aa r X i v : . [ m a t h . G R ] S e p The automorphism group of thebipartite Kneser graph
S. Morteza Mirafzal
Abstract.
Let n and k be integers with n > k, k ≥
1. We denote by H ( n, k ) the bipartite Kneser graph , that is, a graph with the family of k -subsets and ( n − k )-subsets of [ n ] = { , , ..., n } as vertices, in whichany two vertices are adjacent if and only if one of them is a subset of theother. In this paper, we determine the automorphism group of H ( n, k ).We show that Aut ( H ( n, k )) ∼ = Sym ([ n ]) × Z where Z is the cyclicgroup of order 2. Then, as an application of the obtained result, we givea new proof for determining the automorphism group of the Knesergraph K ( n, k ). In fact we show how to determine the automorphismgroup of the Kneser graph K ( n, k ) given the automorphism group ofthe Johnson graph J ( n, k ). Note that the known proofs for determiningthe automorphism groups of Johnson graph J ( n, k ) and Kneser graph K ( n, k ) are independent from each other. Mathematics Subject Classification (2010).
Primary 05C25 Secondary94C15.
Keywords. bipartite Kneser graph, vertex-transitive graph, automor-phism group.
1. Introduction
For a positive integer n >
1, let [ n ] = { , , ..., n } and V be the set of all k -subsets and ( n − k )-subsets of [ n ]. The bipartite Kneser graph H ( n, k ) has V as its vertex set, and two vertices A, B are adjacent if and only if A ⊂ B or B ⊂ A . If n = 2 k it is obvious that we do not have any edges, and insuch a case, H ( n, k ) is a null graph, and hence we assume that n ≥ k + 1.It follows from the definition of the graph H ( n, k ) that it has 2 (cid:0) nk (cid:1) verticesand the degree of each of its vertices is (cid:0) n − kk (cid:1) = (cid:0) n − kn − k (cid:1) , hence it is a regulargraph. It is clear that H ( n, k ) is a bipartite graph. In fact, if V = { v ∈ V ( H ( n, k )) | | v | = k } and V = { v ∈ V ( H ( n, k )) | | v | = n − k } , then { V , V } . S. Morteza Mirafzalis a partition of V ( H ( n, k )) and every edge of H ( n, k ) has a vertex in V and avertex in V and | V | = | V | . It is an easy task to show that the graph H ( n, k )is a connected graph. The bipartite Kneser graph H (2 n + 1 , n ) is known asthe middle cube M Q n +1 = Q n +1 ( n, n + 1) [3] or regular hyperstar graph HS (2( n + 1) , n + 1) [11,13].The regular hyperstar graph Q n +1 ( n, n + 1) has been investigated fromvarious aspects, by various authors and some of the recent works about thisclass of graphs are [3,6,11,13,16,17]. The following figure shows the graph H (5 ,
2) ( Q (2 , { i, j, k } ( { i, j } ) isdenoted by ijk ( ij ). 13 123 124242454545343143 1223234 14145125252353513515Fig 1. The bipartite Kneser graph H(5,2)It was conjectured by Dejter, Erd˝os, and Havel [6] among others, thatthe middle cube Q n +1 ( n, n + 1) is Hamiltonian. Recently, M¨utze and Su[17] showed that the bipartite Kneser graph H ( n, k ) has a Hamilton cycle forall values of k . Among various interesting properties of the bipartite Knesergraph H ( n, k ), we are interested in its automorphism group and we want toknow how this group acts on the vertex set of H ( n, k ). Mirafzal [13] deter-mined the automorphism group of HS (2 n, n ) = H (2 n − , n −
1) and showedthat HS (2 n, n ) is a vertex-transitive non-Cayley graph. Also, he showed that HS (2 n, n ) is arc-transitive.Some of the symmetry properties of the bipartite Kneser graph H ( n, k ), areas follows. Proposition 1.1. [16 , Lemma . The graph H ( n, k ) is a vertex-transitivegraph. Proposition 1.2. [16 , T heorem . The graph H ( n, k ) is a symmetric (orarc-transitive) graph. Corollary 1.3. [16 , Corollary . The connectivity of the bipartite Knesergraph H ( n, k ) is maximum, namely, (cid:0) n − kk (cid:1) . he automorphism group of the bipartite Kneser graph 3 Proposition 1.4. [16 , P roposition . The bipartite Kneser graph H ( n, isa Cayley graph. Theorem 1.5. [16 , T heorem . Let H ( n, be a bipartite Kneser graph.Then, Aut ( H ( n, ∼ = Sym ([ n ]) × Z , where Z is the cyclic group of order . In [16] the authors proved the following theorem.
Theorem 1.6. [16 , T heorem . Let n = 2 k − . Then, for the bipartiteKneser graph H ( n, k − , we have Aut ( H ( n, k )) ∼ = Sym ([ n ]) × Z , where Z is the cyclic group of order . In [16] the authors asked the following question.
Question
Is the above theorem true for all possible values of n, k ( 2 k < n )?In the sequel, we want to answer the above question. We show that theabove theorem is true for all possible values of n, k ( 2 k < n ).In fact, to the best of our knowledge, the present work is the first answeron this problem. We determine the automorphism group of the graph H ( n, k )and show that Aut ( H ( n, k )) ∼ = Sym ([ n ]) × Z , where Z is the cyclic groupof order 2. In the final step of our work, we offer a new proof for determiningthe automorphism group of the Kneser graph K ( n, k ) which we belief thisproof is more elementary than other known proofs of this result. Note thatthe known proofs for determining the automorphism groups of Johnson graph J ( n, k ) and Kneser graph K ( n, k ) are independent from each other. we showhow we can have the automorphism group of the Kneser graph K ( n, k ) inthe hand, if we have the automorphism group of the Johnson graph J ( n, k )in another hand.There are various important families of graphs Γ, in which we know thatfor a particular group G , we have G ≤ Aut (Γ), but showing that in fact wehave G = Aut (Γ), is a difficult task. For example note to the following cases.(1) The Boolean lattice BL n , n ≥
1, is the graph whose vertex set isthe set of all subsets of [ n ] = { , , ..., n } , where two subsets x and y are ad-jacent if their symmetric difference has precisely one element. The hypercube Q n is the graph whose vertex set is { , } n , where two n -tuples are adjacentif they differ in precisely one coordinates. It is an easy task to show that Q n ∼ = BL n , and Q n ∼ = Cay ( Z n , S ), where Z is the cyclic group of order 2,and S = { e i | ≤ i ≤ n } , where e i = (0 , ..., , , , ..., i thposition. It is an easy task to show that the set H = { f θ | θ ∈ Sym ([ n ]) } , f θ ( { x , ..., x n } ) = { θ ( x ) , ..., θ ( x n ) } is a subgroup of Aut ( BL n ), and hence H is a subgroup of the group Aut ( Q n ). We know that in every Cayley graphΓ = Cay ( G, S ), the group
Aut (Γ) contains a subgroup isomorphic withthe group G . Therefore, Z n is a subgroup of Aut ( Q n ). Now, showing that S. Morteza Mirafzal Aut ( Q n ) = < Z n , Sym ([ n ]) > ( ∼ = Z n ⋊ Sym ([ n ])), is not an easy task [14].(2) Let n, k ∈ N with k < n and Let [ n ] = { , ..., n } . The Kneser graph K ( n, k ) is defined as the graph whose vertex set is V = { v | v ⊆ [ n ] , | v | = k } and two vertices v , w are adjacent if and only if | v ∩ w | =0. The Kneser graph K ( n, k ) is a vertex-transitive graph [5]. It is an easy task to show that the set H = { f θ | θ ∈ Sym ([ n ]) } , f θ ( { x , ..., x k } ) = { θ ( x ) , ..., θ ( x k ) } , is a subgroupof Aut ( K ( n, k )) [5]. But, showing that H = { f θ | θ ∈ Sym ([ n ]) } = Aut ( K ( n, k ))is rather a difficult work [5, chapter 7].(3) Let n and k be integers with n > k ≥ n ] = { , , ..., n } . Wenow consider the bipartite Kneser graph Γ = H ( n, k ). Let A, B be m -subsetsof [ n ] and let | A ∩ B | = t . Let θ be a permutation in Sym ([ n ]). It is an easy taskto show that | f θ ( A ) ∩ f θ ( B ) | = t , where f θ ( { x , ..., x m } ) = { θ ( x ) , ..., θ ( x m ) } .Moreover, if A ⊂ B , then f θ ( A ) ⊂ f θ ( B ). Therefore, if θ ∈ Sym ([ n ]), then f θ : V ( H ( n, k )) −→ V ( H ( n, k )) , f θ ( { x , ..., x k } ) = { θ ( x ) , ..., θ ( x k ) } is an automorphism of H ( n, k ) and the mapping, ψ : Sym ([ n ]) −→ Aut ( H ( n, k )), defined by the rule ψ ( θ ) = f θ is an injection.Therefore, the set H = { f θ | θ ∈ Sym ([ n ]) } , is a subgroup of Aut ( H ( n, k ))which is isomorphic with Sym ([ n ]). Also, the mapping α : V (Γ) → V (Γ),defined by the rule, α ( v ) = v c , where v c is the complement of the subset v in[ n ], is an automorphism of the graph B ( n, k ). In fact, if A ⊂ B , then B c ⊂ A c ,and hence if { A,B } is an edge of the graph B ( n, k ), then { α ( A ) , α ( B ) } is anedge of the graph H ( n, k ). Therefore we have, < H, α > ≤ Aut ( H ( n, k )).In this paper, we want to show that for the bipartite Kneser graph H ( n, k ),in fact we have, Aut ( H ( n, k )) = < H, α > ( ∼ = Sym ([ n ]) × Z ).
2. Preliminaries
In this paper, a graph Γ = (
V, E ) is considered as a finite undirected simplegraph where V = V (Γ) is the vertex-set and E = E (Γ) is the edge-set. Forall the terminology and notation not defined here, we follow [1 , , = ( V , E ) and Γ = ( V , E ) are called isomorphic ,if there is a bijection α : V −→ V such that { a, b } ∈ E if and only if { α ( a ) , α ( b ) } ∈ E for all a, b ∈ V . In such a case the bijection α is called anisomorphism. An automorphism of a graph Γ is an isomorphism of Γ withitself. The set of automorphisms of Γ with the operation of composition offunctions is a group, called the automorphism group of Γ and denoted by Aut (Γ).The group of all permutations of a set V is denoted by Sym ( V ) orjust Sym ( n ) when | V | = n . A permutation group G on V is a subgroup of Sym ( V ). In this case we say that G act on V . If X is a graph with vertex-set V , then we can view each automorphism as a permutation of V , and sohe automorphism group of the bipartite Kneser graph 5 Aut ( X ) is a permutation group. If G acts on V , we say that G is transitive (or G acts transitively on V ), when there is just one orbit. This means thatgiven any two elements u and v of V , there is an element β of G such that β ( u ) = v .The graph Γ is called vertex - transitive , if Aut (Γ) acts transitively on V (Γ). For v ∈ V (Γ) and G = Aut (Γ), the stabilizer subgroup G v is thesubgroup of G consisting of all automorphisms that fix v . We say that Γ is symmetric (or arc - transitive ) if, for all vertices u, v, x, y of Γ such that u and v are adjacent, also, x and y are adjacent, there is an automorphism π in Aut (Γ) such that π ( u ) = x and π ( v ) = y .Let n, k ∈ N with k ≤ n , and let [ n ] = { , ..., n } . The Johnson graphJ ( n, k ) is defined as the graph whose vertex set is V = { v | v ⊆ [ n ] , | v | = k } and two vertices v , w are adjacent if and only if | v ∩ w | = k −
1. The Johnsongraph J ( n, k ) is a vertex-transitive graph [5]. It is an easy task to show thatthe set H = { f θ | θ ∈ Sym ([ n ]) } , f θ ( { x , ..., x k } ) = { θ ( x ) , ..., θ ( x k ) } , is asubgroup of Aut ( J ( n, k ))[5]. It has been shown that Aut ( J ( n, k )) ∼ = Sym ([ n ]),if n = 2 k, and Aut ( J ( n, k )) ∼ = Sym ([ n ]) × Z , if n = 2 k , where Z is the cyclicgroup of order 2 [2,9,15].Although, in most situations it is difficult to determine the automor-phism group of a graph Γ and how it acts on the vertex set of Γ, there arevarious papers in the literature, and some of the recent works appear in thereferences [7,8,9,10,12,13,14,15,16,18,19].
3. Main results
Lemma 3.1.
Let n and k be integers with n > k ≥ , and let Γ = (
V, E ) = H ( n, k ) be a bipartite Kneser graph with partition V = V ∪ V , V ∩ V = ∅ ,where V = { v | v ⊂ [ n ] , | v | = k } and V = { w | w ⊂ [ n ] , | w | = n − k } . If f is an automorphism of Γ such that f ( v ) = v for every v ∈ V , then f is theidentity automorphism of Γ .Proof. First, note that since f is a permutation of the vertex set V and f ( V ) = V , then f ( V ) = V . Let w ∈ V be an arbitrary vertex in V .Since f is an automorphism of the graph Γ, then for the set N ( w ) = { v | v ∈ V , v ↔ w } , we have f ( N ( w )) = { f ( v ) | v ∈ V , v ↔ w } = N ( f ( w )). On theother hand, since for every v ∈ V , f ( v ) = v , then f ( N ( w )) = N ( w ), andtherefore N ( f ( w )) = N ( w ). In other words, w and f ( w ) are ( n − k )-subsetsof [ n ] such that their family of k -subsets are the same. Now, it is an easy taskto show that f ( w ) = w . Therefore, for every vertex x in Γ we have f ( x ) = x and thus f is the identity automorphism of Γ. (cid:3) Remark 3.2.
If in the assumptions of the above lemma, we replace with f ( v ) = v for every v ∈ V , then we can show, by a similar discussion, that f is the identity automorphism of Γ. S. Morteza Mirafzal Lemma 3.3.
Let
Γ = (
V, E ) be a connected bipartite graph with partition V = V ∪ V , V ∩ V = ∅ . Let f be an automorphism of Γ such that for afixed vertex v ∈ V , we have f ( v ) ∈ V . Then, f ( V ) = V and f ( V ) = V .Or, for a fixed vertex v ∈ V , we have f ( v ) ∈ V . Then, f ( V ) = V and f ( V ) = V .Proof. In the first step, we show that if w ∈ V then f ( w ) ∈ V . We knowthat if w ∈ V , then d Γ ( v, w ) = d ( v, w ), the distance between v and w in thegraph Γ, is an even integer. Assume d ( v, w ) = 2 l , 0 ≤ l ≤ D , where D isthe diameter of Γ. We prove by induction on l , that f ( w ) ∈ V . If l = 0, then d ( v, w ) = 0, thus v = w , and hence f ( w ) = f ( v ) ∈ V . Suppose that if w ∈ V and d ( v, w ) = 2( k − f ( w ) ∈ V . Assume w ∈ V and d ( v, w ) = 2 k .Then, there is a vertex u ∈ Γ such that d ( v, u ) = 2 k − k −
1) and d ( u, w ) = 2. We know (by the induction assumption) that f ( u ) ∈ V andsince d ( f ( u ) , f ( w )) = 2, therefore f ( w ) ∈ V . Now, it follows that f ( V ) = V and consequently f ( V ) = V . (cid:3) Corollary 3.4.
Let
Γ = H ( n, k ) = ( V, E ) be a bipartite Kneser graph withpartition V = V ∪ V , V ∩ V = ∅ . If f is an automorphism of the graph Γ ,then f ( V ) = V and f ( V ) = V , or f ( V ) = V and f ( V ) = V . In the sequel, we need the following result for proving our main theorem.
Lemma 3.5.
Let l, m, u are positive integers with l > u and m > u . If l > m then (cid:0) lu (cid:1) > (cid:0) mu (cid:1) .Proof. The proof is straightforward. (cid:3)
Theorem 3.6.
Let n and k be integers with n > k ≥ , and let Γ = (
V, E ) = H ( n, k ) be a bipartite Kneser graph with partition V = V ∪ V , V ∩ V = ∅ ,where V = { v | v ⊂ [ n ] , | v | = k } and V = { w | w ⊂ [ n ] , | w | = n − k } . Then, Aut (Γ) ∼ = Sym ([ n ]) × Z , where Z is the cyclic group of order .Proof. Let α : V (Γ) → V (Γ), defined by the rule, α ( v ) = v c , where v c is the complement of the subset v in [ n ]. Also, let H = { f θ | θ ∈ Sym ([ n ]) } , f θ ( { x , ..., x k } ) = { θ ( x ) , ..., θ ( x k ). We have seen already that H ( ∼ = Sym ([ n ]))and < α > ( ∼ = Z ) are subgroups of the group G = Aut (Γ). We can see that α H , and for every θ ∈ Sym ([ n ]), we have, f θ α = αf θ [15]. Therefore, Sym ([ n ]) × Z ∼ = H × < α > ∼ = < H, α > = { f γ α i | γ ∈ Sym ([ n ]) , ≤ i ≤ } = S is a subgroup of G . We now want to show that G = S . Let f ∈ Aut (Γ) = G .We show that f ∈ S . There are two cases(i) There is a vertex v ∈ V such that f ( v ) ∈ V , and hence by Lemma 3.3.we have f ( V ) = V .(ii) There is a vertex v ∈ V such that f ( v ) ∈ V , and hence by Lemma 3.3.we have f ( V ) = V .he automorphism group of the bipartite Kneser graph 7(i) Let f ( V ) = V . Then, for every vertex v ∈ V we have f ( v ) ∈ V ,and therefore the mapping g = f | V : V → V , is a permutation of V where f | V is the restriction of f to V . Let Γ = J ( n, k ) be the Johnson graph withthe vertex set V . Then, the vertices v, w ∈ V are adjacent in Γ if and onlyif | v ∩ w | = k − g = f | V is an automorphism of thegraph Γ .For proving our assertion, it is sufficient to show that if v, w ∈ V are suchthat | v ∩ w | = k − | g ( v ) ∩ g ( w ) | = k −
1. Note that since v, w are k -subsets of [ n ], then if u is a common neighbour of v, w in thebipartite Kneser graph Γ = H ( n, k ), then the set u contains the sets v and w . In particular u contains the ( k + 1)-subset v ∪ w . We now can see that thenumber of vertices u , such that u is adjacent in Γ to both of the vertices v and w , is (cid:0) n − k − n − k − (cid:1) . Note that if t is a positive integer such that k + 1 + t = n − k ,then t = n − k −
1. Now, if we adjoin to the ( k + 1)-subset v ∪ w of [ n ], n − k − v ∪ w in [ n ], then we obtain a subset u of [ n ] such that v ∪ w ⊆ u and u is a ( n − k )-subset of [ n ]. Now, since v and w have (cid:0) n − k − n − k − (cid:1) common neighbours in the graph Γ, then the vertices g ( v ) and g ( w ) must have (cid:0) n − k − n − k − (cid:1) = (cid:0) n − k − k (cid:1) neighbours in Γ, and therefore | g ( v ) ∩ g ( w ) | = k −
1. In fact, if | g ( v ) ∩ g ( w ) | = k − h < k −
1, then h >
1, and hence | g ( v ) ∪ g ( w ) | = k + h . Thus, if t is a positive integer such that k + h + t = n − k , then t = n − k − h . Hence, for constructing a ( n − k )-subset u ⊇ g ( v ) ∪ g ( w ) we must adjoin t = n − k − h elements of the complement of g ( v ) ∪ g ( w ) in [ n ], to the set g ( v ) ∪ g ( w ). Therefore the number of commonneighbours of vertices g ( v ) and g ( w ) in the graph Γ is (cid:0) n − k − hn − k − h (cid:1) = (cid:0) n − k − hk (cid:1) .Note that by Lemma 3.5. it follows that (cid:0) n − k − hk (cid:1) = (cid:0) n − k − k (cid:1) .Our argument shows that the permutation g = f | V is an automorphismof the Johnson graph Γ = J ( n, k ) and therefore by [2 chapter 9, 15] there isa permutation θ ∈ Sym ([ n ]) such that g = f θ .On the other hand, we know that f θ by its natural action on the vertexset of the bipartite Kneser graph Γ = H ( n, k ) is an automorphism of Γ.Therefore, l = f − θ f is an automorphism of the bipartite Kneser graph Γsuch that l is the identity automorphism on the subset V . We now canconclude, by Lemma 3.1. that l = f − θ f , is the identity automorphism of Γ,and therefore f = f θ .In other words, we have proved that if f is an automorphism of Γ = H ( n, k ) such that f ( V ) = V , then f = f θ , for some θ ∈ Sym ([ n ]), and hence f ∈ S .(ii) We now assume that f ( V ) = V . Then, f ( V ) = V . Since themapping α is an automorphism of the graph Γ, then f α is an automorphismof Γ such that f α ( V ) = f ( α ( V )) = f ( V ) = V . Therefore, by what is provedin (i), we have f α = f θ , for some θ ∈ Sym ([ n ]). Now since α is of order 2,then f = f θ α ∈ S = { f γ α i | γ ∈ Sym ([ n ]) , ≤ i ≤ } . (cid:3) S. Morteza MirafzalLet n, k be integers and n > , k < n , [ n ] = { , , ..., n } . Let K ( n, k ) =Γ be a Kneser graph. It is a well known fact that, Aut (Γ) ∼ = Sym ([ n ]) [5, chap7]. In fact, the proof in [5, chap 7] shows that the automorphism group of theKneser graph K ( n, k ) is the group H = { f θ | θ ∈ Sym ([ n ]) } ( ∼ = Sym ([ n ])).The proof of this result, that appears in [5, chap 7], uses the following factwhich is one of the fundamental results in extermal set theory. Fact (Erd˝os-Ko-Rado) If n > k , then α ( K ( n, k )) = (cid:0) n − k − (cid:1) , where α ( K ( n, k )) is the independence number of the Kneser graph K ( n, k ).In the sequel, we provide a new proof for determining the automorphismgroups of Kneser graphs. The main tool which we use in our method isTheorem 3.6. Note that, the main tool which we use for proving Theorem3.6. is the automorphism group of Johnson graph J ( n, k ), which have beenalready obtained [2 chapter 9, 15] by using elementary and relevant facts ofgraph theory and group theory. Theorem 3.7.
Assume n, k are integers and n > , k < n , [ n ] = { , , ..., n } .If K ( n, k ) = Γ is a Kneser graph, then we have Aut (Γ) ∼ = Sym ([ n ]) .Proof. Let g be an automorphism of the graph Γ. We now consider the bi-partite Kneser graph Γ = H ( n, k ) = ( V, E ), with partition V = V ∪ V , V ∩ V = ∅ , where V = { v | v ⊂ [ n ] , | v | = k } and V = { w | w ⊂ [ n ] , | w | = n − k } . We define the mapping f : V → V by the following rule; f ( v ) = ( g ( v ) v ∈ V ( αgα )( v ) v ∈ V It is an easy task to show that f is a permutation of the vertex set V suchthat f ( V ) = V = g ( V ). We show that f is an automorphism of the bipartiteKneser graph Γ . Let { v, w } be an edge of the graph Γ with v ∈ V . Then v ⊂ w , and hence v ∩ w c = v ∩ α ( w ) = ∅ . In other words { v, α ( w ) } is an edgeof the Kneser graph Γ. Now, since the mapping g is an automorphism of theKneser graph Γ, then { g ( v ) , g ( α ( w )) } is an edge of the Kneser graph Γ, andtherefore we have g ( v ) ∩ g ( α ( w )) = ∅ . This implies that g ( v ) ⊂ ( g ( α ( w ))) c = α ( g ( α ( w ))). In other words { g ( v ) , α ( g ( α ( w ))) } = { f ( v ) , f ( w ) } is an edgeof the bipartite Kneser graph Γ . Therefore f is an automorphism of thebipartite Kneser graph H ( n, k ). Now, since f ( V ) = V , then by Theorem 3.6.there is a permutation θ in Sym ([ n ]) such that f = f θ . Then, for every v ∈ V we have g ( v ) = f ( v ) = f θ ( v ), and therefore g = f θ . We now can concludethat Aut ( K ( n, k )) is a subgroup of the group H = { f γ | γ ∈ Sym ([ n ]) } .On the other hand, we can see that H is a subgroup of Aut ( K ( n, k )), andtherefore we have Aut ( K ( n, k )) = H = { f γ | γ ∈ Sym ([ n ]) } ∼ = Sym ([ n ]). (cid:3) he automorphism group of the bipartite Kneser graph 9
4. Conclusion
In this paper, we studied one of the algebraic properties of the bipartiteKneser graph H ( n, k ). We determined the automorphism group of this graphfor all n, k, where 2 k < n (Theorem 3.6). Then, by Theorem 3.6. we offereda new proof for determining the automorphism group of the Kneser graph K ( n, k )(Theorem 3.7).
5. Acknowledgements
The author is thankful to the anonymous reviewers for their valuable com-ments and suggestions.
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