcast variant of the CONGESTED CLIQUE model), where nodes are required to send the same message through all their incident edges.The lower bound construction requires sending Ω(n 2 ) bits through the cut of size O(n 2 ), but the fact that in the broadcast variant of the CONGEST b model every node is required to send the same message via all incident edges, at most O(n b) bits can be transmitted through the cut, yielding a lower bound of Ω( n b ).(In particular, for the broadcast variant of the CONGEST b model, Drucker et al. [6,Theorem 15] proved that detecting a clique K , ≥ 4, requires Ω( n b ) rounds.)Note however that in the (non-broadcast) CONGEST b model, this construction does not give any not-trivial bound, since s |C| = O(1).Our main building block for our lower bound is the construction of (Ω(n 2 ), O(n 3/2 ))-lower-bound graphs (in Section 3.1) that can be used to encode a set disjointness instance of size s = Ω(n 2 ) such that the cut is of size |C| = O(n 3/2 ).By incorporating these bounds in the framework described above, this construction leads to the first non-trivial lower bound of Ω( for the subgraph detection problem in the CONGEST b model for the clique K 4 .This construction can also be extended to detect larger cliques, yielding the lower bound of Ω( Since these are the first superconstant lower bounds for detecting a clique in the CONGEST model and since the best upper bound for these problems is still O(n), the next goal is to understand to what extent these bounds could be improved and whether the existing approach could be used for that task.Do we need Ω( in the CONGEST b model, or maybe we need as many as a linear number of rounds?While we do not know the answer to this question, and in fact, this question is the main open problem left by this paper, we can prove that any better lower bound would require a significantly different approach, going beyond the two-party communication framework in the vertex partition model. Indeed, let us consider the vertex partition model in the two-party communication framework, as defined above.The input consists of an undirected G = (V, E) with an arbitrary vertex partition V = V A ∪ V B .We consider a scenario where Alice is given the subgraph The arguments in our construction of lower-bound graphs in Theorem 9 imply that for some inputs, any two-party communication protocol in the vertex partition model for the problem of listing all cliques in a given graph with n nodes requires communication of Ω( √ n |C|) bits between Alice and Bob.We will prove in Section 4 (Theorem 10) that this lower bound is asymptotically tight in the two-party communication framework in the vertex partition model.We show that there is a two-party communication protocol in the vertex partition model for listing all cliques that uses O( √ n |C|) communication rounds, where C is the set of shared edges between Alice and Bob.This shows that we cannot obtain stronger lower bounds for the K -detection problem, for = O( √ n), in the CONGEST model using the two-party communication framework in the vertex partition model.

Related works
As a fundamental primitive, subgraph detection and listing in the CONGEST model has been recently receiving attention from multiple authors, focusing mainly on randomized complexity.However, despite major efforts, for the CONGEST model, relatively little is known about the complexity of the subgraph detection problem.
Rather surprisingly, prior to our work, no non-trivial results about the complexity of clique K ( ≥ 4) detection in the CONGEST model have been known.While there is a trivial lower bound of a constant number of rounds, and as we mentioned earlier, one can easily solve the problem in O(n) rounds in the CONGEST model, no sublinear upper bounds nor superconstant lower bounds have been known.
In a recent breakthrough in this area, Izumi and Le Gall [11] raised some hopes that maybe these problems could be solved in a sublinear number of rounds in the CONGEST model.They considered the subgraph detection problem for the smallest interesting subgraph H, the triangle K 3 , and presented a very clever algorithm that detects a triangle in O(n 2/3 ) rounds.Further, Izumi and Le Gall [11] also showed that the related problem of finding all triangles (triangle listing) can be solved in O(n 3/4 ) rounds.There is no non-trivial lower bound for the triangle detection problem, though it is known (cf.[11,16]) that the more complex triangle listing problem requires Ω(n 1/3 / log n) rounds, even in the CONGESTED CLIQUE model.It can also be shown that the problem of listing all triangles such that each node v learns all triangles that it is part of significantly harder than the general triangle listing problem and requires Ω(n/ log n) rounds [11,Proposition 4.4].While rather disappointingly, we do not know how to extend any of these upper bounds to other cliques K with ≥ 4, the work of Izumi and Le Gall [11] raises hope that detecting cliques K could potentially be solved in a sublinear number of rounds.
In fact, even for K 3 , we do not even know whether detecting a triangle K 3 can be solved in a polylogarithmic or even a constant number of rounds in the CONGEST model (the lower bound of Ω(n 1/3 / log n) rounds in the CONGESTED CLIQUE model (cf.[11,16]) holds only for a more complex problem of detecting all triangles).Even et al. [7] noted that the problem is significantly simpler for trees, and designed a randomized color-coding algorithm that detects any constant-size tree on nodes in O( ) rounds.
As for lower bounds for the subgraph detection problem in the CONGEST model, until very recently, the only hardness results known in the literature have been for cycles.For any fixed ≥ 4, there is a polynomial lower bound for detecting the -cycle C in the CONGEST model [6], where it has been shown that detecting C requires (ex(n, C )/ log n) rounds, where ex(n, C ) is the Turán number for cycles, that is, the largest possible number of edges in a C -free graph over n vertices.In particular, for odd-length cycles (of length 5 or more), the lower bound of [6] is Ω(n/ log n), and it is Ω( √ n/ log n) for = 4. Very recently, Korhonen and Rybicki [14] improved the lower bound for all even-length cycles to Ω( √ n/ log n).Further, Gonen and Oshman [10] extended these lower bounds for C -freeness to some related classes of graphs, though still with some cyclic underlying structure.(As mentioned above, we note that Drucker et al. [6] presented lower bounds for other graphs, but this was in a broadcast variant of the CONGESTED CLIQUE model, where nodes are required to send the same message on all their edges.In particular, for the broadcast variant of the CONGESTED CLIQUE model, Drucker et al. [6] proved that detecting a clique K , ≥ 4, requires Ω(n/ log n) rounds.) The only lower bound for the subgraph detection problem for H significantly other than cycles, is a very recent work of Fischer et al. [8], who demonstrated that the subgraph detection problem is hard even for some subgraphs H of constant size.In particular, for any constant ≥ 2, there is a graph H with a constant number of vertices and edges such that the problem of finding H in a network of size n requires time Ω(n 2− 1 /b) in the CONGEST model, where b is the bandwidth of each communication links.
There has also been some recent research for the deterministic subgraph detection problem in the CONGEST model.For example, Drucker et al. [6] designed an O( √ n) round algorithm for C 4 detection, and Even et al. [7] and Korhonen and Rybicki [14] obtained path and tree detection algorithms requiring only a constant number of rounds.Korhonen and Rybicki [14] considered also deterministic subgraph detection (for paths, cycles, trees, pseudotrees, and on d-degenerate graphs) in the weaker broadcast CONGEST model, where nodes send the same message to all neighbors in each communication round.In the CONGESTED CLIQUE model, deterministic subgraph detection algorithms were given by Dolev et al. [5] and Censor-Hillel et al. [3].
We summarize earlier results together with our new results in Table 1.

Property testing of H-freeness
Since there have been so few positive results for the original subgraph detection problem, recently there have been some advances in a relaxation of this problem, a closely related (and significantly simpler) problem of testing subgraphs freeness in the framework of property testing for distributed computations (see, e.g., [1,7]).In the property testing setting, an algorithm has to decide, with probability at least 2 3 , if the input graph is (a) H-free (i.e., does not contain a subgraph isomorphic to H) or (b) ε-far from being H-free (that is, the goal is to distinguish whether the input graph G is H-free or one needs to modify more than ε|E(G)| edges of G to obtain a graph that is H-free); in the intermediate case, the algorithm can perform arbitrarily (see e.g., [3,7] for more details).Property testing of H-freeness in the CONGEST model has received a lot of attention lately (see, e.g., [1,2,7,8,9]).In particular, it has been shown [7] that testing H-freeness can be done in O(1/ε) round in the CONGEST model for any constant-size graph H containing an edge (x, y) such that any cycle in H contains at least one of x, y.This implies testing in O(1/ε) rounds of any cycle C k , and of any subgraph H on five (or less) vertices except K 5 .Further, for any ≥ 5, K -freeness can be tested in O((ε • |E(G)|) [7] show that testing if the input graph is T -free for a tree T on vertices can be done in O( 1+ 2 /ε ) rounds the CONGEST model.
2 Lower bound results (detecting a clique requires Ω( √ n) rounds) In this section we prove our hardness results showing that any algorithm in the CONGEST b model that detects a K with probability at least ) rounds, for every ≥ 4. Our lower bound for the complexity of detecting K in the CONGEST model relies on a reduction to the two-party communication complexity lower bound for the set disjointness problem (cf.Theorem 1 in Section 1.2), which we implement with the help of lower-bound graphs (cf.Section 2.1).

Lower-bound graphs
Our reduction to the two-party communication complexity lower bound for the set disjointness problem relies on a notion of a lower-bound graph (cf. Figure 1).2. The edge set E is the union of (not necessarily disjoint) sets E 1 , E 2 , . . ., E k such that, for every i, 3. For every i, j, Center: Graph G as in the proof of Theorem 3 obtained from the set disjointness instance with X = (1, 0, 0, 1) and Y = (0, 1, 1, 1).Graph G contains a K 4 if and only if the set disjointness instance evaluates to 1. Right: The highlighted edges form a K 4 .

Using lower-bound graphs and set disjointness to prove the hardness of clique detection
With the notion of lower-bound graphs at hand, we can formalize our reduction to the two-party communication complexity lower bound for set disjointness to obtain the following central theorem.
Theorem 3. Let G be a (k, m)-lower-bound graph.Then, detecting a K 4 in the CONGEST b model with probability at least 2 3 requires Ω k mb rounds.Proof.Let A be an algorithm in the CONGEST b model for K 4 detection, that is, such that with probability at least 2  3 , if G contains a K 4 then at least one node outputs 1 and if G contains no copy of K 4 then no node outputs 1.We will show that A can be used to solve the two-party set disjointness problem for instances of size k.
Consider a set disjointness instance (X, Y ) of size k.Let G = (A, B, E) be a (k, m)-lower-bound graph, let E 1 , E 2 , . . ., E k be the edge partition as in Item 2 of Definition 2, and let H A = (A, E A ) and H B = (B, E B ) be the graphs associated with G (Item 4 in Definition 2).Alice constructs the set E A ⊆ E A such that for every i with We first show that the graph Indeed, since by Item 4 of Definition 2, the graphs H A and H B are bipartite (and thus the subgraphs G [A] and G [B] are bipartite too), any copy of K 4 in G must consist of two vertices from A and two vertices from B. Let a 1 , a 2 be any pair of distinct vertices in A and b 1 , b 2 be any pair of distinct vertices in B. Observe that if there is no The simulation of A on G is executed as follows.Suppose that A runs in r rounds.Alice simulates vertices A and Bob simulates vertices B. In round i, Alice sends all messages from A with destinations in B to Bob, and Bob sends all messages from B with destinations in A to Alice.Since the cut between A and B is of size m, Alice and Bob exchange messages with overall mb bits per round.Thus, overall they communicate rmb bits.Since the algorithm allows them to solve set disjointness, by Theorem 1, we have rmb = Ω(k).Thus, A requires Ω( k mb ) rounds.
In Theorem 9 in Section 3, we prove the existence of a (Ω(n 2 ), O(n 3/2 ))-lower-bound graph.By combining Theorem 9 with Theorem 3, we obtain the following main result.

Detection of K for ≥ 5
The lower bound construction given in Theorem 3 can be extended to the task of detecting K , for ≥ 5 (see also Figure 2).To this end, we add a clique on − 4 new nodes to graph G (from the proof of Theorem 3) and connect each of these nodes to every vertex in A ∪ B. Observe that this increases the cut between A and B by n( − 4) edges.For = O( √ n), there are only O(n 3/2 ) additional edges, which implies that the same lower bound as for K 4 holds.If = ω( √ n), then the number of additional edges is significant, since the size of the cut increases by more than a constant factor.In this case, the round complexity is Ω( n 2 n( −4) b ) = Ω( n b ).Similarly as before, the encoded set disjointness instance evaluates to 1 if and only if G contains a clique of size .We thus conclude with the following theorem.

Lower-bound graph construction
In this section, we construct our main technical tool and prove the existence of a (Ω(n 2 ), O(n 3/2 ))-lower-bound graph, see Definition 2. We will show in Theorem 9 that Algorithm 1 below constructs a (Ω(n 2 ), O(n 3/2 ))-lowerbound graph with high probability (observe that a non-zero probability already suffices to prove the existence of such a graph).Let K be the family of sets {a For S ⊆ A ∪ B, let K(S) ⊆ K be the family of subsets K with S ⊆ K.

Peeling Process:
Let A ⊆ A and B ⊆ B be a uniform random sample of A and B, respectively, where every vertex is included with probability return H := (A, B, K∈H E K ).
In the peeling phase, we greedily compute a subset H ⊆ K such that at the end, the graph induced by the edges of H is a (Ω(n2 ), O(n3/2 ))-lower bound graph.When inserting a set K = {a 1 , a 2 , b 1 , b 2 } ∈ K into H, we make sure that the following three properties are fulfilled: 1. We ensure that later on we will never add a To this end, when inserting K into H, for every K ∈ K that contains the same pair of A-vertices (or B-vertices), we add its pair of B vertices (resp.pair of A vertices) to set F B (resp.F A ), indicating that this is a forbidden pair.Then, when inserting an element of K into H, we make sure that its pairs of A and B vertices are not forbidden.

Analysis of Algorithm 1
Our analysis relies on some basic properties of the structure of subgraphs of random graphs (for a more complete treatment of related problems, see, e.g., [12,Chapter 3]).We prove three high probability claims about the construction in Algorithm 1: that the random graph G contains many copies of K 2,2 (Lemma 6), that only a small fraction of pairs of A vertices are contained in more than six copies of K 2,2 (Lemma 7), and finally that the resulting graph H contains Ω(n 2 ) copies of K 2,2 (Lemma 8).With these three claims at hand, we will complete the analysis to prove in Theorem 9 that with high probability, the output of Algorithm 1 is a (Ω(n 2 ), O(n 3/2 ))lower-bound graph.
We begin with a proof that in Algorithm 1, the random graph G contains many copies of K 2,2 .
Lemma 6. Suppose that p ≥ 1 n .Then there is a constant C such that 2 : We distinguish the following cases: A quick sanity check shows that t 0 + t where the last equality holds for every p ≥ 1 n .We apply Chebyshev's inequality and obtain: for some constant C.

Definition 2 .
Let G = (A, B, E) be a bipartite graph with |A| = |B| = n and let k, m be integers.Then G is called a (k, m)-lower-bound graph if: 1. |E| ≤ m.

Figure 2 :
Figure 2: Extension of our lower bound for K 4 detection to K detection, for ≥ 5. We add a clique K −4 on − 4 new vertices to the graph G and connect every vertex of the clique to every other vertex of G .Then the resulting graph contains a clique on vertices if and only if the encoded set disjointness instance evaluates to 1, i.e., x i = y i = 1, for some i.

Theorem 4 .
Every algorithm in the CONGEST b model that detects a K 4 with probability at least 2 3 requires Ω( √ n/b) rounds.

3. 1 1 √nAlgorithm 1 . 1 √ n . 1 .
Construction of a (Ω(n 2 ), O(n 3/2 ))-lower-bound graphWe proceed as follows.We start our construction with a bipartite random graph G = (A, B, E) with |A| = |B| = n, where every potential edge ab between a ∈ A and b ∈ B is included with probability p = .Observe that for anya 1 , a 2 ∈ A (a 1 = a 2 ) and b 1 , b 2 ∈ B (b 1 = b 2 ), the probability that G[{a 1 , a 2 , b 1 , b 2 }] is isomorphic to a K 2,2 is p 4 .We therefore expect G to contain n Construction of a (Ω(n 2 ), O(n 3/2 ))-lower-bound graph: Input: Integer n, let p = Random Graph: Let G = (A, B, E) with |A| = |B| = n be the bipartite random graph where for every a ∈ A, b ∈ B the edge ab is included in E with probability p.

2 2 p 4 ,
We will compute the expectation and the variance of |K| and then use Chebyshev's inequality to bound the probability that |K| deviates substantially from its expectation.Let X be the family of all sets{a 1 , a 2 , b 1 , b 2 } with a 1 , a 2 ∈ A, a 1 = a 2 , b 1 , b 2 ∈ B, b 1 = b 2, and for X ∈ X let χ(X) be the indicator variable of the event "G[X] is isomorphic to K 2,2 ".Then:E|K| = X∈X P [χ(X) = 1] = |X |p 4 = n since K 2,2 contains 4 edges.To bound the variance V|K|, we use the identity V|K| = E|K| 2 − (E|K|)

Table 1 :
Prior (randomized) results for the problem of detecting a given subgraph H, or for listing all copies of H, in the CONGEST model (less relevant results (upper bounds) for the CONGESTED CLIQUE model are omitted; note that lower bounds for CONGESTED CLIQUE hold also for CONGEST and lower bounds for broadcast CONGESTED CLIQUE do not imply any bounds for CONGEST). CONGEST 2 .4. Define two graphs associated with G, H A = (A, E A ) and H B = (B, E B ). H A is the graph on vertex set A, where a 1 , a 2 ∈ A are adjacent if and only if there exists an index i with A(E i ) = {a 1 , a 2 }.Similarly, H B is the graph on vertex set B, where b 1 , b 2 ∈ B are adjacent if and only if there exists an index j with B(E j ) = {b 1 , b 2 }.Then, we require that H A and H B are bipartite.Figure 1: Left: Example of a (4, 12)-lower-bound graph G = (A, B, E).The dotted edges are the edges of the associated graphs H A and H B (observe that H A and H B form cycles of lengths 4, which are bipartite).For 1 ≤ i ≤ 4, let E i be the edge set of subgraph G[{a i , a (i mod 4)+1 , b i , b (i mod 4)+1 }].Observe that E = i≤4 E i , and, for every i, G[E i ] is isomorphic to K 2,2 .Observe further that for 1 + t 21 + t 22 + t 3 + t 4 = n 2 4 .We thus obtain: V|K| = E|K| 2 − (E|K|) 2 = p 8 (t 0 + t 1 + t 2,1 ) + p 7 t 2,2 + p 6 t 3 + p 4 t 4 − n 2 ≤ p 7 t 2,2 + p 6 t 3 + p 4 t 4 = O(p 7 n 6 ) ,