J. Indones. Math. Soc. (MIHMI) Vol. xx, No. xx (20xx), pp. xx–xx.
Survey of edge antimagic labelings of graphs ˇa1 , Edy Tri Baskoro2 , Mirka Miller3 , Joe Martin Bac 3 Ryan , Rinovia Simanjuntak2 , Kiki A. Sugeng3 Abstract. An (a, d)-edge-antimagic total labeling of G is a one-to-one mapping f taking the vertices and edges onto 1, 2, . . . , |V (G)| + |E(G)| so that the edge-weights w(uv) = f (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 Pn , n ≥ 3, cycles, wheels, and complete graphs Kn , n ≥ 3. They conjecture that every connected graph, except K2 , is antimagic. 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 Received dd-mm-yyyy, Accepted dd-mm-yyyy. 2000 Mathematics Subject Classification: Key words and Phrases:
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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 K2 , are antimagic. 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)-vertexantimagic 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 literature (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 introduced 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]. In this paper we extend the survey paper [26] and summarize the results concerning (a, d)-EAT and super (a, d)-EAT labelings. We provide several conjectures and open problems for further research.
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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 uv∈E(G)
|E(G)|−1
w(uv) =
X
(a + id) = a|E(G)| +
i=0
|E(G)|(|E(G)| − 1)d . 2
(1)
In the computation of the edge-weights of G, each edge label is used once and the label of vertex ui is used deg(ui ) times, i = 1, 2, . . . , |V (G)|, where deg(ui ) is the degree of vertex ui . The sum of all vertex labels and edge labels used to calculate the edge-weights is thus equal to |V |+|E|
X j=1
|V | X
|V | (|V | + |E|)(|V | + |E| + 1) X + (deg(ui )−1)f (ui ). j+ (deg(ui )−1)f (ui ) = 2 i=1 i=1
(2) Combining (1) and (2) gives |V |
|E|(|E| − 1)d (|V | + |E|)(|V | + |E| + 1) X a|E| + = + (deg(ui ) − 1)f (ui ). (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, (iv) |E(G)| ≡ 3 (mod 4), |V (G)| ≡ 0 (mod 4), a even, and d odd. 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.
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Thus a + (|E| − 1)d ≤ 3|V | + 3|E| − 3, d≤
3|V | + 3|E| − 9 |E| − 1
(4)
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) labeling 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 Cn be the cycle with V (Cn ) = {vi : 1 ≤ i ≤ n} and E(Cn ) = {vi vi+1 : 1 ≤ i ≤ n − 1} ∪ {vn v1 }. It follows from (4) that for every cycle Cn there is no (a, d)-EAT labeling with d > 5. Kotzig and Rosa [17] showed that the cycles Cn , n ≥ 3, are edge-magic ((a, 0)-EAT in our terminology) with the common edge-weight 3n + 1 (for n odd), 5n 2 + 2 (for n ≡ 2 (mod 4)) and 3n (for n ≡ 0 (mod 4)). An edge-magic labeling for Cn with the common edge-weight 5n+3 (for n odd) and 5n 2 2 + 2 (for n even) was 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 Cn has (2n + 2, 1)-EAT and (3n + 2, 1)-EAT labelings. Theorem 4 (25) Every even cycle C2k has (4k + 2, 2)-EAT and (4k + 3, 2)-EAT labelings. Theorem 5 (25) Every odd cycle C2k+1 , k ≥ 1, has (3k + 4, 3)-EAT and (3k + 5, 3)-EAT labelings. Theorem 6 (3) Every odd cycle C2k+1 , k ≥ 1, has (3k + 4, 2)-EAT, (5k + 5, 2)EAT, (2k + 4, 4)-EAT and (2k + 5, 4)-EAT labelings. Theorem 7 (23) Every odd cycle C2k+1 , k ≥ 1, has (4k + 4, 1)-EAT, (6k + 5, 1)EAT, (4k + 4, 2)-EAT and (4k + 5, 2)-EAT labelings.
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Theorem 8 (8) Every cycle Cn , 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 Pn be the path with V (Pn ) = {vi : 1 ≤ i ≤ n} and E(Pn ) = {vi vi+1 : 1 ≤ i ≤ n − 1}. Applying Equation (4) to the paths, we obtain that for every path Pn 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 Pn has (2n + 2, 1)-EAT, (3n, 1)-EAT, (n + 4, 3)-EAT and (2n + 2, 3)-EAT labelings. Theorem 11 (23) Every odd path P2k+1 , k ≥ 1, has (4k + 4, 1)-EAT, (6k + 5, 1)EAT, (4k + 4, 2)-EAT and (4k + 5, 2)-EAT labelings. Theorem 12 Every odd path P2k+1 , k ≥ 1, has (3k + 4, 2)-EAT, (5k + 4, 2)-EAT, (2k + 4, 4)-EAT and (2k + 6, 4)-EAT labelings. Theorem 13 (3) Every even path P2k , k ≥ 1, has (3k + 3, 2)-EAT and (5k + 1, 2)EAT labelings. Theorem 14 (29) Every even path P2k , k ≥ 1, has (2k + 4, 4)-EAT labeling. Theorem 15 (29) Every path Pn , n ≥ 2, has (6, 6)-EAT labeling. We propose the following open problem. Open Problem 2 Find (a, 5)-EAT labelings for paths Pn , 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
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Theorem 16 If a graph G(V, E) is super (a, d)-EAT then d ≤
2|V (G)|+|E(G)|−5 . |E(G)|−1
The next theorem is useful for extending labelings. 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 Cn , n ≥ 3, is super (a, d)-EAT then d < 3. Theorem 18 Let Cn , n ≥ 3, be super (a,d)-EAT. Then (i) if n is even, then d = 1 and a = 2n + 2, (ii) if n is odd, then either d = 0 and a = 5n+3 2 , or d = 1 and a = 2n + 2, or d = 2 and a = 3n+5 2 . The edge-magic labeling for an odd cycle Cn ¡with common edge-weight ¢ described by Godbold and Slater in [15], is super 5n+3 , 0 -EAT. 2
5n+3 2 ,
Theorem 19 (5) The cycle Cn 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
We shall write Cnt to mean the graph constructed from a cycle Cn by joining two vertices whose distance in the cycle is t. For n ≥ 4, 2 ≤ t ≤ n − 2, the graph Cnt is of course also the graph Cnn−t . This section provides the values of t for which there exists a super (a, d)-EAT labeling of Cnt . If n is odd we can restrict our attention to t either odd or even, while if n is even we will pay attention only to t at most n2 . Suppose the endpoints 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. Theorem 20 (6) Let Cnt , n ≥ 4, t ≥ 2, be super (a, d)-EAT. Then (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.
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The following results for super (a, 0)-EAT labelings were obtained by MacDougall and Wallis [21] and for super (a, d)-EAT labelings, d ∈ {1, 2}, by Baˇca and Murugan [6]. Theorem 21 (21,6) For n odd, n = 2k + 1 ≥ 5, and for all possible values t, every graph Cnt has (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. Theorem 22 (21,6) For n ≡ 0 (mod 4), n ≥ 4, the graph Cnt has (i) a super ( 5n 2 + 2, 0)-EAT labeling, and (ii) a super ( 3n 2 + 2, 2)-EAT labeling, for all t ≡ 2 (mod 4). Theorem 23 (21,6) For n = 10 and for n ≡ 2 (mod 4), n ≥ 18, the graph Cnt has (i) super ( 5n 2 + 2, 0)-EAT labeling, and (ii) super ( 3n 2 + 2, 2)-EAT labeling, for all t ≡ 3 (mod 4) and for t = 2 and t = 6. Theorem 24 (21,6) For n odd, n ≥ 5, and for all possible values of t, the graph Cnt has a super (2n + 2, 1)-EAT labeling. Theorem 25 (21,6) For n even, n ≥ 6, and for t odd, t ≥ 3, the graph Cnt has a super (2n + 2, 1)-EAT labeling. Theorem 26 (21,6) For n ≡ 0 (mod 4), n ≥ 4, and for t ≡ 2 (mod 4), t ≥ 2, the graph Cnt has a super (2n + 2, 1)-EAT labeling. The paper [6] concludes with the following conjecture. Conjecture 1 There is a super (2n + 2, 1)-EAT labeling of Cnt for (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 Fn is a set of n triangles having a common central vertex, and otherwise disjoint. If the friendship graph Fn , 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 Fn has (a, 1)-EAV labeling if and only if n ∈ {1, 3, 4, 5, 7}.
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From Lemma 1 and Theorem 17 we obtain Theorem 27 (7) For n ∈ {1, 3, 4, 5, 7}, the friendship graph Fn has super (a, 0)EAT and super (a, 2)-EAT labelings. Moreover, Baˇca et al. [7] proved that Theorem 28 (7) Every friendship graph Fn , n ≥ 1, has super (a, 1)-EAT labeling. For further investigation, we propose the following open problem. Open Problem 3 For the friendship graph Fn , determine if there is a super (a, 0)EAT or super (a, 2)-EAT labeling, for n > 7.
3.4. Fans
A fan Fn , n ≥ 2, is a graph obtained by joining all vertices of path Pn to a further vertex called the centre. It follows from Theorem 16 that if Fn , n ≥ 2, is super (a, d)-EAT then d < 3. Lemma 2 (7) The fan Fn has (3, 1)-EAV labeling if and only if 2 ≤ n ≤ 6. Figueroa-Centeno, Ichishima and Muntaner-Batle [12] showed that fan Fn 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 Fn is super (a, d)-EAT total if 2 ≤ n ≤ 6 and d ∈ {0, 1, 2}. Open Problem 4 For the fan Fn , determine if there is a super (a, 1)-EAT or super (a, 2)-EAT labeling for n > 6.
3.5. Wheels
A wheel Wn , n ≥ 3, is a graph obtained by joining all vertices of cycle Cn to a further vertex called the centre. If wheel Wn , n ≥ 3, is super (a, d)-EAT then d < 2. Enomoto, Llado, Nakamigawa and Ringel [11] proved that a wheel graph Wn is not super edge-magic (super (a, 0)-EAT). In [7] it is proved that: Theorem 30 (7) The wheel Wn has super (a, d)-EAT labeling if and only if d = 1 and n 6≡ 1 (mod 4).
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u2
v1
v2
u3
v3
u4
un−1
un
v4
vn−1
vn
Figure 1: Ladder Ln = Pn × P2 . 3.6. Ladders
Let Ln = Pn × P2 be a ladder with V (Ln ) = {ui , vi : 1 ≤ i ≤ n} and E(Ln ) = {ui ui+1 , vi vi+1 : 1 ≤ i ≤ n − 1} ∪ {ui vi : 1 ≤ i ≤ n}. See Fig.1. From Theorem 16 it follows that if ladder Ln , n ≥ 2, is super (a, d)-EAT, then the parameter d ≤ 73 . Figueroa-Centeno, Ichishima and Muntaner-Batle [12] proved that the ladder ¡ ¢ Ln has n+5 2 , 1 -edge-antimagic vertex labeling. Then, in the light of Theorem 17, the next theorem holds. Theorem 31 (27) The ladder Ln = Pn × P2 is super (a, d)-EAT if n is odd and d ∈ {0, 1, 2}. Theorem 32 (27) The ladder Ln = Pn × P2 is super (a, 1)-EAT if n is even. It is easily verified that L2 is not super (a, 0)-EAT. Figueroa-Centeno, Ichishima 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 Ln = Pn × P2 is super (a, d)-EAT if n is even and d ∈ {0, 2}. 0
Another variation of a ladder graph is specified as follows. A ladder Ln , n ≥ 2, is a graph obtained by completing the ladder Ln = Pn × P2 by edges ui vi+1 for 1 ≤ i ≤ n − 1. 0
Lemma 3 (27) The ladder Ln , n ≥ 2, has (a, 1)-EAV labeling. Sugeng et al. proved that 0
Theorem 33 (27) The ladder Ln , n ≥ 2, has super (a, d)-EAT labeling if and only if d ∈ {0, 1, 2}. 3.7. Generalized Prisms
The generalized prism can be defined as the cartesian product Cm × Pn of a cycle on m vertices with a path on n vertices. Let V (Cm × Pn ) = {vi,j : 0 ≤
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i ≤ m − 1 and 1 ≤ j ≤ n} be the vertex set and E(Cm × Pn ) = {vi,j vi+1,j : 0 ≤ i ≤ m − 1 and 1 ≤ j ≤ n} ∪ {vi,j vi,j+1 : 0 ≤ i ≤ m − 1 and 1 ≤ j ≤ n − 1} be the edge set, where i is taken modulo m. Clearly, |V (Cm × Pn )| = mn and |E(Cm × Pn )| = m(2n − 1). See Fig.2. If the generalized prism is super (a, d)-EAT then, by Theorem 16, d < 3. v0,n
vm−1,n
v0,n−1
v1,n
vm−1,n−1 v1,n−1
v0,2 vm−2,n
vm−1,2
v2,n v1,2
vm−2,n−1 v0,1 vm−2,2
vm−1,1
v2,n−1 v1,1 v2,2
vm−2,1
v2,1
vm−3,n vm−3,n−1
v3,2 vm−3,1
v3,1
v3,n−1
v3,n
vm−3,2
Figure 2: Generalized prism Cm × Pn . Lemma 4 (27) The generalized prism Cm × Pn 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 Cm × Pn has super (a, d)-EAT labeling. Note that Figueroa-Centeno, Ichishima and Muntaner-Batle [12] have shown that the generalized prism Cm × Pn 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 Cm × Pn , for m even. Theorem 35 (27) If m is even, m ≥ 4, n ≥ 2, then the generalized prism Cm ×Pn has super (a, 1)-EAT labeling. Lemma 5 (27) For prism Cm × P2 , 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 Cm × P2 , we obtain the following Theorem 36 (27) The prism Cm × P2 has super (a, d)-EAT labeling if and only if (i) d ∈ {0, 1, 2} and m is odd, m ≥ 3, or (ii) d = 1 and m is even, m ≥ 4.
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What can be said about super (a, d)-EAT labeling of Cm × Pn for the remaining 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 Cm × Pn has super (a, d)-EAT labeling.
3.8. Generalized Antiprisms
A generalized antiprism Anm can be obtained by completing the generalized prism Cm × Pn by edges vi,j+1 vi+1,j for 0 ≤ i ≤ m − 1, 1 ≤ j ≤ n − 1, with indices i taken modulo m. Let V (Anm ) = V (Cm ×Pn ) = {vi,j : 0 ≤ i ≤ m−1 and 1 ≤ j ≤ n} be the vertex set of Anm and E(Anm ) = E(Cm × Pn ) ∪ {vi,j+1 vi+1,j : 0 ≤ i ≤ m − 1 and 1 ≤ j ≤ n − 1} be the edge set of Anm , where i is taken modulo m. In [27], it is shown that Theorem 37 (27) The generalized antiprism Anm , m ≥ 3, n ≥ 2, is super (a, d)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 ≤ b(n − 1)/2c, consists of an outer n-cycle y0 , y1 , . . . , yn−1 , a set of n spokes yi xi , 0 ≤ i ≤ n − 1, and n edges xi xi+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 ≤ b(n − 1)/2c, is super (a, d)-EAT then d < 3. Theorem 38 (5) Let the generalized Petersen graph P (n, m), n ≥ 3, 1 ≤ m ≤ b(n − 1)/2c, be super (a, d)-EAT. Then (i) if n is even, then d = 1 and a = 4n + 2, (ii) if n is odd, then either d = 0 and a = 11n+3 , or d = 1 and a = 4n + 2, 2 or d = 2 and a = 5n+5 . 2 Figueroa-Centeno, Ichishima and Muntaner-Batle [12] and Fukuchi [13] constructed (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 ≤ 5n+5 , 0 -EAT labeling and super , 2 -EAT labeling. m ≤ 2, has super 11n+3 2 2 Furthermore, Ngurah and Baskoro [22] proved Theorem 40 (22) Every generalized Petersen graph P (n, m), n ≥ 3, 1 ≤ m < has a super (4n + 2, 1)-EAT labeling.
n 2,
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y0 y1 y8 x0 x8 x1 y7
y2
x7 x2 x6 x3 x5
y6
x4
y5
y3
y4 y0 y1
y8 x0 x8 x1 y7
y2
x7 x2 x6 x3 y3 x5
y6
y5
x4
y4
Figure 3: Generalized Petersen graph P (9, 4) and P (9, 3).
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n−1 Theorem 41 (5) For ¡ 11n+3 ¢ n odd, n ≥ 3, every generalized ¡ 5n+5 ¢Petersen graph P (n, 2 ) has a super , 0 -EAT labeling and super 2 2 , 2 -EAT labeling.
Baˇca et al. [5] put forward the following Conjecture 4 There is a super (a, d)-EAT labeling for the generalized Petersen graph P (n, m) for n odd, n ≥ 9, d ∈ {0, 2} and 3 ≤ m ≤ n−3 2 .
3.10. Complete Bipartite Graphs
Let Kn,n be the complete bipartite graph with V (Kn,n ) = {xi : 1 ≤ i ≤ n} ∪ {yj : 1 ≤ j ≤ n} and E(Kn,n ) = {xi yj : 1 ≤ i ≤ n and 1 ≤ j ≤ n}. Using Theorem 16, if the complete bipartite graph Kn,n , n ≥ 4, is super (a, d)-EAT then d < 2, while if Kn,n , 2 ≤ n ≤ 3, is super (a, d)-EAT then d < 3. Theorem 42 (7) Every complete bipartite graph Kn,n , n ≥ 2, has super (a, 1)EAT labeling. Theorem 43 (11) A complete bipartite graph Km,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 Kn,n . For K2,2 and K3,3 it was proved that Theorem 44 (7) For complete bipartite graph Kn,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 Kn,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 Kn , n ≥ 3, is super (a, d)-EAT then d ≤ 2. In [7], Baˇca et al. proved that Theorem 46 (7) The complete graph Kn , 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.
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Let xo denote the central vertex of a star Sn , n ≥ 1, and xi , 1 ≤ i ≤ n, be its leaves. Theorem 16 provides an upper bound on the parameter d, i.e., if a star Sn , n ≥ 1, is super (a, d)-EAT then d ≤ 3. In [28], Sugeng et al. proved that Lemma 6 (28) Every star Sn , n ≥ 1, has (a, 1)-EAV labeling. In light of the preceding lemma and Theorem 17, the next result follows immediately. Theorem 47 (28) The star Sn , 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 completing an edge labeling by a special procedure, it was shown that Theorem 48 (28) The star Sn , n ≥ 1, has super (a, 1)-EAT labeling. To completely characterize super (a, d)-EAT labeling of Sn , it only remains to consider the case d = 3. In [28], it is proved that Theorem 49 (28) For the star Sn , 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 Sn 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 S1 ∪ S2 ∪ . . . ∪ Sr , where each Si is a star with central vertex ci and ni leaves for i = 1, 2, . . . , r, and the leaves of Si include ci−1 and ci+1 for i = 2, 3, . . . , r − 1. We denote the caterpillar as Sn1 ,n2 ,...,nr , where the vertex set is V (Sn1 ,n2 ,...,nr ) r−1 S j = {ci : 1 ≤ i ≤ r} ∪ {xi : 2 ≤ j ≤ ni − 1} ∪ {xj1 : 1 ≤ j ≤ n1 − 1} ∪ {xjr : 2 ≤ i=2
j ≤ nr } and the edge set is E(Sn1 ,n2 ,...nr ) = {ci ci+1 : 1 ≤ i ≤ r − 1} ∪
r−1 S i=2
{ci xji :
2 ≤ j ≤ ni − 1} ∪ {c1 xj1 : 1 ≤ j ≤ n1 − 1} ∪ {cr xjr : 2 ≤ j ≤ nr }. See Fig.4. r r P P |V (Sn1 ,n2 ,...,nr )| = ni − r + 2 and |E(Sn1 ,n2 ,...,nr )| = ni − r + 1. i=1
i=1
From Theorem 16, it follows that if a caterpillar Sn1 ,n2 ,...,nr is super (a, d)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.
15
Survey of edge antimagic labelings of graphs x11
x21
x31
x2r
x1n1 −1
x3r
x4r
xrnr
c2
c1
cr
x22
x32
x42
x2n2 −1
Figure 4: Caterpillar Sn1 ,n2 ,...,nr . 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 number of vertices. Let Sn1 ,n2 ,...,nr be a caterpillar and N1 =
b r2 c+1
P
i=1
n2i−1 and N2 =
b r2 c
P
i=1
n2i , where
b 2r c denotes the greatest integer smaller than or equal to 2r . The following theorems give results for super (a, 3)-edge-antimagicness of caterpillar Sn1 ,n2 ,...,nr . Theorem 54 (28) If r is even and N1 = N2 or |N1 − N2 | = 1 then the caterpillar Sn1 ,n2 ,...,nr has super (a, 3)-EAT labeling. Theorem 55 (28) If r is odd and N1 = N2 or N1 = N2 + 1 then the caterpillar Sn1 ,n2 ,...,nr has super (a, 3)-EAT labeling. For the caterpillar Sn1 ,n2 ,...,nr , r odd and N2 = N1 + 1, we have not found any super (a, 3)-EAT labeling. So, we propose the following open problem. Open Problem 5 For the caterpillar Sn1 ,n2 ,...,nr , determine if there is a super (a, 3)-EAT labeling, for r odd and N2 = N1 + 1. It is an easy consequence of the Theorem 54 that a double star Sm,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 Sm,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 Sn1 ,n2 ,...,nr , N1 6= N2 and |N1 − N2 | 6= 1, there is no super (a, 3)-EAT labeling.
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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 which edge-antimagic total labelings have been considered but not yet found. Many researchers have studied edge-magic total and super edge-magic total labelings for many families of disconnected graphs (see the general survey of Gallian [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.
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ˇa: Department of Applied Mathematics Technical University, Letn´ Martin Bac a 9, 042 00 Koˇsice, Slovak Republic. E-mail:
[email protected].
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Edy Tri Baskoro: Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia. E-mail:
[email protected].
Mirka Miller: School of Information Technology and Mathematical Sciences, University of Ballarat, Australia. E-mail:
[email protected].
Joe Ryan: School of Information Technology and Mathematical Sciences, University of Ballarat, Australia. E-mail:
[email protected].
Rinovia Simanjuntak: Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia. E-mail:
[email protected].
Kiki A. Sugeng: School of Information Technology and Mathematical Sciences, University of Ballarat, Australia. E-mail:
[email protected].
