SURVEY OF EDGE ANTIMAGIC LABELINGS OF GRAPHS
J. Indones. Math. Soc. (MIHMI) Vol. 12, No. 1 (2006), pp. 113–130.
SURVEY OF EDGE ANTIMAGIC LABELINGS
1 OF GRAPHS
2
3 3 M. Baˇ ca , E.T. Baskoro , M. Miller , J. Ryan , 2 3 Abstract.R. Simanjuntak , and K.A. Sugeng
An (a, d)-edge-antimagic total labeling of G is a one-to-one mapping f taking fthe vertices and edges onto 1, 2, . . . , |V (G)| + |E(G)| so that the edge-weights w(uv) =
(u) + f (v) + f (uv) : uv ∈ E(G), form an arithmetic progression with initial term a and
common difference d. Such a labeling is called super if the smallest possible labels appear
on the vertices. In this paper we survey what is known about edge-antimagic total and
super edge-antimagic total labelings and provide a summary of current conjectures and
open problems.1. INTRODUCTION All graphs in this paper are finite, undirected, and simple. A graph G has vertex set V (G) and edge set E(G). We follow the notation and terminology of
Wallis [31] and West [33].
A labeling of a graph is any mapping that sends some set of graph elements to a set of positive integers. If the domain is the vertex-set or the edge-set, the labelings are called, respectively, vertex labelings or edge labelings. Moreover, if the domain is V (G) ∪ E(G) then the labelings are called total labelings.
Hartsfield and Ringel in [16] introduced the concept of an antimagic graph. In their terminology, a graph G(V, E) is called antimagic if its edges are labeled with labels 1, 2, . . . , |E(G)| in such a way that all vertex-weights are pairwise distinct, where a vertex-weight of vertex v is the sum of labels of all edges incident with v.
Hartsfield and Ringel [16] point out that among the antimagic graphs are paths P n , n ≥ 3, cycles, wheels, and complete graphs K n , n ≥ 3. They conjecture that every connected graph, except K 2 , is antimagic.
Received 13 August 2005, Revised 28 October 2005, Accepted 10 Nov 2005. 2000 Mathematics Subject Classification Key words and Phrases : 05C78.
: Edge-antimagic total labeling, Super edge-antimagic total labeling.
M. Baˇ ca, et al.
Alon, Kaplan, Lev, Roditty and Yuster [2] used probabilistic arguments with tools from analytic number theory to show that this conjecture is true for all graphs having minimum degree Ω(log |V (G)|). They also prove that if G is a graph with |V (G)| ≥ 4 vertices and maximum degree ∆(G) ≥ |V (G)| − 2 then G is antimagic and that all complete partite graphs, except K , are antimagic.
2 It is not difficult to produce many antimagic labelings for most graphs. Thus Bodendiek and Walther [10] introduced further restriction on the vertex-weights.
They defined the concept of an (a, d)-antimagic labeling as an edge labeling in which the vertex-weights form an arithmetic progression starting from a and having common difference d. In [18] this labeling is called (a, d)-vertex-antimagic edge labeling.
For a graph G(V, E), an injective mapping from V (G) ∪ E(G) onto the set {1, 2, . . . , |V (G)| + |E(G)|} is an (a, d)-vertex-antimagic total labeling if the set of all vertex-weights is {a, a + d, a + 2d, . . . , a + (|V (G)| − 1)d}. The (a, d)-vertex- antimagic total labeling was introduced by Baˇca et al. [4] as a natural extension of the notion of vertex-magic total labeling defined by MacDougall et al. in [19] and [20]. Results about (a, d)-antimagic labelings and (a, d)-vertex-antimagic total labelings can be found in the general survey of Gallian [14].
In this paper we concentrate on a variation of antimagic labeling, where we consider the sum of all labels associated with an edge. We define edge-weight of an edge uv under a vertex labeling to be the sum of the vertex labels corresponding to the vertices u and v. Under a total labeling, we also add the label of the edge uv.
By an (a, d)-edge-antimagic vertex ((a, d)-EAV) labeling we mean a one-to-one mapping f from V (G) onto {1, 2, . . . , |V (G)|} such that the set of edge-weights of all edges in G, {f (u) + f (v) : uv ∈ E(G)} is {a, a + d, a + 2d, . . . , a + (|E(G)| − 1)d}.
The equivalent notion of a strongly (a, d)-indexable labeling was defined by Hegde in his Ph.D. thesis (see Acharya and Hegde [1]). An (a, d)-edge-antimagic total ((a, d)-EAT) labeling is defined as a one-to-one mapping f from V (G)∪E(G) onto the set {1, 2, . . . , |V (G)|+|E(G)|} so that the set
{f (u)+f (uv)+f (v) : uv ∈ E(G)} is equal to {a, a+d, a+2d, . . . , a+(|E(G)|−1)d}, for two integers a > 0 and d ≥ 0.
The (a, 0)-EAT labelings are usually called edge-magic labelings in the liter- ature (see [12], [17], [24] and [30]). An (a, d)-EAT labeling f is called super if it has the property that the vertex labels are the integers 1, 2, . . . , |V (G)|, that is, the smallest possible labels, and f (E(G)) = {|V (G)| + 1, |V (G)| + 2, . . . , |V (G)| + |E(G)|}.
Definitions of (a, d)-EAT labeling and super (a, d)-EAT labeling were intro- duced by Simanjuntak et al. [25]. These labelings are natural extensions of the notion of edge-magic labeling, defined by Kotzig and Rosa [17], where edge-magic labeling is called magic valuation, and the notion of super edge-magic labeling, which was defined by Enomoto, Llado, Nakamigawa and Ringel [11].
Edge antimagic labelings of graphs
In this paper we extend the survey paper [26] and summarize the results con- cerning (a, d)-EAT and super (a, d)-EAT labelings. We provide several conjectures and open problems for further research.
2. (a, d)-EDGE-ANTIMAGIC TOTAL LABELINGS Assume that graph G has an (a, d)-EAT labeling f . The sum of all the edge-weights is X |E(G)|−1 X
|E(G)|(|E(G)| − 1)d w(uv) = (a + id) = a|E(G)| + . (1) uv ∈E(G) =0 i
2 In the computation of the edge-weights of G, each edge label is used once and the label of vertex u is used deg(u ) times, i = 1, 2, . . . , |V (G)|, where deg(u ) i i i is the degree of vertex u . The sum of all vertex labels and edge labels used to i calculate the edge-weights is thus equal to
|V |+|E| |V | X X X |V |
(|V | + |E|)(|V | + |E| + 1) + j+ (deg(u i )−1)f (u i ) = (deg(u i )−1)f (u i ). j i
2 i
=1 =1 =1
(2) Combining (1) and (2) gives X |V |
|E|(|E| − 1)d (|V | + |E|)(|V | + |E| + 1) a|E| + + = (deg(u i ) − 1)f (u i ). (3)
2
2 i
=1
Using parity considerations of the left hand and the right hand sides of (3) we have: Proposition 1.
[25] A graph with all vertices of odd degrees cannot have an (a, d)- EAT labeling with a and d both even. Proposition 2. [25] Let G be a graph with all vertices of odd degrees. If |E(G)| ≡ 0 (mod 4) and |V (G)| ≡ 2 (mod 4) then G has no (a, d)-EAT labeling. Proposition 3. [25] Suppose G is a graph whose every vertex has an odd degree. Then in the following cases G has no (a, d)-EAT labeling.
(i) |E(G)| ≡ 1 (mod 4), |V (G)| ≡ 0 (mod 4), and a even, (ii) |E(G)| ≡ 1 (mod 4), |V (G)| ≡ 2 (mod 4), and a odd, (iii) |E(G)| ≡ 2 (mod 4), |V (G)| ≡ 2 (mod 4), and d odd,
M. Baˇ ca, et al.
For an (a, d)-EAT labeling, the minimum possible edge-weight is at least 1 + 2 + 3. Consequently a ≥ 6. The maximum possible edge-weight is no more than (|V | + |E| − 2) + (|V | + |E| − 1) + (|V | + |E|) = 3|V | + 3|E| − 3.
Thus a + (|E| − 1)d ≤ 3|V | + 3|E| − 3, 3|V | + 3|E| − 9 d ≤ (4)
|E| − 1 and we have an upper bound for the parameter d for an (a, d)-EAT labeling of G. Next we present some relationships between (a, d)-EAV labeling, (a, d)-EAT labeling and other kinds of labelings, namely, edge-magic vertex ((k, 0)-EAV) la- beling and edge-magic total ((k, 0)-EAT) labeling. Theorem 1.
[3] If G has an edge-magic vertex labeling with magic constant k then G has a (k + |V | + 1, 1)-EAT labeling. Theorem 2. [3] Let G be a graph which admits total labeling and whose edge labels constitute an arithmetic progression with difference d. Then the following are equivalent.
(i) G has an edge-magic total labeling with magic constant k, (ii) G has a (k − (|E| − 1)d, 2d)-EAT labeling. Let C n be the cycle with V (C n ) = {v i : 1 ≤ i ≤ n} and E(C n ) = {v i v i +1 : 1 ≤ i ≤ n − 1} ∪ {v n v
1 }. It follows from (4) that for every cycle C n there is no (a, d)-EAT labeling with d > 5.
Kotzig and Rosa [17] showed that the cycles C n , n ≥ 3, are edge-magic ((a, 0)-EAT in our terminology) with the common edge-weight 3n + 1 (for n odd),
5n
- 2 (for n ≡ 2 (mod 4)) and 3n (for n ≡ 0 (mod 4)). An edge-magic labeling
2 5n+3 5n
for C n with the common edge-weight (for n odd) and + 2 (for n even) was
2
2
described by Godbold and Slater in [15]. Explicit constructions that show that all cycles are edge-magic have been found by Berkman, Parnas and Roditty [9].
For d ≥ 1, the following results are known. Theorem 3.
[25] Every cycle C n has (2n+2, 1)-EAT and (3n+2, 1)-EAT labelings. Theorem 4. [25] Every even cycle C has (4k + 2, 2)-EAT and (4k + 3, 2)-EAT
2k labelings.
Theorem 5.
[25] Every odd cycle C 2k+1 , k ≥ 1, has (3k+4, 3)-EAT and (3k+5, 3)- EAT labelings. Theorem 6.
[3] Every odd cycle C , k ≥ 1, has (3k + 4, 2)-EAT, (5k + 5, 2)-
2k+1 EAT, (2k + 4, 4)-EAT and (2k + 5, 4)-EAT labelings.
Theorem 7. [23] Every odd cycle C , k ≥ 1, has (4k + 4, 1)-EAT, (6k + 5, 1)-
2k+1 EAT, (4k + 4, 2)-EAT and (4k + 5, 2)-EAT labelings. Edge antimagic labelings of graphs Theorem 8.
[8] Every cycle C n , n ≥ 3, has (2n + 2, 3)-EAT and (n + 4, 3)-EAT labelings. Open Problem 1.
Find (a, d)-EAT labelings for even cycles with d ∈ {4, 5} and for odd cycles with d = 5. Let P be the path with V (P ) = {v : 1 ≤ i ≤ n} and E(P ) = {v v : n n i n i i +1 1 ≤ i ≤ n − 1}. Applying Equation (4) to the paths, we obtain that for every path
P n there is no (a, d)-EAT labeling with d > 6. Theorem 9.
[30] All paths are edge-magic ((a, 0)-EAT). Theorem 10.
[3] Every path P n has (2n + 2, 1)-EAT, (3n, 1)-EAT, (n + 4, 3)-EAT and (2n + 2, 3)-EAT labelings. Theorem 11. [23] Every odd path P , k ≥ 1, has (4k + 4, 1)-EAT, (6k + 5, 1)-
2k+1 EAT, (4k + 4, 2)-EAT and (4k + 5, 2)-EAT labelings.
Theorem 12. Every odd path P , k ≥ 1, has (3k + 4, 2)-EAT, (5k + 4, 2)-EAT,
2k+1 (2k + 4, 4)-EAT and (2k + 6, 4)-EAT labelings.
Theorem 13.
[3] Every even path P 2k , k ≥ 1, has (3k + 3, 2)-EAT and (5k + 1, 2)- EAT labelings. Theorem 14.
[29] Every even path P , k ≥ 1, has (2k + 4, 4)-EAT labeling.
2k Theorem 15. [29] Every path P n , n ≥ 2, has (6, 6)-EAT labeling.
We propose the following open problem. Open Problem 2. Find (a, 5)-EAT labelings for paths P n , for the feasible values of a.
3. SUPER (a, d)-EDGE-ANTIMAGIC TOTAL LABELINGS
We start this section by a necessary condition for a graph to be super (a, d)- EAT which will provide an upper bound on the parameter d. The minimum possible edge-weight in a super (a, d)-EAT labeling is at least
1+2+(|V (G)|+1) = |V (G)|+4. Consequently, a ≥ |V (G)|+4. On the other hand, the maximum possible edge-weight is at most (|V (G)| − 1) + |V (G)| + (|V (G)| + |E(G)|) = 3|V (G)| + |E(G)| − 1.
Thus a + (|E(G)| − 1)d ≤ 3|V (G)| + |E(G)| − 1 and we have the following
2|V (G)|+|E(G)|−5 Theorem 16. If a graph G(V, E) is super (a, d)-EAT then d ≤ .
|E(G)|−1
M. Baˇ ca, et al.
Theorem 17.
[3] If G has an (a, d)-EAV labeling then (i) G has a super (a + |V | + 1, d + 1)-EAT labeling, (ii) G has a super (a + |V | + |E|, d − 1)-EAT labeling.
3.1. Cycles It follows from Theorem 16 that if C n , n ≥ 3, is super (a, d)-EAT then d < 3. Theorem 18.
Let C n , n ≥ 3, be super (a,d)-EAT. Then (i) if n is even, then d = 1 and a = 2n + 2,
5n+3
(ii) if n is odd, then either d = 0 and a = , or
2 3n+5
d = 1 and a = 2n + 2, or d = 2 and a = .
2 5n+3
The edge-magic labeling for an odd cycle C n with common edge-weight , ¡
2 5n+3
described by Godbold and Slater in [15], is super , 0 ¢-EAT.
2 Theorem 19.
[5] The cycle C n has super (a, d)-EAT labeling if and only if either (i) d ∈ {0, 2} and n is odd, n ≥ 3, or (ii) d = 1 and n ≥ 3.
3.2. Cycles with Chord t We shall write C to mean the graph constructed from a cycle C n by joining n two vertices whose distance in the cycle is t. For n ≥ 4, 2 ≤ t ≤ n − 2, the graph t n
−t C is of course also the graph C . n n
This section provides the values of t for which there exists a super (a, d)-EAT t labeling of C . If n is odd we can restrict our attention to t either odd or even, n n while if n is even we will pay attention only to t at most . Suppose the endpoints
2
of the chord receive labels x and y. The following result (in the light of Theorem 16) provides the values a and x + y under a super (a, d)-edge-antimagic total labeling. t Theorem 20. [6] Let C , n ≥ 4, t ≥ 2, be super (a, d)-EAT. Then n (i) if d = 0 and n = 2k, then x + y = 2k + 1 and a = 5k + 2, (ii) if d = 0 and n = 2k + 1, then either x + y = k + 1 and a = 5k + 4, or x + y = 3k + 3 and a = 5k + 5, (iii) if d = 1 then x + y = n + 1 and a = 2n + 2, (iv) if d = 2 and n = 2k, then x + y = 2k + 1 and a = 3k + 2, (v) if d = 2 and n = 2k + 1, then either x + y = k + 1 and a = 3k + 3, or x + y = 3k + 3 and a = 3k + 4.
The following results for super (a, 0)-EAT labelings were obtained by Mac- Dougall and Wallis [21] and for super (a, d)-EAT labelings, d ∈ {1, 2}, by Baˇca and Murugan [6].
Theorem 21. t [6, 21] For n odd, n = 2k + 1 ≥ 5, and for all possible values t,
Edge antimagic labelings of graphs
(i) super (a, 0)-EAT labeling with a = 5k + 4 or a = 5k + 5, and (ii) super (a, 2)-EAT labeling with a = 3k + 3 or a = 3k + 4. t Theorem 22.
[6, 21] For n ≡ 0 (mod 4), n ≥ 4, the graph C has n
5n
(i) a super ( + 2, 0)-EAT labeling, and
2 3n
(ii) a super ( + 2, 2)-EAT labeling,
2 for all t ≡ 2 (mod 4). t Theorem 23.
[6, 21] For n = 10 and for n ≡ 2 (mod 4), n ≥ 18, the graph C has n
5n
(i) super ( + 2, 0)-EAT labeling, and
2 3n
(ii) super ( + 2, 2)-EAT labeling,
2 for all t ≡ 3 (mod 4) and for t = 2 and t = 6.
Theorem 24. [6, 21] For n odd, n ≥ 5, and for all possible values of t, the graph t C has a super (2n + 2, 1)-EAT labeling. n t Theorem 25.
[6, 21] For n even, n ≥ 6, and for t odd, t ≥ 3, the graph C has a n super (2n + 2, 1)-EAT labeling. Theorem 26. t [6, 21] For n ≡ 0 (mod 4), n ≥ 4, and for t ≡ 2 (mod 4), t ≥ 2, the graph C has a super (2n + 2, 1)-EAT labeling. n The paper [6] concludes with the following conjecture. t Conjecture 1. There is a super (2n + 2, 1)-EAT labeling of C for n
(i) n ≡ 0 (mod 4) and for t ≡ 0 (mod 4), and (ii) n ≡ 2 (mod 4) and for t even.
3.3. Friendship Graphs The friendship graph F n is a set of n triangles having a common central vertex, and otherwise disjoint. If the friendship graph F n , n ≥ 1, is super (a, d)-EAT then, from Theorem 16, it follows that d < 3. The following result characterizes (a, 1)- edge-antimagicness of friendship graphs.
Lemma 1.
[7] The friendship graph F n has (a, 1)-EAV labeling if and only if n ∈ {1, 3, 4, 5, 7}.
M. Baˇ ca, et al.
From Lemma 1 and Theorem 17 we obtain Theorem 27. [7] For n ∈ {1, 3, 4, 5, 7}, the friendship graph F n has super (a, 0)- EAT and super (a, 2)-EAT labelings.
Moreover, Baˇca et al. [7] proved that Theorem 28. [7] Every friendship graph F n , n ≥ 1, has super (a, 1)-EAT labeling. For further investigation, we propose the following open problem. Open Problem 3. For the friendship graph F n , determine if there is a super (a, 0)-EAT or super (a, 2)-EAT labeling, for n > 7.
3.4. Fans A fan F n , n ≥ 2, is a graph obtained by joining all vertices of path P n to a further vertex called the centre. It follows from Theorem 16 that if F n , n ≥ 2, is super (a, d)-EAT then d < 3.
Lemma 2.
[7] The fan F n has (3, 1)-EAV labeling if and only if 2 ≤ n ≤ 6. Figueroa-Centeno, Ichishima and Muntaner-Batle [12] showed that fan F n is super edge-magic (super (a, 0)-EAT) if and only if 2 ≤ n ≤ 6. Then, in light of
Lemma 2, we get the following Theorem 29.
[7] The fan F n is super (a, d)-EAT total if 2 ≤ n ≤ 6 and d ∈ {0, 1, 2}. Open Problem 4. For the fan F n , determine if there is a super (a, 1)-EAT or super (a, 2)-EAT labeling for n > 6.
3.5. Wheels A wheel W n , n ≥ 3, is a graph obtained by joining all vertices of cycle C n to a further vertex called the centre. If wheel W n , n ≥ 3, is super (a, d)-EAT then d < 2. Enomoto, Llado, Nakamigawa and Ringel [11] proved that a wheel graph W n is not super edge-magic (super (a, 0)-EAT).
In [7] it is proved that: Theorem 30. [7] The wheel W n has super (a, d)-EAT labeling if and only if d = 1 and n 6≡ 1 (mod 4).
u u u u u u 1 2 Edge antimagic labelings of graphs 3 4 n n−1 v v v v v 1 2 3 4 n v n−1 Figure 1: Ladder L n = P n × P .
2
3.6. Ladders Let L n = P n × P be a ladder with V (L n ) = {u i , v i : 1 ≤ i ≤ n} and
2 E(L n ) = {u i u i , v i v i : 1 ≤ i ≤ n − 1} ∪ {u i v i : 1 ≤ i ≤ n}. See Fig.1. From
- 1 +1
Theorem 16 it follows that if ladder L n , n ≥ 2, is super (a, d)-EAT, then the
7 parameter d ≤ .
3 Figueroa-Centeno, Ichishima and Muntaner-Batle [12] proved that the ladder
¡ n +5 L n has , 1 ¢-edge-antimagic vertex labeling. Then, in the light of Theorem 17,
2 the next theorem holds.
Theorem 31. [27] The ladder L n = P n × P is super (a, d)-EAT if n is odd and
2 d ∈ {0, 1, 2}.
Theorem 32.
[27] The ladder L n = P n × P 2 is super (a, 1)-EAT if n is even. It is easily verified that L is not super (a, 0)-EAT. Figueroa-Centeno, Ichishima
2 and Muntaner-Batle [12] have found super (a, 0)-EAT labelings for n = 4 and n = 6.
They suspect that a super (a, 0)-EAT labeling might be found for larger even values of n. Thus the following conjecture may hold. Conjecture 2. The ladder L n = P n × P is super (a, d)-EAT if n is even and
2 d ∈ {0, 2}. ′
Another variation of a ladder graph is specified as follows. A ladder L , n n ≥ 2, is a graph obtained by completing the ladder L n = P n × P by edges u i v i
2 +1 for 1 ≤ i ≤ n − 1. ′ Lemma 3. [27] The ladder L , n ≥ 2, has (a, 1)-EAV labeling. n
Sugeng et al. proved that ′ Theorem 33. [27] The ladder L , n ≥ 2, has super (a, d)-EAT labeling if and only n if d ∈ {0, 1, 2}.
M. Baˇ ca, et al.
3.7. Generalized Prisms The generalized prism can be defined as the cartesian product C m × P n of a cycle on m vertices with a path on n vertices. Let V (C m × P n ) = {v i,j : 0 ≤ i ≤ m − 1 and 1 ≤ j ≤ n} be the vertex set and E(C m × P n ) = {v i,j v i : 0 ≤
- 1,j
i ≤ m − 1 and 1 ≤ j ≤ n} ∪ {v i,j v i,j : 0 ≤ i ≤ m − 1 and 1 ≤ j ≤ n − 1}
+1
be the edge set, where i is taken modulo m. Clearly, |V (C m × P n )| = mn and |E(C m × P n )| = m(2n − 1). See Fig.2. If the generalized prism is super (a, d)-EAT then, by Theorem 16, d < 3. v m−1,n v v v 0,n 0,n−1 1,n v m−2,n v 1,2 m−2,n−1 v m−2,2 v v m−1,n−1 v v m−1,2 2,n v v 0,1 v v
0,2
1,1 v v 1,n−1 2,n−1 v v m−3,n m−3,n−1 v m−3,2 v m−3,1 v v m−2,1 m−1,1 2,1 v 3,1 v 2,2 v 3,n−1 3,2 v v 3,n Figure 2: Generalized prism C m × P n .Lemma 4.
[27] The generalized prism C m × P n has (a, 1)-EAV labeling if m is odd, m ≥ 3 and n ≥ 2. Theorem 34. [27] If m is odd, m ≥ 3, n ≥ 2 and d ∈ {0, 1, 2}, then the generalized prism C m × P n has super (a, d)-EAT labeling.
Note that Figueroa-Centeno, Ichishima and Muntaner-Batle [12] have shown that the generalized prism C m × P n is super edge-magic (super (a, 0)-EAT) if m is odd and n ≥ 2. The next theorem shows super (a, 1)-edge-antimagicness of C m × P n , for m even.
Theorem 35.
[27] If m is even, m ≥ 4, n ≥ 2, then the generalized prism C m × P n has super (a, 1)-EAT labeling. Lemma 5.
[27] For prism C m × P
2 , m even, m ≥ 4, there is no super (a, 0)-EAT labeling and no super (a, 2)-EAT labeling.
Applying Theorems 34 and 35 and Lemma 5 for prism C m × P , we obtain
2
the following Theorem 36. [27] The prism C m × P has super (a, d)-EAT labeling if and only if
2
(i) d ∈ {0, 1, 2} and m is odd, m ≥ 3, or
Edge antimagic labelings of graphs
What can be said about super (a, d)-EAT labeling of C m × P n for the remain- ing cases if m is even and d ∈ {0, 2}? Sugeng et al. [27] propose the following Conjecture 3.
If m is even, m ≥ 4, n ≥ 3 and d ∈ {0, 2}, then the generalized prism C m × P n has super (a, d)-EAT labeling.
3.8. Generalized Antiprisms n A generalized antiprism A can be obtained by completing the generalized m prism C m × P n by edges v i,j v i for 0 ≤ i ≤ m − 1, 1 ≤ j ≤ n − 1, with indices i n +1 +1,j taken modulo m. Let V (A ) = V (C m ×P n ) = {v i,j : 0 ≤ i ≤ m−1 and 1 ≤ j ≤ n} n n m be the vertex set of A and E(A ) = E(C m × P n ) ∪ {v i,j +1 v i +1,j : 0 ≤ i ≤ m m n m − 1 and 1 ≤ j ≤ n − 1} be the edge set of A , where i is taken modulo m. m
In [27], it is shown that n Theorem 37. [27] The generalized antiprism A , m ≥ 3, n ≥ 2, is super (a, d)- m EAT if and only if d = 1.
3.9. Generalized Petersen Graphs Watkins [32] defined a generalized Petersen graph as follows: The generalized
Petersen graph P (n, m), n ≥ 3 and 1 ≤ m ≤ ⌊(n − 1)/2⌋, consists of an outer n-cycle y , y , . . . , y n , a set of n spokes y i x i , 0 ≤ i ≤ n − 1, and n edges x i x i ,
1 −1
- m
0 ≤ i ≤ n − 1, with indices taken modulo n. See Fig.3. From Theorem 16, it follows that if generalized Petersen graph P (n, m), n ≥ 3, 1 ≤ m ≤ ⌊(n − 1)/2⌋, is super (a, d)-EAT then d < 3. Theorem 38. [5] Let the generalized Petersen graph P (n, m), n ≥ 3, 1 ≤ m ≤ ⌊(n − 1)/2⌋, be super (a, d)-EAT. Then
(i) if n is even, then d = 1 and a = 4n + 2,
11n+3
(ii) if n is odd, then either d = 0 and a = , or d = 1 and a = 4n + 2,
2 5n+5
or d = 2 and a = .
2 Figueroa-Centeno, Ichishima and Muntaner-Batle [12] and Fukuchi [13] con-
structed (a, 1)-EAV labelings for the generalized Petersen graphs P (n, 1) and P (n, 2) which, together with Theorem 17, give the following result. Theorem 39. [5] Every generalized Petersen graph P (n, m), n odd, n ≥ 3, 1 ≤ ¡
11n+3 5n+5 m ≤ 2, has super , 0 ¢-EAT labeling and super ¡ , 2 ¢-EAT labeling.
2
2 Furthermore, Ngurah and Baskoro [22] proved n Theorem 40.
[22] Every generalized Petersen graph P (n, m), n ≥ 3, 1 ≤ m < ,
2 has a super (4n + 2, 1)-EAT labeling. n −1 Theorem 41.
[5] For n odd, n ≥ 3, every generalized Petersen graph P (n, ) ¡
2 11n+3 5n+5 M. Baˇ ca, et al. y y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8
x
x 1 x 2 x 3 x 4 x 8 x 7 x 6 x 5 y y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8x
x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 Figure 3: Generalized Petersen graph P (9, 4) and P (9, 3). Edge antimagic labelings of graphsBaˇca et al. [5] put forward the following Conjecture 4.
There is a super (a, d)-EAT labeling for the generalized Petersen n
−3 graph P (n, m) for n odd, n ≥ 9, d ∈ {0, 2} and 3 ≤ m ≤ .
2
3.10. Complete Bipartite Graphs Let K n,n be the complete bipartite graph with V (K n,n ) = {x i : 1 ≤ i ≤ n} ∪ {y j : 1 ≤ j ≤ n} and E(K n,n ) = {x i y j : 1 ≤ i ≤ n and 1 ≤ j ≤ n}. Using
Theorem 16, if the complete bipartite graph K n,n , n ≥ 4, is super (a, d)-EAT then d < 2, while if K n,n , 2 ≤ n ≤ 3, is super (a, d)-EAT then d < 3. Theorem 42.
[7] Every complete bipartite graph K n,n , n ≥ 2, has super (a, 1)-EAT labeling. Theorem 43.
[11] A complete bipartite graph K m,n is super edge-magic (super (a, 0)-EAT) if and only if m = 1 or n = 1. Theorem 43 asserts that, for n ≥ 2, there is no super (a, 0)-EAT labeling of
K n,n . For K 2,2 and K 3,3 it was proved that Theorem 44. [7] For complete bipartite graph K n,n , 2 ≤ n ≤ 3, there is no super (a, 2)-EAT labeling.
From Theorems 42, 43 and 44, it follows that Theorem 45. [7] The complete bipartite graph K n,n has super (a, d)-EAT labeling if and only if d = 1 and n ≥ 2.
3.11. Complete Graphs From Theorem 16, it follows that if the complete graph K n , n ≥ 3, is super
(a, d)-EAT then d ≤ 2. In [7], Baˇca et al. proved that Theorem 46.
[7] The complete graph K n , n ≥ 3, has super (a, d)-EAT labeling if and only if (i) d = 0 and n = 3, or (ii) d = 1 and n ≥ 3, or (iii) d = 2 and n = 3.
3.12. Stars Let x o denote the central vertex of a star S n , n ≥ 1, and x i , 1 ≤ i ≤ n, be its leaves. Theorem 16 provides an upper bound on the parameter d, i.e., if a star
S , n ≥ 1, is super (a, d)-EAT then d ≤ 3. In [28], Sugeng et al. proved that n Lemma 6.
M. Baˇ ca, et al.
In light of the preceding lemma and Theorem 17, the next result follows immediately. Theorem 47.
[28] The star S n , n ≥ 1, has super (a, 0)-EAT labeling and super (a, 2)-EAT labeling. Applying the construction of (a, 1)-EAV labeling from Lemma 6 and com- pleting an edge labeling by a special procedure, it was shown that
Theorem 48.
[28] The star S n , n ≥ 1, has super (a, 1)-EAT labeling. To completely characterize super (a, d)-EAT labeling of S n , it only remains to consider the case d = 3. In [28], it is proved that
Theorem 49. [28] For the star S n , n ≥ 3, there is no super (a, 3)-EAT labeling.
Thus from Lemma 6 and Theorems 47, 48 and 49, it follows that Theorem 50.
[28] The star S n has super (a, d)-EAT labeling if and only if (i) d ∈ {0, 1, 2} and n ≥ 1, or (ii) d = 3 and 1 ≤ n ≤ 2.
3.13. Caterpillars A caterpillar is a graph derived from a path by hanging any number of leaves from the vertices of the path. The caterpillar can be seen as a sequence of stars
S ∪ S ∪ . . . ∪ S r , where each S i is a star with central vertex c i and n i leaves for
1
2 i = 1, 2, . . . , r, and the leaves of S i include c i and c i for i = 2, 3, . . . , r − 1.
−1 +1
We denote the caterpillar as S n ,n ,...,n r , where the vertex set is V (S n ,n ,...,n r ) r 1 2 1 2 S j j
−1 j
= {c i : 1 ≤ i ≤ r} ∪ {x : 2 ≤ j ≤ n i − 1} ∪ {x : 1 ≤ j ≤ n − 1} ∪ {x : 2 ≤ i
1 r i
1 =2
r
r −1 S jj ≤ n r } and the edge set is E(S n ,n ,...n ) = {c i c i +1 : 1 ≤ i ≤ r − 1} ∪ {c i x : 1 2 i i j j =2 2 ≤ j ≤ n i − 1} ∪ {c x : 1 ≤ j ≤ n − 1} ∪ {c r x : 2 ≤ j ≤ n r }. See Fig.4.
1 1 r P P r r
1 |V (S n ,n ,...,n r )| = n i − r + 2 and |E(S n ,n ,...,n r )| = n i − r + 1. 1 2 i i 1 2 =1 =1 r
From Theorem 16, it follows that if a caterpillar S n ,n ,...,n is super (a, d)- 1 2 EAT then d ≤ 3. Kotzig and Rosa [17] (see also Wallis [31]) proved that caterpillars have super (a, 1)-EAV labeling. This result, together with Theorem 17, gives Theorem 51.
All caterpillars are super (a, 0)-EAT and super (a, 2)-EAT. Theorem 52.
[28] Every caterpillar with even number of vertices has super (a, 1)- EAT labeling. Theorem 53.
[28] There is a super (a, 1)-EAT labeling for a caterpillar with odd
x x x 1 1 1 x x r r 2 1 3 n Edge antimagic labelings of graphs 1 r r 1 − x x x 1 2 3 4 n r c 1 c c 2 r x x 2 2 3 2 x 2 4 x n 2 2 − 1 Figure 4: Caterpillar S r . n ,n ,...,n r r 1 2 ⌊ ⌋+1 ⌊ ⌋ 2 P P 2 Let S n ,n ,...,n r be a caterpillar and N = n and N = n , where 1 2
1 2i−1 i i 2 2i r =1 =1 r
⌊ ⌋ denotes the greatest integer smaller than or equal to . The following theorems
2 2 give results for super (a, 3)-edge-antimagicness of caterpillar S n ,n ,...,n r . 1 2 Theorem 54. [28] If r is even and N = N or |N − N | = 1 then the caterpillar
1
2
1
2 S n ,n ,...,n r has super (a, 3)-EAT labeling. 1 2 Theorem 55. r
[28] If r is odd and N
1 = N
2 or N 1 = N 2 + 1 then the caterpillar S n ,n ,...,n has super (a, 3)-EAT labeling. 1 2 For the caterpillar S n ,n ,...,n r , r odd and N = N + 1, we have not found 1 22
1 any super (a, 3)-EAT labeling. So, we propose the following open problem.
Open Problem 5. For the caterpillar S r , determine if there is a super n ,n ,...,n 1 2 (a, 3)-EAT labeling, for r odd and N = N + 1.
2
1 It is an easy consequence of the Theorem 54 that a double star S m,n , m, n ≥ 2,
can have super (a, 3)-EAT labeling if m = n or |m − n| = 1. For other cases, it was proved in [28] that Theorem 56. [28] For the double star S m,n , m 6= n and |m − n| 6= 1, there is no super (a, 3)-EAT labeling.
Sugeng et al. [28] suggest the following Conjecture 5. For the caterpillar S n ,n ,...,n r , N 6= N and |N − N | 6= 1, there 1
2
1
2
1
2 is no super (a, 3)-EAT labeling.
4. CONCLUSION In the foregoing sections we presented results concerning (a, d)-EAT and super
(a, d)-EAT labeling for a variety of families of connected graphs. However, there are many graphs which were not been studied, and several families of graphs for
M. Baˇ ca, et al.
Many researchers have studied edge-magic total and super edge-magic total labelings for many families of disconnected graphs (see the general survey of Gal- lian [14]) and obtained a wealth of results. We believe that these results can be extended to (a, d)-EAT or super (a, d)-EAT labelings.
Acknowledgement The authors would like to thank the referee for his valuable comments. REFERENCES
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M. Baˇ ca, et al.
M. Baˇ ca : Department of Applied Mathematics Technical University, Letn´ a 9, 042 00 Koˇsice,
Slovak Republic. E-mail: Martin.Baca@tuke.sk.E.T. Baskoro : Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia. E-mail: ebaskoro@math.itb.ac.id.
M. Miller : School of Information Technology and Mathematical Sciences, University of Ballarat, Australia. E-mail: m.miller@ballarat.edu.au.
J. Ryan : School of Information Technology and Mathematical Sciences, University of Ballarat, Australia. E-mail: joe.ryan@ballarat.edu.au.
R. Simanjuntak : Department of Mathematics, Institut Teknologi Bandung, Bandung
40132, Indonesia. E-mail: rino@math.itb.ac.id.K.A. Sugeng : School of Information Technology and Mathematical Sciences, University
of Ballarat, Australia. E-mail: k.sugeng@ballarat.edu.au.