aa r X i v : . [ m a t h . C O ] F e b Note on group distance magic graphs G × C n Sylwia Cichacz ∗ Faculty of Applied MathematicsAGH University of Science and TechnologyAl. Mickiewicza 30, 30-059 Krak´ow, Poland
August 7, 2018
Abstract
A Γ-distance magic labeling of a graph G = ( V, E ) with | V | = n isa bijection f from V to an Abelian group Γ of order n such that theweight w ( x ) = P y ∈ N G ( x ) f ( y ) of every vertex x ∈ V is equal to thesame element µ ∈ Γ, called the magic constant .In this paper we will show that if G is a graph of order n = 2 p (2 k +1)for some natural numbers p , k such that deg( v ) ≡ c (mod 2 p +2 ) forsome constant c for any v ∈ V ( G ), then there exists a Γ-distance magiclabeling for any Abelian group Γ of order 4 n for the direct product G × C . Moreover if c is even then there exists a Γ-distance magiclabeling for any Abelian group Γ of order 8 n for the direct product G × C . Keywords:MSC:
All graphs considered in this paper are simple finite graphs. We use V ( G ) forthe vertex set and E ( G ) for the edge set of a graph G . The neighborhood N ( x ) or more precisely N G ( x ), when needed, of a vertex x is the set ofvertices adjacent to x , and the degree d ( x ) of x is | N ( x ) | , the size of theneighborhood of x . By C n we denote a cycle on n vertices.A distance magic labeling (also called sigma labeling ) of a graph G = ( V, E ) of order n is a bijection l : V → { , , . . . , n } with the property ∗ The author was supported by National Science Centre grant nr 2011/01/D/ST/04104. µ (called the magic constant ) such that P y ∈ N G ( x ) l ( y ) = µ for every x ∈ V . If a graph G admits a distance magiclabeling, then we say that G is a distance magic graph ([4]). The sum P y ∈ N G ( x ) l ( y ) is called the weight of the vertex x and denoted by w ( x ).The concept of distance magic labeling has been motivated by the con-struction of magic squares. It is worth mentioning that finding an r -regulardistance magic labeling turns out equivalent to finding equalized incompletetournament EIT( n, r ) [9]. In an equalized incomplete tournament EIT( n, r )of n teams with r rounds, every team plays exactly r other teams and thetotal strength of the opponents that team i plays is k . Thus, it is easy tonotice that finding an EIT( n, r ) is the same as finding a distance magic la-beling of any r -regular graph on n vertices. For a survey, we refer the readerto [4].The following observations were independently proved: Observation 1.1 ([16, 17, 20, 21]) . Let G be an r -regular distance magicgraph on n vertices. Then µ = r ( n +1)2 . Observation 1.2 ([16, 17, 20, 21]) . No r -regular graph with r odd can be adistance magic graph. We recall two out of four standard graph products (see [12]). Both, the lexicographic product G ◦ H and the direct product G × H are graphs withthe vertex set V ( G ) × V ( H ). Two vertices ( g, h ) and ( g ′ , h ′ ) are adjacent in: • G ◦ H if and only if either g is adjacent with g ′ in G or g = g ′ and h is adjacent with h ′ in H ; • G × H if g is adjacent to g ′ in G and h is adjacent to h ′ in H .The graph G ◦ H is also called the composition and denoted by G [ H ] (see[11]). The product G × H is also known as Kronecker product , tensor prod-uct , categorical product and graph conjunction is the most natural graphproduct. The direct product is commutative, associative, and it has sev-eral applications, for instance it may be used as a model for concurrency inmultiprocessor systems [18]. Some other applications can be found in [15].Some graphs which are distance magic among (some) products can beseen in [1, 2, 3, 5, 6, 8, 17, 19]. Theorem 1.3 ([17]) . Let r ≥ , n ≥ , G be an r -regular graph and C n bethe cycle of length n . The graph G ◦ C n admits a distance magic labeling ifand only if n = 4 . heorem 1.4 ([17]) . Let G be an arbitrary regular graph. Then G ◦ K n isdistance magic for any even n . Theorem 1.5 ([2]) . Let G be an arbitrary regular graph. Then G × C isdistance magic. The following problem was posted in [4].
Problem 1.6 ([4]) . If G is a non-regular graph, determine if there is adistance magic labeling of G ◦ C . The similar problem for the direct product was stated in [1]:
Problem 1.7 ([1]) . If G is a non-regular graph, determine if there is adistance magic labeling of G × C . Moreover it was proved that:
Theorem 1.8 ([1]) . Let m and n be two positive integers such that m ≤ n .The graph K m,n × C is a distance magic graph if and only if the followingconditions hold:1. m + n ≡ and2. m ≥ √ n +1) − − − n. Froncek in [8] defined the notion of group distance magic graphs , i.e. thegraphs allowing the bijective labeling of vertices with elements of an Abeliangroup resulting in constant sums of neighbor labels.
Definition 1.9. A Γ -distance magic labeling of a graph G = ( V, E ) with | V | = n is a bijection f from V to an Abelian group Γ of order n such thatthe weight w ( x ) = P y ∈ N G ( x ) f ( y ) of every vertex x ∈ V is equal to the sameelement µ ∈ Γ , called the magic constant. A graph G is called a groupdistance magic graph if there exists a Γ -distance magic labeling for everyAbelian group Γ of order | V ( G ) | . The connection between distance magic graphs and Γ-distance magicgraphs is as follows. Let G be a distance magic graph of order n with themagic constant µ ′ . If we replace the label n in a distance magic labeling forthe graph G by the label 0, then we obtain a Z n -distance magic labeling forthe graph G with the magic constant µ ≡ µ ′ (mod n ). Hence every distancemagic graph with n vertices admits a Z n -distance magic labeling. Althougha Z n -distance magic graph on n vertices is not necessarily a distance magicgraph (see [8]), it was proved that Observation 1.2 also holds for a Z n -distance magic labeling ([7]). 3 bservation 1.10 ([7]) . Let r be a positive odd integer. No r -regular graphon n vertices can be a Z n -distance magic graph. The following theorem was proved in [8]:
Theorem 1.11 ([8]) . The Cartesian product C m (cid:3) C k , m, k ≥ , is a Z mk -distance magic graph if and only if km is even. Froncek also showed that the graph C k (cid:3) C k has a ( Z ) k -distance magiclabeling for k ≥ µ = (0 , , . . . ,
0) ([8]).Cichacz proved:
Theorem 1.12 ([6]) . Let G be a graph of order n and Γ be an Abeliangroup of order n . If n = 2 p (2 k + 1) for some natural numbers p , k and deg( v ) ≡ c (mod 2 p +1 ) for some constant c for any v ∈ V ( G ) , then thereexists a Γ -distance magic labeling for the graph G ◦ C . It seems that the direct product is the natural choice among (standard)products to deal with Γ-distance magic graphs and group distance magicgraphs in general. The reason for this is that the direct product is suitableproduct if we observe graphs as categories. Hence it should perform wellwith the product of (Abelian) groups, what the below theorem illustrates.
Theorem 1.13 ([3]) . If an r -regular graph G is Γ -distance magic and an r -regular graph G is Γ -distance magic, then the direct product G × G is Γ × Γ -distance magic. However, dealing with this product is also most difficult in many respectsamong standard products. For instance, G × H does not need to be con-nected, even if both factors are. More precisely, G × H is connected if andonly if both G and H are connected and at least one of them is non-bipartite[22]. The main open problem concerning the direct product is the famousHedetniemi’s conjecture. Hedetniemi conjectured that for all graphs G and H , χ ( G × H ) = min { χ ( G ) , χ ( H ) } , [13].Anholcer at al. proved the following theorems: Theorem 1.14 ([3]) . If G is an r -regular graph of order n , then lexico-graphic product G ◦ C is a group distance magic graph. Theorem 1.15 ([3]) . If G is an r -regular graph of order n , then directproduct G × C is a group distance magic graph. Theorem 1.16 ([3]) . If G is an r -regular graph of order n for some even r , then direct product G × C is a group distance magic graph. In this paper we prove the analogous theorems to Theorem 1.12 for directproduct G × C n for n ∈ { , } . 4 Direct product G × C We start with the following lemma.
Lemma 2.1.
Let G be a graph of order n and Γ be an arbitrary Abeliangroup of order n such that Γ ∼ = Z p × A for p ≥ and some Abeliangroup A of order n p − . If deg( v ) ≡ c (mod 2 p ) for some constant c andany v ∈ V ( G ) , then there exists a Γ -distance magic labeling for the graph G × C . Proof.
Let V ( G ) = { x , x , . . . , x n − } be the vertex set of G , let C = u u u u u , and H = G × C . Let v i,j = ( x i , u j ) for i = 0 , , . . . , n − j = 0 , , ,
3. Notice that if x p x q ∈ E ( G ), then v q,j ∈ N H ( v p,j ) if and only if j ∈ { i − , i + 1 } (where the sum on the second suffix is taken modulo 4).Using the isomorphism φ : Γ → A × Z p , we identify every g ∈ Γ withits image φ ( g ) = ( a i , w ), where a i ∈ A and w ∈ Z p , i = 0 , , . . . , n p − − H in the following way: f ( v i,j ) = (cid:26) (cid:0) a ⌊ i · − p +2 ⌋ , (2 i + j ) mod 2 p − (cid:1) for j = 0 , , (0 , p − − f ( v i,j − ) for j = 2 , i = 0 , , . . . , n − j = 0 , , , if ( v i, ) + f ( v i, ) = f ( v i, ) + f ( v i, ) = (0 , p − . Since deg( v ) ≡ c (mod 2 p ) for any v ∈ V ( G ), therefore the weight of every x ∈ V ( H ) is w ( x ) = (0 , − c ). Lemma 2.2.
Let G be a graph of order n and Γ be an arbitrary Abeliangroup of order n such that Γ ∼ = Z × Z × A for some Abelian group A oforder n . If all vertices of G have even degrees or all vertices of G have odddegrees, then there exists a Γ -distance magic labeling for the graph G × C . Proof.
Let V ( G ) = { x , x , . . . , x n − } be the vertex set of G , let C = u u u u u , and H = G × C . Let v i,j = ( x i , u j ) for i = 0 , , . . . , n − j = 0 , , ,
3. Recall that if x p x q ∈ E ( G ), then v q,j ∈ N H ( v p,j ) if and onlyif j ∈ { i − , i + 1 } (where the sum on the second suffix is taken modulo4). Since all vertices of G have even degrees or all vertices of G have odddegrees, thus deg( v ) ≡ c (mod 2) for some constant c and any v ∈ V ( G )5sing the isomorphism φ : Γ → A × Z × Z , we identify every g ∈ Γ withits image φ ( g ) = ( a i , j , j ), where j , j ∈ Z and a i ∈ A , i = 0 , , . . . , n − H in the following way: f ( v i,j ) = ( a i , ,
0) for j = 0 , ( a i , ,
0) for j = 1 , ( − a i , ,
1) for j = 2 , ( − a i , ,
1) for j = 3for i = 0 , , . . . , n − j = 0 , , , i = 0 , . . . , n − f ( v i, ) + f ( v i, ) = f ( v i, ) + f ( v i, ) = (0 , , . Therefore, for every x ∈ V ( H ), w ( x ) = (0 , c, c ) . Theorem 2.3.
Let G be a graph of order n . If n = 2 p (2 k + 1) for somenatural numbers p , k and deg( v ) ≡ c (mod 2 p +2 ) for some constant c forany v ∈ V ( G ) , then there exists a group distance magic labeling for the graph G × C . Proof.
The fundamental theorem of finite Abelian groups states that the finiteAbelian group Γ can be expressed as the direct sum of cyclic subgroups ofprime-power order. This implies that Γ ∼ = Z α × Z p α × Z p α × . . . × Z p αmm for some α >
0, where 4 n = 2 α Q mi =1 p α i i and p i for i = 1 , . . . , m are notnecessarily distinct primes.Suppose first that Γ ∼ = Z × Z ×A for some Abelian group A of order n , thenwe are done by Lemma 2.2. Observe now that the assumption deg( v ) ≡ c (mod 2 p +2 ) and the unique (additive) decomposition of any natural number c into powers of 2 imply that there exist constants c , c , . . . , c p such thatdeg( v ) ≡ c i (mod 2 i ), for i = 1 , , . . . , p + 1, for any v ∈ V ( G ). Hence ifΓ ∼ = Z α × A for some 2 ≤ α ≤ p + 2 and some Abelian group A of order n α , then we obtain by Lemma 2.1 that there exists a Γ-distance magiclabeling for the graph G × C . 6he following observation shows that in the general case the conditionon the degrees of the vertices v ∈ V ( G ) in Theorem 2.3 is not necessary forthe existence of a Γ-distance magic labeling of a graph G × C : Observation 2.4.
Let G = K p,q,t be a complete tripartite graph with allpartite set odd, then G × C is a group distance magic graph. Proof.
Since n = p + q + t is odd Γ ∼ = A × Z × Z or Γ ∼ = A × Z for some Abeliangroup A of order n = p + q + t If Γ ∼ = A × Z × Z for some Abelian group A of order n = p + q + t , then there exists a Γ-distance magic labeling for thegraph K p,q,t × C by Lemma 2.2. Suppose now that Γ ∼ = A × Z for someAbelian group A of order p + q + t . Let K p,q,t have the partition vertex sets A = { x , x , . . . , x p − } , B = { y , y , . . . , y q − } and C = { u , u , . . . , u t − } and let C = v v v v v . Without loosing generality we can assume that p ≡ q (mod 4) (by Pigeonhole Principle). Using the isomorphism φ : Γ →A × Z , we identify every g ∈ Γ with its image φ ( g ) = ( a i , j ), where j ∈ Z and a i ∈ A , i = 0 , , . . . , p + q + t − K p,q,t × C in the following way: f ( x i , v j ) = (cid:26) ( a i , j ) for j = 0 , , (0 , − f ( x i , v j − ) for j = 2 , i = 0 , , . . . , p − j = 0 , , , f ( y i , v j ) = (cid:26) ( a p + i , j ) for j = 0 , , (0 , − f ( y i , v j − ) for j = 2 , i = 0 , , . . . , q − j = 0 , , , f ( u i , v j ) = ( a p + q + i , j ) for j = 0 , , (0 , − f ( u i , v j − ) for j = 2 , , if t ≡ p (mod 4)(0 , − f ( u i , v j − ) for j = 2 , , if t + 2 ≡ p (mod 4)for i = 0 , , . . . , t − j = 0 , , , f ( x i , v j ) + f ( x i , v j +2 ) = f ( y l , v j ) + f ( y l , v j +2 ) = (0 , , i = 0 , . . . , p − l = 0 , . . . , q − j = 0 , f ( u i , v j ) + f ( u i , v j +2 ) = (cid:26) (0 , , if t ≡ p (mod 4) , (0 , , if t + 2 ≡ p (mod 4) , for every i = 0 , . . . , t − j = 0 ,
1. Since t ∈ { , } we obtain that w ( x ) = (0 ,
2) for every x ∈ V ( K p,q,t × C ).Notice that a graph K , α × C is group distance magic for any α ∈ N by Theorem 2.3, although a graph K , α × C is distance magic if andonly if α = 0 by Theorem 1.8.It is worthy to mention, that it was proved that if m is odd and n iseven, then a graph K m,n ◦ C is group distance magic (see [6]), however it isnot longer true in the case of direct product, what shows the bellow lemma.Recall that for a group Γ an involution ι ∈ Γ is a such element that ι = 0and 2 ι = 0. Lemma 2.5.
Let m and n be two positive integers such that m is odd and n is even, the graph K m,n × C is not a Γ -distance magic graph for any group Γ of order m + 4 n having exactly one involution ι . Proof.
Since there exists exactly one involution ι ∈ Γ notice that Γ = Z ×A for some group A of order m + n . Let K m,n have the partition vertex sets A = { x , x , . . . , x m − } and B = { y , y , . . . , y n − } and let C = v v v v v .Suppose that ℓ is a Γ-distance magic labeling of the graph K m,n × C and µ = w ( x ), for all vertices x ∈ V ( K m,n × C ). Notice that K m,n × C ∼ = 2 K m, n .We can assume that ( x i , v j ) , ( x i , v j +2 ) , ( y l , v j +1 ) , ( y l , v j +3 ) ∈ V ( K j p, q ) for i = 0 , , . . . , p − l = 0 , , . . . , q − j = 0 ,
1. It is easy to observe that: µ = m − X i =0 ( ℓ ( x i , v ) + ℓ ( x i , v )) = m − X i =0 ( ℓ ( x i , v ) + ℓ ( x i , v )) == n − X l =0 ( ℓ ( y l , v ) + ℓ ( y l , v )) = n − X l =0 ( ℓ ( y l , v ) + ℓ ( y l , v )) . Thus 4 µ = P g ∈ Γ g = ι . Since m + n is odd and Γ = Z × A such anelement µ ∈ Γ does not exist, a contradiction.
Corollary 2.6.
Let m and n be two positive integers such that m is oddand n is even, the graph K m,n × C is not a Z m +4 n -distance magic graph. roof. Since there exists exactly one involution ι = 2 n + 2 m in Z m +4 n weare done by Observation 2.5.We finish this section with the following theorem. Lemma 2.7.
Let m and n be two positive integers such that m is odd and n is even, the graph K m,n × C is a Γ -distance if and only if Γ ∼ = Z × Z × A for a group A of order m + n . Proof.
Since | V ( K m,n × C ) | = 4( m + n ) and m is even and n is oddΓ ∼ = Z × Z × A or Γ = Z × A for some group A of order m + n . IfΓ = Z ×A then there does not exist a Γ-labeling of K m,n × C by Lemma 2.5.Suppose now that Γ ∼ = Z × Z × A ∼ = A × Z × Z for a group A of order m + n . Let K m,n have the partition vertex sets A = { x , x , . . . , x m − } and B = { y , y , . . . , y n − } and let C = v v v v v . Using the iso-morphism φ : Γ → A × Z × Z , we identify every g ∈ Γ with its image φ ( g ) = ( a i , j , j ), where j , j ∈ Z and a i ∈ A , i = 0 , , . . . , m + n −
1. Since m + n is odd without loosing the generality that a = 0 and a n +1 = − a n = 0.Label the vertices of K m,n × C in the following way: f ( x , v ) = (0 , , f ( x , v ) = (0 , , f ( x , v ) = (0 , , f ( x , v ) = (0 , , f ( y , v ) = ( a n , , f ( y , v ) = ( a n , , f ( y , v ) = ( − a n , , f ( y , v ) = ( − a n , , f ( y , v ) = ( a n +1 , , f ( y , v ) = ( a n +1 , , f ( y , v ) = ( − a n +1 , , f ( y , v ) = ( − a n +1 , , f ( x i , v j ) = ( a i , ,
0) for j = 0 , ( a i , ,
0) for j = 1 , ( − a i , ,
1) for j = 2 , ( − a i , ,
1) for j = 3for i = 1 , , . . . , m − j = 0 , , , f ( y i , v j ) = ( a m + i , ,
0) for j = 0 , ( a m + i , ,
0) for j = 1 , ( − a m + i , ,
1) for j = 2 , ( − a m + i , ,
1) for j = 3for i = 2 , , . . . , n − j = 0 , , , f ( x i , v j ) + f ( x i , v j +2 ) = f ( y l , v j ) + f ( y l , v j +2 ) = (0 , , , for every i = 1 , , . . . , m − l = 2 , . . . , n − j = 0 , w ( x i , v j ) = ( n − , ,
1) + (0 , ,
1) = (0 , ,
1) for i = 0 , , . . . , m − j = 0 , , , w ( y l , v j ) = ( m − , , , ,
1) =(0 , ,
1) for i = 0 , , . . . , m − j = 0 , , , G × C In this section we show that some direct products G × C are group distancemagic. Used constructions are similar to those by Anholcer at al. in [3]. Westart with the following lemma: Lemma 3.1.
Let G be a graph of order n and Γ be an arbitrary Abeliangroup of order n such that Γ ∼ = Z × Z × A for some Abelian group A oforder n . If all vertices of G have even degrees, then there exists a Γ -distancemagic labeling for the graph G × C . Proof.
Let V ( G ) = { x , x , . . . , x n − } be the vertex set of G , let C = u u . . . u u , and H = G × C . Let v i,j = ( x i , u j ) for i = 0 , , . . . , n − j = 0 , , . . . ,
7. Notice that if x p x q ∈ E ( G ), then v q,j ∈ N H ( v p,i ) if and onlyif j ∈ { i − , i + 1 } (where the sum on the first suffix is taken modulo 8).Since all vertices of G have even degrees, thus deg( v ) = 2 l v any v ∈ V ( G ).Using the isomorphism φ : Γ → A × Z × Z , we identify every g ∈ Γ withits image φ ( g ) = ( a i , j , j ), where j , j ∈ Z and a i ∈ A , i = 0 , , . . . , n − j ∈ { , , . . . , n − } we set f ( v i,j ) = ( a i + j , , , if i ∈ { , } , ( a i + j − , , , if i ∈ { , } , (0 , , − f ( v i,j − ) , if i ∈ { , , , } . Clearly, f : V ( C × G ) → Γ is a bijection and f ( v i,j ) + f ( v i,j +2 ) = (0 , y j ),where y j ∈ { (1 , , (1 , } , and so 2 y j = (0 , i ∈{ , , . . . , n − } and j ∈ { , , . . . , } we get w ( v i,j ) = X x p ∈ N G ( x i ) ( f ( v p,j − ) + f ( v p,j +1 )) = X x p ∈ N G ( x i ) (0 , y j − ) == l v i,j (0 , ,
0) = (0 , , G × C is Γ-distance magic. Theorem 3.2.
Let G be a graph of order n . If n = 2 p (2 k + 1) for somenatural numbers p , k and deg( v ) ≡ c (mod 2 p +2 ) for some constant c forany v ∈ V ( G ) , then there exists a group distance magic labeling for the graph G × C .Proof. Let V ( G ) = { x , x , . . . , x n − } be the vertex set of G , let C = u u . . . u u , and H = G × C . Let v i,j = ( x i , u j ) for i = 0 , , . . . , n − j = 0 , , . . . ,
7. Notice that if x p x q ∈ E ( G ), then v q,j ∈ N H ( v p,j ) if and onlyif j ∈ { i − , i + 1 } (where the sum on the first suffix is taken modulo 8).Recall that the assumption deg( v ) ≡ c (mod 2 p +2 ) and the unique (ad-ditive) decomposition of any natural number c into powers of 2 imply thatthere exist constants c , c , . . . , c p such that deg( v ) ≡ c i (mod 2 i ), for i = 1 , , . . . , p + 1, for any v ∈ V ( G ).We are going to consider three cases, depending on the structure of Γ. Case 1: Γ ∼ = Z × Z ×A for some Abelian group of order 2 n .There exists a Γ-distance magic labeling of G × C by Lemma 3.1. Case 2: Γ ∼ = Z ×A for some Abelian group A of order 2 n .Using the isomorphism φ : Γ → A × Z , we identify every g ∈ Γ with itsimage φ ( g ) = ( a i , w ), where w ∈ Z and a i ∈ A , i = 0 , , . . . , n − i ∈ { , , . . . , n − } we define f ( v i,j ) = ( a i + j , , if j ∈ { , } , ( a i + j − , , if j ∈ { , } , (0 , − f ( v i,j − ) , if j ∈ { , , , } . Again f : V ( G × C ) → Γ is obviously a bijection and f ( v i,j ) + f ( v i,j +2 ) =( y i , y j ∈ { , } , and thus 2 y j = 2. Since deg( v ) ≡ c (mod 2 )for any v ∈ V ( G ), for every i ∈ { , , . . . , n − } and j ∈ { , , . . . , } we get w ( v i,j ) = X x p ∈ N G ( x i ) ( f ( v p,j − )+ f ( v p,j +1 )) = X x p ∈ N G ( x i ) (0 , y j − ) = c (0 ,
2) = (0 , c )and G × C is Γ-distance magic. Case 3: Γ ∼ = Z α ×A for 2 < α ≤ p and some Abelian group A of order n α − . 11sing the isomorphism φ : Γ → A × Z α , we identify every g ∈ Γ with itsimage φ ( g ) = ( a i , w ), where w ∈ Z α and a i ∈ A for i ∈ { , , . . . , n α − − } .For i ∈ { , , . . . , n α − − } define the following labeling f : f ( v i,j ) = (cid:0) a ⌊ i · − α +3 ⌋ , (2 i + j )(mod 2 α − ) (cid:1) , if j ∈ { , } , (cid:0) , α − ) + f ( v i,j − (cid:1) , if j ∈ { , } , (0 , α − − f ( v i,j − ) , if j ∈ { , , , } . As in previous cases f : V ( G × C ) → Γ is a bijection and f ( v i,j )+ f ( v i,j +2 ) =(0 , y j ) for some y j ∈ { α − − , α − } . Thus 2( f ( v i,j ) + f ( v i,j +2 )) =(0 , y j ) = (0 , − j ∈ { , , . . . , } and i ∈ { , , . . . , n − } weget w ( v i,j ) = X x p ∈ N G ( x i ) ( f ( v p,j − ) + f ( v p,j +1 )) = X x p ∈ N G ( x i ) (0 , y j − ) == c α (0 , −
2) = (0 , − c α )and G × C is Z α ×A -distance magic since r is even.In the proof of the below observation we use similar methods to thosepresented in [5]. Observation 3.3.
Let m and n be two positive integers such that m ≤ n .If the graph K m,n × C is a distance magic graph, then the conditions hold: • m + n ≡ and • m ≥ √ n +1) − − − n. Proof.
Let K m,n have the partition vertex sets A = { x , x , . . . , x m − } and B = { y , y , . . . , y n − } and let C = v v v . . . v v . Suppose that ℓ is a dis-tance magic labeling of the graph K m,n × C and µ = w ( x ), for all vertices x ∈ V ( K m,n × C ). We can assume that ( x i , v j ) , ( x i , v j +2 ) , ( y l , v j +1 ) , ( y l , v j +3 ) ∈ V ( K j p, q ) for i = 0 , , . . . , p − l = 0 , , . . . , q − i = 0 , , . . . ,
7. It iseasy to observe that: µ = m − X i =0 ( ℓ ( x i , v ) + ℓ ( x i , v )) = m − X i =0 ( ℓ ( x i , v ) + ℓ ( x i , v )) == m − X i =0 ( ℓ ( x i , v ) + ℓ ( x i , v )) = m − X i =0 ( ℓ ( x i , v ) + ℓ ( x i , v )) =12 n − X l =0 ( ℓ ( y l , v ) + ℓ ( y l , v )) = n − X l =0 ( ℓ ( y l , v ) + ℓ ( y l , v )) == n − X l =0 ( ℓ ( y l , v ) + ℓ ( y l , v )) = n − X l =0 ( ℓ ( y l , v ) + ℓ ( y l , v )) == X x ∈ V ( K m,n × C ) l ( x )4 = n +8 m X i =1 i m + 8 n )(8 m + 8 n + 1)16 , which implies that m + n ≡ P m − i =0 ( ℓ ( x i , v ) + ℓ ( x i , v ) + . . . + ℓ ( x i , v ) + ℓ ( x i , v )) ≤ P mi =1 ( i + 8 n ) = 4 m (8 m + 16 n + 1), thus µ ≤ m (8 m + 16 n + 1). Whichimplies ( m + n )(8 m + 8 n + 1) ≤ m (8 m + 16 n + 1) and therefore:2[2 m + (2 n + 18 )] ≥ n (8 n + 1) + 2(2 n + 18 ) = (4 n + 14 ) − . That is: 1 ≥ (16 n + 1) − m + 8 n + 12 ) , Therefore, either 2 = 2(16 n + 1) − (16 m + 16 n + 1) or m ≥ √ n +1) − − − n. Suppose that 1 ≥ n + 1) − (16 m + 16 n + 1) , then the diophantineequation 2 = 2 x − y needs to have a solution such that x and y are bothodd, a contradiction.Notice that a graph K , α × C is group distance magic for any α ∈ N by Theorem 3.2, although a graph K , α × C is not distance magic for α > References [1] M. Anholcer, S. Cichacz, A. G˝orlich,
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