Distance labelings: a generalization of Langford sequences
aa r X i v : . [ m a t h . C O ] J un DISTANCE LABELINGS: A GENERALIZATION OF LANGFORD SEQUENCES
S. C. L ´OPEZ AND F. A. MUNTANER-BATLE
Abstract.
A Langford sequence of order m and defect d can be identified with a labeling of thevertices of a path of order 2 m in which each labeled from d up to d + m − k are at distance k . In this paper, we introduce twogeneralizations of this labeling that are related to distances. Key words:
Langford sequence, distance l -labeling, distance J -labeling δ -sequence and δ -set. Introduction
For the graph terminology not introduced in this paper we refer the reader to [9, 10]. For m ≤ n ,we denote the set { m, m + 1 , . . . , n } by [ m, n ]. A Skolem sequence [5, 6] of order m is a sequenceof 2 m numbers ( s , s , . . . , s m ) such that (i) for every k ∈ [1 , m ] there exist exactly two subscripts i, j ∈ [1 , m ] with s i = s j = k , (ii) the subscripts i and j satisfy the condition | i − j | = k . Thesequence (4 , , , , , , ,
1) is an example of a Skolem sequence of order 4. It is well known thatSkolem sequences of order m exist if and only if m ≡ hooked Skolem sequence of order m , where there existsa zero at the second to last position of the sequence containing 2 m + 1 elements. Later on, in 1981,Abrham and Kotzig [1] introduced the concept of extended Skolem sequence , where the zero is allowedto appear in any position of the sequence. An extended Skolem sequence of order m exists for every m . The following construction was given in [1]:(1) ( p m , p m − , p m − , . . . , , , , . . . , p m − , p m , q m , q m − , q m − , . . . , , , , , . . . , q m − , q m ) , where p m and q m are the largest even and odd numbers not exceeding m , respectively. Notice thatfrom every Skolem sequence we can obtain two trivial extended Skolem sequences just by adding azero either in the first or in the last position.Let d be a positive integer. A Langford sequence of order m and defect d [8] is a sequence ( l , l , . . . , l m )of 2 m numbers such that (i) for every k ∈ [ d, d + m −
1] there exist exactly two subscripts i, j ∈ [1 , m ]with l i = l j = k , (ii) the subscripts i and j satisfy the condition | i − j | = k . Langford sequences, for d = 2, where introduced in [4] and they are referred to as perfect Langford sequences . Notice that, aLangford sequence of order m and defect d = 1 is a Skolem sequence of order m . Bermond, Browerand Germa on one side [2], and Simpson on the other side [8] characterized the existence of Langfordsequences for every order m and defect d . Theorem 1.1. [2, 8]
A Langford sequence of order m and defect d exists if and only if the followingconditions hold: (i) m ≥ d − , and (ii) m ≡ or (mod ) if d is odd; m ≡ or (mod ) if d iseven. For a complete survey on Skolem-type sequences we refer the reader to [3].
Date : October 5, 2018.
Distance labelings.
Let L = ( l , l , . . . , l m ) be a Langford sequence of order m and defect d .Consider a path P with V ( P ) = { v i : i = 1 , , . . . , m } and E ( P ) = { v i v i +1 : i = 1 , , m − } .Then, we can identify L with a labeling f : V ( P ) → [ d, d + m −
1] in such a way that, (i) for every k ∈ [ d, d + m −
1] there exist exactly two vertices v i , v j ∈ [1 , m ] with f ( v i ) = f ( v j ) = k , (ii) thedistance d ( v i , v j ) = k . Motivated by this fact, we introduce two notions of distance labelings, one ofthem associated with a positive integer l and the other one associated with a set of positive integers I .Let G be a graph and let l be a positive integer. Consider any function f : V ( G ) → [0 , l ]. We saythat f is a distance labeling of length l (or distance l -labeling ) of G if the following two conditionshold, (i) either f ( V ( G )) = [0 , l ] or f ( V ( G )) = [1 , l ] and (ii) if there exist two vertices v i , v j with f ( v i ) = f ( v j ) = k then d ( v i , v j ) = k . Clearly, a graph can have many different distance labelings.We denote by λ ( G ), the labeling length of G , the minimum length l for which a distance l -labeling of G exists. We say that a distance l -labeling of G is proper if for every k ∈ [1 , l ] there exist at leasttwo vertices v i , v j of G with f ( v i ) = f ( v j ) = k . We also say that a proper distance l -labeling of G is regular of degree r (for short r -regular ) if for every k ∈ [1 , l ] there exist exactly r vertices v i , v i , . . . , v i r with f ( v i ) = f ( v i ) = . . . = f ( v i r ) = k . Clearly, if a graph G admits a proper distance l -labelingthen l ≤ D ( G ), where D ( G ) is the diameter of G .Let G be a graph and let J be a set of nonnegative integers. Consider any function f : V ( G ) → J .We say that f is a distance J -labeling of G if the following two conditions hold, (i) f ( V ( G )) = J and (ii) for any pair of vertices v i , v j with f ( v i ) = f ( v j ) = k we have that d ( v i , v j ) = k . We saythat a distance J -labeling is proper if for every k ∈ J \ { } there exist at least two vertices v i , v j with f ( v i ) = f ( v j ) = k . We also say that a proper distance J -labeling of G is regular of degree r (for short r -regular ) if for every k ∈ J \ { } there exist exactly r vertices v i , v i , . . . , v i r with f ( v i ) = f ( v i ) = . . . = f ( v i r ) = k . Clearly, a distance l -labeling is a distance J -labeling in whicheither J = [0 , l ] or J = [1 , l ]. Thus, the notion of a J -labeling is more general than the notion of a l -labeling.In this paper, we provide the labeling length of some well known families of graphs. We also studythe inverse problem, that is, for a given pair of positive integers l and r we ask for the existence ofa graph of order lr with a regular l -labeling of degree r . Finally, we study a similar question whenwe deal with J -labelings. The organization of the paper is the following one. Section 2 is devoted to l -labelings, we start calculating the labeling length of complete graphs, paths, cycles and some othersfamilies. The inverse problem is studied in the second part of the section. Section 3 is devoted to theinverse problem in J -labelings. There are many open problem that remain to be solve, we end thepaper by presenting some of them. 2. Distance l -labelings We start this section by providing the labeling length of some well known families of graphs.
Proposition 2.1.
The complete graph K n has λ ( K n ) = 1 .Proof. By assigning the label 1 to all vertices of K n , we obtain a distance -labeling of it. ✷ Proposition 2.2.
The path P n has λ ( P n ) = ⌊ n/ ⌋ .Proof. By previous comment, we know that a Skolem sequence of order m exists if m ≡ ⌊ n/ ⌋ -labeling when n , ISTANCE LABELINGS: A GENERALIZATION OF LANGFORD SEQUENCES 3 ⌊ n/ ⌋ . Thus, we have that λ ( P n ) ≤ ⌊ n/ ⌋ . Since, there are not three vertices in the path which areat the same distance, this lower bound turns out to be an equality. ✷ The sequence that appears in (1) also works for constructing proper distance labelings of cycles. Thus,we obtain the next result.
Proposition 2.3.
Let n be a positive integer. The cycle C n has λ ( C n ) = ⌊ n/ ⌋ .Proof. Since, except for C there are not three vertices in the cycle which are at the same distance, we havethat λ ( C n ) ≥ ⌊ n/ ⌋ . The sequence that appears in (1) allows us to construct a (proper) distance ⌊ n/ ⌋ -labeling of C n when n is odd. Moreover, if n is even we can obtain a distance ⌊ n/ ⌋ -labeling of C n from the sequence that appears in (1) just by removing the end odd label. ✷ Proposition 2.4.
The star K ,k has λ ( K ,k ) = 2 , when k ≥ , and λ ( K ,k ) = 1 otherwise.Proof. For k ≥
3, consider a labeling f that assigns the label 1 to the central vertex and to one of its leaves,and that assigns label 2 to the other vertices. Then f is a (proper) distance 2-labeling of K ,k . For1 ≤ k ≤
2, the sequences 1 − − −
1, where 0 is assigned to a leaf, give a (proper) distance1-labeling of K , and K , , respectively. ✷ Proposition 2.5.
Let m and n be integers with ≤ m ≤ n . Then, λ ( K m,n ) = m . In particular, thegraph K m,n admits a proper distance l -labeling if and only if, ≤ m ≤ .Proof. Let X and Y be the stable sets of K m,n , with | X | ≤ | Y | . We have that D ( K m,n ) = 2, however themaximum number of vertices that are mutually at distance 2 is n . Thus, by assigning label 2 to allvertices, except one, in Y , 1 to the remaining vertex in Y and to one vertex in X , 0 to another vertexof X we still have left m − X to label. ✷ Proposition 2.6.
Let n and k be positive integers with k ≥ . Let S nk be the graph obtained from K ,k by replacing each edge with a path of n + 1 vertices. Then λ ( S nk ) = n − , if k = n − , n − , if k = n, n, if k > n. Proof.
Suppose that S nk admits a distance l -labeling with l < n . Then, all the labels assigned to leavesshould be different. Moreover, although each even label could appear k -times, one for each of the k paths that are joined to the star K ,k , odd labels appear at most twice (either in the same or intwo of the original forming paths). Thus, at least 2 n − S nk . The following construction provides a distance 2( n − S nk , when k = n − − − . . . − n − k = n , weneed to introduce a new odd label, which corresponds to 2 n −
1. Finally, when k > n , we cannotcomplete a distance l -labeling without using 2 n labels. Fig. 3 provides a proper 2 n -labeling that canbe generalized in that case. ✷ S. C. L ´OPEZ AND F. A. MUNTANER-BATLE
Fig. 1 and 2 show proper distance labelings of S and S , respectively, that have been obtained byusing the above constructions, and then, combining pairs of paths (whose end odd labels sum up to8) for obtaining a proper distance 8-labeling and 9-labeling, respectively. b b b b b b b b b b b b b b b b b b b b b Figure 1.
A proper distance 8-labeling of S . b b b b b b b b b b b b b b b b b b b b b b b b b b Figure 2.
A proper distance 9-labeling of S . b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Figure 3.
A proper distance 10-labeling of S .The case k < n − S nk obtained by assigning the labels in the sequence 0 − − − . . . − n − o ) − s i − s i − s io to the verticesof the path P i , i = 1 , . . . , k , where 0 is the label assigned to the central vertex of S nk , and { s ij } j =1 ,...,oi =1 ,...,k is the (multi)set of odd labels. By considering the patern 1 −
1, 3 − − −
3, 5 − − − − − S nk admits an l -distance labelingwith l ∈ { n − o ) , n − o ) + 1 } and ⌊ n − k + 1 ⌋ ≤ o ≤ ⌊ n + 22 k + 1 ⌋ . Thus, according to the proof of Proposition 2.6, we strongly suspect that2( n − o ) ≤ λ ( S nk ) ≤ n − o ) + 1 , where ⌊ (2 n − / (2 k + 1) ⌋ ≤ o ≤ ⌊ (2 n + 2) / (2 k + 1) ⌋ . Proposition 2.7.
For n ≥ , let W n be the wheel of order n + 1 . Then λ ( W n ) = ⌈ n/ ⌉ . Proof.
Except for W , all wheels have D ( W n ) = 2. The maximum number of vertices that are mutually atdistance 2 is ⌊ n/ ⌋ and all of them are in the cycle. Thus, by assigning label 2 to all these vertices, 0to one vertex of the cycle and 1 to the central vertex and to one vertex of the cycle, we still have tolabel ⌈ n/ ⌉ − ✷ . ISTANCE LABELINGS: A GENERALIZATION OF LANGFORD SEQUENCES 5
Proposition 2.8.
For n ≥ , let F n be the fan of order n + 1 . Then λ ( F n ) = ⌊ n/ ⌋ . Proof.
Except for F , all fans have D ( F n ) = 2. The maximum number of vertices that are mutually atdistance 2 is ⌈ n/ ⌉ and all of them are in the path. Thus, by assigning label 2 to all these vertices, 0to one vertex of the path, 1 to the central vertex and to one vertex of the path when n is even and totwo vertices when n is odd, we still have to label ⌈ n/ ⌈− ✷ The inverse problem.
For every positive integer l , there exists a graph G of order l with atrivial l -labeling that assigns a different label in [1 , l ] to each vertex. In this section, we are interestedon the existence of a graph G that admits a proper distance l -labeling.We are now ready to state and prove the next result. Theorem 2.1.
For every pair of positive integers l and r , there exists a graph G of order lr with aregular l -labeling of degree r .Proof. We give a constructive proof. Assume first that l is odd. Let G be the graph obtained from thecomplete graph K r by identifying r − K r with one of the end vertices of a path of length ⌊ l/ ⌋ and the remaining vertex of K r with the central vertex of the graph S ⌊ l/ ⌋ r +1 . That is, G is obtainedfrom K r by attaching 2 r paths of length ⌊ l/ ⌋ to its vertices, r + 1 to a particular vertex v of K r and exactly one path to each of the remaining vertices F = { v , v , . . . , v r } of K r . Now, consider thelabeling f of G that assigns 1 to the vertices of K r , the sequence 1 − − . . . − l to the vertices of thepaths attached to F and one of the paths attached to v , and the sequence 1 − − − . . . − ( l −
1) tothe remaining paths. Then f is a regular l -labeling of degree r of G . Assume now that l is even. Let G be the graph obtained in the above construction for l −
1. Then, by adding a leave to each vertexof G labeled with l − G ′ that admits a regular l -labeling f ′ of degree r . Thelabeling f ′ can be obtained from the labeling f of G , defined above, just by assigning the label l tothe new vertices. ✷ b b b b b b b b b b b b b b b b b b b b Figure 4.
A regular 5-labeling of degree 4 of a graph G .Notice that, the graph provided in the proof of Theorem 2.1 also has λ ( G ) = l . Fig. 4 and 5 showexamples for the above construction. The pattern provided in the proof of the above theorem, for r = 2, can be modified in order to obtain the following lower bound for the size of a graph G as inTheorem 2.1. Proposition 2.9.
For every positive integer l there exists a graph of order l and size ( l +2)( l +1) / − that admits a regular distance l -labeling of degree . S. C. L ´OPEZ AND F. A. MUNTANER-BATLE b b b b b b b b b b b b b b b b b b b b b b b b Figure 5.
A regular 6-labeling of degree 4 of a graph G ′ . Proof.
Let G be the graph of order 2 l and size ( l + 1) l/ l −
1, obtained from K l +1 and the path P l byidentifying one of the end vertices u of P l with a vertex v of K l +1 . Let f be the labeling of G thatassigns the sequence 1 − − − . . . − l to the vertices of P l and 1 − − − . . . − l to the vertices of K l +1 in such a way that f ( u ≡ v ) = 1. Then, f is a 2-regular l -labeling of G . ✷ Thus, a natural question appears.
Question 2.1.
Can we find graphs that admit a regular distance l -labeling of degree which are moredense that the one of Proposition 2.9? Distance J -labelings It is clear from definition that to say that a graph admits a (proper) distance l -labeling is the sameas to say that the graph admits a (proper) distance [0 , l ]-labeling. That is, we relax the condition onthe labels, the set of labels is not necessarily a set of consecutive integers. In this section, we studywhich kind of sets J can appear as a set labels of a graph that admits a distance J -labeling.The following easy fact is obtained from the definition. Lemma 3.1.
Let G be a graph with a proper distance J -labeling f . Then J ⊂ [0 , D ( G )] , where D ( G ) is the diameter of G . The inverse problem: distance J -labelings obtained from sequences. We start witha definition. Let S = ( s , s , . . . , s , s , . . . , s , . . . , s l , . . . , s l ) be a sequence of nonnegative integerswhere, (i) s i < s j whenever i < j and (ii) each number s i appears k i times, for i = 1 , , . . . , l . We saythat S is a δ -sequence if there is a simple graph G that admits a partition of the vertices V ( G ) = ∪ li =1 V i such that, for all i ∈ { , , . . . , l } , | V i | = k i , and if u, v ∈ V i then d G ( u, v ) = s i . The graph G is saidto realize the sequence S .Let Σ = { s < s < . . . < s l } be a set of nonnegative integers. We say that Σ is a δ -set with n degreesof freedom or a δ n -set if there is a δ -sequence S of the form S = ( s , s , . . . , s , s , . . . , s , . . . , s l , . . . , s l ),in which the following conditions hold: (i) all except n numbers different from zero appear at leasttwice, and (ii) if s = 0 then 0 appears exactly once in S . We say that any graph realizing S also realizes Σ. If n = 0 we simply say that Σ is a δ -set . Let us notice that an equivalent definition fora δ -set is the following one. We say that Σ is a δ -set if there exists a graph G that admits a properdistance Σ-labeling. Proposition 3.1.
Let
Σ = { s < s < . . . < s l } be a set such that s i − s i − ≤ , for i = 1 , , . . . , l .Then Σ is a δ -set. Furthermore, there is a caterpillar of order l that realizes Σ . ISTANCE LABELINGS: A GENERALIZATION OF LANGFORD SEQUENCES 7
Proof.
We claim that for each set Σ = { s < s < . . . < s l } such that s i − s i − ≤ l that admits a 2-regular distance Σ-labeling in which the label s l is assigned toexactly two leaves. The proof is by induction. For l = 1, the path P admits a 2-regular distance { } -labeling, and for l = 2, the star K , and the path P admit a 2-regular distance { , } -labelingand a 2-regular distance { , } -labeling, respectively. Assume that the claim is true for l and letΣ = { s < s < . . . < s l +1 } such that s i − s i − ≤
2. Let Σ ′ = Σ \ { s l +1 } . By the inductionhypothesis, there is a caterpillar G ′ of order 2 l that admits a regular distance Σ-labeling of degree 2in which the label s l is assigned to leaves, namely, u and u . Let u ∈ V ( G ′ ) be the (unique) vertex in G ′ adjacent to u . Then, if s l +1 − s l = 2, the caterpillar obtained from G ′ by adding two new vertices v and v and the edges u i v i , for i = 1 ,
2, admits a regular distance Σ-labeling of degree 2 in whichthe label s l +1 is assigned to leaves { v , v } . Otherwise, if s l +1 − s l = 1 then the caterpillar obtainedfrom G ′ by adding two new vertices v and v and the edges uv and u v admits a regular distanceΣ-labeling of degree 2 in which the label s l +1 is assigned to leaves { v , v } . This proves the claim. Tocomplete the proof, we only have to consider the vertex partition of G defined by the vertices thatreceive the same label. ✷ Proposition 3.1 provides us with a family of δ -sets, in which, if we order the elements of each δ -set, weget that the differences between consecutive elements are upper bounded by 2. This fact may lead usto get the idea that the differences between consecutive elements in δ -sets cannot be too large. Thisis not true in general and we show it in the next result. Theorem 3.1.
Let { k < k < . . . < k n } be a set of positive integers. Then there exists a δ -set Σ = { s < s < . . . < s l } and a set of indices { ≤ j < j < . . . < j n } , with j n < l − , such that s j +1 − s j = k , s j +1 − s j = k , . . . , s j n +1 − s j n = k n . Moreover, s can be chosen to be any positive integer.Proof. Choose any number d ∈ N and choose any Langford sequence of defect d . We let d = s . (Noticethat if d = 1 then the sequence is actually a Skolem sequence). Let this Langford sequence be L .Next, choose a Langford sequence L with defect max L + k . Next, choose a Langford sequence L with defect max L + k . Keep this procedure until we have used all the values k , k , . . . , k n . At thispoint create a new sequence L , where L is the concatenation of L , L , . . . , L n +1 and label the verticesof the path P r , r = P n +1 i =1 | L i | , with the elements of L keeping the order in the labeling induced bythe sequence L . This shows the result. ✷ The next result shows that there are sets that are not δ -sets. Proposition 3.2.
The set
Σ = { , } is not a δ -set.Proof. The proof is by contradiction. Assume to the contrary that Σ = { , } is a δ -set. That is to say,we assume that there exists a sequence S consisting of k copies of 2 and k copies of 3 that is a δ -sequence. Let G be a graph that realizes S and V ∪ V the partition of V ( G ) defined as follows: if u, v ∈ V i then d G ( u, v ) = i + 1, for i = 1 ,
2. It is clear that V must be formed by the leaves of a starwith center some vertex a ∈ V . Since a is at distance 1 of any vertex in V , it follows that a mustbe in V and furthermore, all vertices adjacent to a must be in V . Thus, there are no two adjacentvertices in the neighborhood of a . At this point, let b ∈ V \ { a } . Then, there is a path of the form a, u , u , b , where u ∈ V and hence, u , b ∈ V . This contradicts the fact that d G ( u , b ) = 1. ✷ The above proof works for any set of the form Σ = { , n } , for n ≥
3. Thus, in fact, Proposition 3.2can be generalized as follows.
S. C. L ´OPEZ AND F. A. MUNTANER-BATLE
Proposition 3.3.
The set
Σ = { , n } is not a δ -set. Notice that, although Σ = { , n } is not a δ -set, it is a δ -set, since we can consider a star in whichthe center is labeled with n and the leaves with 2.The next result gives a lower bound on the size of δ -sets in terms of the maximum of the set. Theorem 3.2.
Let Σ be a δ -set with s = max Σ . Then, | Σ | ≥ ⌈ ( s + 1) / ⌉ .Proof. Let Σ be a δ -set with s = max Σ. Let G be a graph that realizes Σ and let V ( G ) = ∪ i ∈ Σ V i bethe partition defined as follows: if u, v ∈ V i then d G ( u, v ) = i . Let a , a ∈ V s . At this point, let P = a = b b . . . b s +1 = a be a path of length s starting at a and ending at a . We claim thatthere are no three vertices in V ( P ) belonging to the same set V j , j ∈ Σ. We proceed by contradiction.Assume to the contrary that there exist vertices u, v and w ∈ V ( P ) such that u, v, w ∈ V j . That is, d G ( u, v ) = d G ( u, w ) = d G ( v, w ) = j . However, it is clear that the above cannot happen if we take thedistances in the path. That is to say, the following is impossible: d P ( u, v ) = d P ( v, w ) = d P ( v, w ) = j .Without loss of generality assume that d p ( u, v ) = k = j . Clearly, k > j . Otherwise, we have that d G ( u, v ) ≤ k instead of d G ( u, v ) = j . Let P ′ be a path in G of length j that joins u and v . Then,the subgraph of G obtained from P by substituting the subpath of P joining u and v by P ′ containsa subpath of length strictly smaller than s . Thus, we obtain a contradiction. Hence, each set in thepartition of V ( G ) can contain at most two vertices of P . Since | V ( P ) | = s + 1, it follows that we needat least ⌈ ( s + 1) / ⌉ sets in the partition of V ( G ). Therefore, we obtain that | Σ | ≥ ⌈ ( s + 1) / ⌉ . ✷ It is clear that the above proof cannot be improved in general, since from Proposition 3.1 we get thatthe any set of the form { , , . . . , n +1 } is a δ -set and |{ , , . . . , n +1 }| = ⌈ (2 n +2) / ⌉ . Furthermore,Proposition 3.3 is an immediate consequence of the above result. It is also worth to mention that thereare sets which meet the bound provided in Theorem 3.2, however they are not δ -sets. For instance,the set { , } considered in Lemma 3.2. From this fact, we see that we cannot characterize δ -setsfrom, only, a density point of view. Next we want to propose the following open problem. Open problem 3.1.
Characterize δ -sets. Let Σ be a set. By construction, a path of order | Σ | in which each vertex receives a different labelingof Σ defines a distance | Σ | -labeling. That is, every set is a δ | Σ | -set. So, according to that, we proposethe next problem. Open problem 3.2.
Given a set Σ is there any construction that provides the minimum r such that Σ is a δ r -set. Thus, the above problem is a bit more general than Problem 3.1.
Acknowledgements
The research conducted in this document by the first author has been supportedby the Spanish Research Council under project MTM2011-28800-C02-01 and symbolically by theCatalan Research Council under grant 2014SGR1147.
References [1] J. Abrham and A. Kotzig, Skolem sequences and additive permutations, Discrete Math. 37 (1981) 143–146.[2] J.C. Bermond, A.E. Brouwer and A. Germa, Syst`emes de triples et diff´erences associ´ees, Proc. Colloque C.N.R.S.- Probl´emes combinatoires et th´eorie des graphes. Orsay 1976, 35–38.[3] N. Franceti´c and E. Mendelsohn, A survey of Skolem-type sequences and Rosa’s use of them, Math. Slovaca 59(2009) 39–76.[4] C.D. Langford, Problem, Mathematical Gazette 42 (1958) 228.[5] R.S. Nickerson and D.C.B. Marsh, Problem e1845, The American Mathematical Monthly 74 (1967) no. 5,591–592.
ISTANCE LABELINGS: A GENERALIZATION OF LANGFORD SEQUENCES 9 [6] T. Skolem, On certain distributions of integers into pairs with given differences, Conf. Number. Math. Winnipeg,1971, 31–42.[7] T. Skolem, Some remarks on the triple systems of Steiner, Math Scand. 6 (1958) 273–280.[8] J.E. Simpson, Langford sequences: perfect and hooked, Discrete Math. 44 (1983) 97–104.[9] W.D. Wallis, Magic graphs, Birkha¨user, Boston, 2001.[10] D.B. West, Introduction to graph theory, Prentice Hall, INC. Simon & Schuster, A Viacom Company upperSaddle River, NJ07458, 1996.
Departament de Matem`atica Aplicada IV, Universitat Polit`ecnica de Catalunya. BarcelonaTech, C/EsteveTerrades 5, 08860 Castelldefels, Spain
E-mail address : [email protected] Graph Theory and Applications Research Group, School of Electrical Engineering and Computer Science,Faculty of Engineering and Built Environment, The University of Newcastle, NSW 2308 Australia
E-mail address ::