Graphs whose Kronecker covers are bipartite Kneser graphs
aa r X i v : . [ m a t h . C O ] A ug GRAPHS WHOSE KRONECKER COVERS ARE BIPARTITE KNESERGRAPHS
TAKAHIRO MATSUSHITA
Abstract.
We show that there are k simple graphs whose Kronecker covers are isomor-phic to the bipartite Kneser graph H ( n, k ), and that their chromatic numbers coincidewith χ ( K ( n, k )) = n − k + 2. We also determine the automorphism groups of thesegraphs. Introduction A covering map of graphs is a surjective graph homomorphism p : ˜ G → G such that p maps the neighborhood of each vertex v in ˜ G bijectively onto the neighborhood of p ( v ).The Kronecker cover of G is the categorical product K × G by K (see Section 2). When G is connected and non-bipartite, then its Kronecker cover is the unique cover which isbipartite, and when G is bipartite, its Kronecker cover is the disjoint union of two copiesof G . Kronecker covers are fundamental objects in covering theory of graphs, and haveappeared in different branches of combinatorics (see [1], [6], and [7]).It is known that different graphs may have isomorphic Kronecker covers. Then it isnatural to classify the all possible graphs whose Kronecker covers are isomorphic to a givenbipartite graph. Such a problem was actually written in [3], and was settled in the cases ofhypercubes [2] and generalized Petersen graphs [4].The purpose in this paper is to classify the all graphs whose Kronecker covers are iso-morphic to the bipartite Kneser graph H ( n, k ). Moreover, we determine the automorphismgroups of them and chromatic numbers. Here we recall the definition of H ( n, k ). Let n and k be positive integers with n > k . The Kneser graph K ( n, k ) is the graph consisting of k -subsets of [ n ] = { , · · · , n } , where two k -subsets are adjacent if and only if they have nointersection. The bipartite Kneser graph H ( n, k ) is the Kronecker cover of K ( n, k ). Theorem 1.
Suppose that n > k and k ≥ . Then there are k simple graphs G ( n, k ) , G ( n, k ) , · · · , G k − ( n, k ) , where G ( n, k ) = K ( n, k ) , K × G i ( n, k ) ∼ = H ( n, k ) , but any two of them are not isomorphic. Mathematics Subject Classification.
Primary 05C15; Secondary 55U10.
Key words and phrases.
Kneser graphs, bipartite Kneser graphs, Kronecker covers, neighborhoodcomplexes.
Here we recall some previous works concerning Theorem 1. Imrich and Pisanski [3]constructed a graph G such that G = K (5 ,
2) but K × G ∼ = H (5 , KG ′ n,k such that KG ′ n,k = K ( n, k ) but K × KG ′ n,k = H ( n, k ), when k ≥
2. In fact, the graph KG ′ n,k is the graph G ( n, k ) in our sense, and G i ( n, k ) is ageneralization of it.Next we study the automorphism groups of G i ( n, k ). Let Z denote the cyclic groupof order 2, and S n the symmetric group of [ n ] = { , · · · , n } . The automorphism group of G i ( n, k ) is described as follows. Here we recall that the automorphism group of K ( n, k ) isisomorphic to S n . Theorem 2.
For i = 0 , , · · · , k − , there is a group isomorphism Aut( G i ( n, k )) ∼ = ( Z i ⋊ ϕ S i ) × S n − i . Here the action ϕ : S i → Aut( Z i ) is defined by ϕ ( x , · · · , x i ) = ( x σ − (1) , · · · , x σ − ( i ) ) , and Z i ⋊ ϕ S i is the semi-direct product of groups with respect to ϕ . Finally, we study the chromatic number of G i ( n, k ). Recall that χ ( K ( n, k )) = n − k + 2is called the Kneser conjecture and is proved by Lov´asz. In his outstanding proof, Lov´aszintroduced the neighborhood complex N ( G ) of a graph G , and the connectivity of N ( G )gives a lower bound for the chromatic number χ ( G ) of G . Since the Kronecker cover of G determines the isomorphism type of N ( G ) (see [7]) and Lov´asz determines the connectivityof N ( K ( n, k )), we have the same lower bound for χ ( G i ( n, k )). Using this, we can determinethe chromatic number of G i ( n, k ). Theorem 3.
The chromatic number of G i ( n, k ) for i = 0 , , · · · , k − coincides with χ ( K ( n, k )) = n − k + 2 . The rest in this paper is organized as follows. In Section 2, we recall some terminologyand facts concerning Kronecker coverings, and prove Theorem 1 and Theorem 2. In Section3, we review some facts of neighbohood complexes and prove Theorem 3.
Acknowledgement.
The author is supported by JSPS KAKENHI 19K14536.2.
Proofs of Theorem 1 and Theorem 2
We first fix our notation and terminology, and review some facts concerning Kroneckercovers. A graph is a pair G = ( V ( G ) , E ( G )) consisting of a set V ( G ) together with a sym-metric binary relation E ( G ) of V ( G ). We write v ∼ G w or simply v ∼ w to mean that v and w are adjacent in G . A map f : V ( G ) → V ( H ) is a graph homomorphism if v ∼ G w implies f ( v ) ∼ H f ( w ). An n -coloring is a graph homomorphism from G to K n . An isomor-phism is a graph homomorphism having an inverse which is a graph homomorphism. An automorphism of G is an isomorphism from G to G . Let Aut( G ) denote the automorphismgroup of G . An involution of G is an automorphism α of G such that α = id G . RAPHS WHOSE KRONECKER COVERS ARE BIPARTITE KNESER GRAPHS 3 A bigraph [1] is a graph X equipped with a 2-coloring ε : G → K . For a pair X and Y of bigraphs, a graph homomorphism f : V ( X ) → V ( Y ) is even if εf = ε , and odd if εf ( x ) = ε ( x ) for every x ∈ V ( X ).For a pair G and H be graphs. The categorical product G × H is the graph whose vertexset is V ( G ) × V ( H ), where ( v, w ) ∼ G × H ( v ′ , w ′ ) if and only if v ∼ G v ′ and w ∼ H w ′ . The Kronecker cover is the categorical product K × G . Note that the Kronecker cover K × G has a 2-coloring K × G → K , ( i, v ) i , and has an odd involution (1 , v ) ↔ (2 , v ). Infact, every bigraph X equipped with an odd involution α is isomorphic to the Kroneckercover over a certain graph X/α defined as follows.Let X be a bigraph with an odd involution α . Define the quotient graph X/α by V ( X/α ) = (cid:8) { x, α ( x ) } | x ∈ V ( X ) (cid:9) ,E ( X/α ) = { ( σ, τ ) ∈ V ( X/α ) × V ( X/α ) | ( σ × τ ) ∩ E ( X ) = ∅} . In other words, σ ∼ X/α τ if and only if there is x ∈ σ and y ∈ τ such that x ∼ X y . Notethat X/α is not simple in general. In fact,
X/α is simple if and only if there is no vertex x in X such that x ∼ X α ( x ). The graph homomorphism( ε, π ) : X → K × ( X/α ) , x ( ε ( x ) , π ( x ))is an even isomorphism (see [7]). The following two lemmas are known (see [3] and Theorem3.1 of [7]) and easily proved. Lemma 4.
Let X and Y be bigraphs, α and β odd involutions of X and Y respectively,and f : X → Y a (not necessarily even) graph homomorphism satisfying f α = βf . Thenthere is a unique graph homomorphism f : X/α → Y /β satisfying πf = f π . If f is anisomorphism, then f is an isomorphism. Lemma 5.
Let X and Y be bigraphs, and α and β odd involutions of X and Y , re-spectively. For every graph homomorphism f : X/α → Y /β , there is a unique even graphhomomorphism ˜ f : X → Y such that π ˜ f = f π . If f is an isomorphism, then ˜ f is also anisomorphism. Here we mention two important applications of these lemmas. Two odd involutions α and β of X are conjugate if there is an automorphism f such that f α = βf . Similarly, α and β are evenly conjugate if there is an automorphism f such that f α = βf . Using thisterminology, we have the following classification result. Note that in the following corollary,the implication (3) ⇒ (1) is known (see Proposition 3 of [3] for example). Proposition 6.
Let α and β be odd involutions in a bigraph X . Then the following areequivalent. (1) X/α and
X/β are isomorphic. (2) α and β are evenly conjugate. TAKAHIRO MATSUSHITA (3) α and β are conjugate.Proof. Suppose that there is an isomorphism f : X/α → X/β . It follows from (2) of Theorem4 that there is an even isomorphism ˜ f : X → X satisfying f α = βf . It is clear that (2)implies (3). It follows from Lemma 4 that (3) implies (1). (cid:3) Proposition 7.
Let α be an odd involution of a bigraph X . Then Aut(
X/α ) is isomorphicto the subgroup of Aut( X ) consisting of even elements commuting with α .Proof. Let Γ be the subgroup of Aut( X ) consisting of even elements commuting with α .Define the group homomorphisms Φ : Γ → Aut(
X/α ) and Ψ : Aut(
X/α ) → Γ as follows.Let f ∈ Γ. Since f α = αf , Lemma 4 implies that f induces an isomorphism Φ( f ) = f : X/α → Y /β . On the other hand, let g ∈ Aut(
X/α ). It follows from Lemma 5 that thereis a unique even automorphism ˜ g : X → X satisfying ˜ gα = α ˜ g , and put Ψ( g ) = ˜ g ∈ Γ.These correspondences are group homomorphisms and Ψ is the inverse of Φ. (cid:3)
Now we study the automorphism group of K × G . For a pair of graphs G and H , wehave a monomorphism Aut( G ) × Aut( H ) → Aut( G × H ) which sends ( f, g ) to f × g . Here f × g is the autormorphism sending ( v, w ) to ( f ( v ) , f ( w )). Since Aut( K ) = Z , there is amonomorphism Z × Aut( G ) → Aut( K × G )( ∗ )In general, this monomorphism is not an isomorphism (see Remark 11 for example). How-ever, when G = K ( n, k ), this monomorphism is an isomorphism: Theorem 8 (Mirafzal [8]) . If n > k , then, the group homomorphism Z × Aut( K ( n, k )) → Aut( H ( n, k )) described in ( ∗ ) is an isomorphism. In particular, Aut( H ( n, k )) ∼ = Z × S n . When the monomorphism ( ∗ ) is an isomorphism, then the classification of the graphswhose Kronecker covers are K × G is simpler. Here we write τ to indicate the non-trivialinvolution of K . Proposition 9.
Let G be a graph and suppose that the monomorphism ( ∗ ) is an isomor-phism. Then the following hold: (1) For every odd involution α of K × G , there is an involution α ′ of G with α = τ × α ′ . (2) Let α ′ and β ′ be involutions of G . Then τ × α ′ and τ × β ′ are evenly conjugate ifand only if α ′ and β ′ are conjugate, i.e. there is f ∈ Aut( G ) with f α ′ = β ′ f .Proof. Since the monomorphism ( ∗ ) is an isomorphism, every involution α of K × G iswritten by id K × α ′ or τ × α ′ for some α ′ ∈ Aut( G ). Since α is an involution, we have that α ′ is an involution. The involution id K × α ′ is even and τ × α ′ is odd. This follows (1). (2)follows from the fact that every even automorphism of K × G is written as id K × f forsome f ∈ Aut( G ) under our assumption. (cid:3) RAPHS WHOSE KRONECKER COVERS ARE BIPARTITE KNESER GRAPHS 5
We are now ready to prove Theorem 1.
Proof of Theorem 1.
For i = 0 , , · · · , [ n/ σ i ∈ S n to be the composite of transpo-sitions (1 , , · · · (2 i − , i ) . By the classification of conjugacy classes of S n , every element in S n of order 2 is conjugateto some σ i , and i = j implies that σ i and σ j are not conjugate. Define α i to be the oddinvolution τ × σ i of K × G , and put G i = G i ( n, k ) = H ( n, k ) /α i . Then G = K ( n, k ), i = j implies G i = G j , and for every odd involution α of H ( n, k ), there is i with G i ∼ = H ( n, k ) /α .To complete the proof, we prove that G i is simple if and only if i < k .Suppose i ≥ k . Then put v = { , , · · · , k − } ∈ K ( n, k ). Then v ∼ σ i ( v ) in K ( n, k )implies that α i (1 , v ) ∼ (2 , σ i ( v )) in H ( n, k ). Thus G i is not simple. On the other hand,suppose i < k . Then for each v ∈ V ( K ( n, k )), we have that σ i ( v ) ∩ v = ∅ and hence v σ i ( v )for every v ∈ V ( K ( n, k )). This means that ( i, v ) ( τ ( i ) , σ i ( v )) for every ( i, v ) ∈ V ( H ( n, k )).This means that G i is simple, and completes the proof. (cid:3) Before giving the proof of Theorem 2, we introduce the following notation: Let G be agroup and x an element in G . We write Z G ( x ) to indicate the subgroup of G consisting ofthe elements in G , which commute with x . Proof of Theorem 2.
Since ( ∗ ) is an isomorphism, for every automorphism f of H ( n, k ),there is a unique f ∈ Aut( K ( n, k )) ∼ = S n satisfying f = id K × f . Thus Proposition 7implies that the automorphism group of G i ( n, k ) = X/ ( τ × σ i ) is isomorphic to Z S n ( σ i ).Hence the following proposition completes the proof. (cid:3) Proposition 10.
For m = 0 , , · · · , [ n/ , there is a following isomorphism: Z S n ( σ m ) = ( Z m ⋊ ϕ S m ) × S n − m Here the action ϕ : S m → Aut( Z m ) is defined by ϕ ( σ )(( x i ) i ) = ( x σ − ( i ) ) i .Proof. Every element σ in Z S n ( σ m ) does not send an element of { , · · · , m } to { m +1 , · · · , n } , and hence we have Z S n ( σ i ) ∼ = Z S m ( σ m ) × S n − m . In S m , σ m is conjugate withthe element τ = (1 , m + 1) · · · ( m, m ) . Thus it suffices to show Z S m ( τ ) = Z m ⋊ ϕ S m .First we define the group homomorphism Φ : Z m ⋊ ϕ S m → Z S m ( τ ). For i = 1 , · · · , m ,set ε i = ( i, n + i ) ∈ S m . For σ ∈ S m , then define ˜ σ ∈ S m by˜ σ ( i ) = ( σ ( i ) ( i = 1 , · · · , m ) σ ( i − m ) + m ( i = m + 1 , · · · , m ) . Let Φ : Z m ⋊ ϕ S m → Z S m ( τ ) be the map which sends (( x , · · · , x m ) , σ ) to ε x · · · ε x m m ˜ σ .Using the relation ε i ˜ σ = ˜ σε σ − ( i ) , we have that Φ is a group homomorphism. TAKAHIRO MATSUSHITA
Since Φ is injective, it suffices to show that Φ is surjective. Let σ ∈ Z m ( τ ). We identify { , · · · , n } with Z n , and for i = 1 , · · · , n , define k i ∈ Z n by σ ( i ) = i + k i . Since σ and τ commute, we have k i = k n + i . This means that σ gives rise to a permutation the family ofsets s = { , n + 1 } , s = { , n + 2 } , · · · , s n = { n, n } . Define σ ′ ∈ S m by σ ( s i ) = s ˜ σ ( i ) . For i = 1 , · · · , m , define x i ∈ Z as follows: • If σ ( i ) = σ ′ ( i ), then x σ ( i ) = 0. • If σ ( i ) = σ ′ ( i ) + n , then x σ ( i ) = 1.Then we have Φ(( x , · · · , x m ) , σ ′ ) = σ . This completes the proof. (cid:3) Remark . If i >
0, the group homomorphism Z × ( Z i ⋊ ϕ S i ) × S n − i ∼ = Z × Aut( G i ( n, k )) → Aut( H ( n, k )) ∼ = Z × S n described by ( ∗ ) is not an isomorphism in the case of G = G i ( n, k ).3. Proof of Theorem 3
The purpose in this section is to prove Theorem 3. Namely, we want to show that χ ( G i ( n, k ))) = n − k + 2 if n > k . We note that the proof given here is a straightforwardgeneralization of the proof of χ ( G ( n, k )) = n − k + 2 in [7].Recall that Lov´asz [5] introduces neighborhood complexes of graphs to determine χ ( K ( n, k )).We first review the definition and facts concerning neighborhood complexes. Let G be agraph, and v a vertex in G . Then the neighborhood complex N ( G ) is the simplicial complexwhose simplex is a subset of V ( G ) having a common neighbor. Lov´asz showed the followingtwo theorems in his proof of Kneser’s conjecture: Theorem 12. If N ( G ) is m -connected, then χ ( G ) ≥ m + 3 . Theorem 13.
The neighborhood complex N ( K ( n, k )) of K ( n, k ) is ( n − k − -connected. On the other hand, the author noted in [7] that the Kronecker cover of a graph G determines the neighborhood complex of G : Lemma 14 (Theorem 1.2 of [7]. See also [1]) . Let G and H be graphs. If K × G ∼ = K × H ,then their neighborhood complexes N ( G ) and N ( H ) are isomorphic. Combining the above results, we have the following corollary:
Corollary 15.
The neighborhood complex N ( G i ( n, k )) is ( n − k − -connected. In par-ticular, the inequality χ ( G i ( n, k )) ≥ n − k + 2 holds. We now complete the proof of χ ( G i ( n, k )) = n − k + 2. This is proved by induction on n . First, note that G i (2 k, k ) is a disjoint union of copies of K , and hence it is clear that RAPHS WHOSE KRONECKER COVERS ARE BIPARTITE KNESER GRAPHS 7 χ ( G i (2 k, k )) = 2. Suppose that n > k and χ ( G i ( n − , k )) = n − k + 1. A vertex in G i ( n, k ) which is not contained in G i ( n, k ) is written as { (1 , s ) , (2 , σ i s ) } , where s is a k -subset of [ n ] containing n . Note that σ i ∈ S n fixes n . Since G i ( n − , k ) isan induced subgraph of G i ( n, k ), we have that n − k + 2 ≤ χ ( G i ( n, k )) ≤ χ ( G i ( n − , k )) + 1 = n − k + 2 . This completes the proof.
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Proc.: Math. Sci. (2019),Article number 34.
Department of Mathematical Sciences, University of the Ryukyus, Nishihara-cho, Okinawa903-0213, Japan
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