On graphs with α - and b -edge consecutive edge magic labelings

ijc


Introduction
Let G be a graph of order m and size n; the graph G is said to be edge-magic if there exists a bijection f : V (G) ∪ E(G) → {1, 2, . . . , m + n} such that f (u) + f (v) + f (uv) = k for all uv ∈ E(G), where k is a constant. We refer to k as the valence of f ; some authors called k the magic constant of f . This type of total labeling was originally introduced by Kotzig and Rosa [8]. Enomoto et al. [5] said that an edge-magic labeling is super when the set of vertex labels is {1, 2, . . . , m}. The proof of the following lemma can be found in [6]. www.ijc.or.id An outline of the proof of this result is the following. Assuming that for a graph G of order m and size n, there exists a bijection f : V (G) → {1, 2, . . . , m} such that {f (u)+f (v) : uv ∈ E(G)} is formed by n consecutive integers, then f can be extended to a super edge-magic labeling by Figure 1 we show an example of this labeling for the graph 2C 8 ∪ P 2 ; in this example m = 18, n = 17, and the valence of the labeling is k = 46. Sugeng and Miller [12] said that an edge-magic labeling is b-edge consecutive when the set of edge labels is {b + 1, b + 2, . . . , b + n} for some b ∈ {0, 1, . . . , n}. Based on this definition we can see that any super edge-magic labeling of a graph of order m is an m-edge consecutive edge-magic labeling. Let f be a b-edge consecutive edge-magic labeling of a graph of order m and size n. The dual labeling of f (also called the complementary of f ) is the labeling f defined as f (x) = m + n + 1 − f (x) for every x ∈ V (G) ∪ E(G). It is well-known that f is an edge-magic labeling; moreover, f is indeed a b-edge consecutive labeling because f is b-edge consecutive; therefore, {m + n + 1 − f (x) : x ∈ E(G)} is a set of n consecutive integers. We must observe that if k is the valence of f , then the valence of f is 3(m + n + 1) − k.
Two results about b-edge consecutive edge-magic labelings, proven in [12], that are relevant in this work are:  In [13], Sugeng and Silavan extended the results in [12], by providing several classes of trees that admit this type of total labeling. They showed that regular caterpillars, firecrackers, caterpillarlike trees, path-like trees, and banana trees, are b-edge consecutive edge-magic graphs; we must note that the word regular has a different connotation for each variety of tree considered in [13].
Let G be a bipartite graph of order m and size n which stable sets are S 1 and S 2 . Rosa 1 [10] defined an α-labeling of G as an injective function f : As an immediate application of this kind of labeling, Rosa proved that there exists a cyclic decomposition of K 2tn+1 into copies of any graph of size n that admits an α-labeling. An α-graph is any graph that can be α-labeled. Among other results, Rosa showed that all caterpillars are α-trees. Several papers have followed Rosa' seminal work and multiple classes of α-trees are known. For example, in [9], Kotzig proved that almost all trees can be α-labeled; other two families of α-trees are the path-like trees [1] and the triangular trees [4]. Gallian [7] devotes an entire section of his survey to this labeling.
Suppose that G is an α-graph. Let S 1 and S 2 be the stable sets of G, where s 1 = |S 1 | and s 2 = |S 2 |. Assuming that f is an α-labeling of G that assigns the label 1 to a vertex of S 1 , then the valence of the m-edge consecutive edge-magic labeling obtained from f is k places the label 1 on an element of S 2 . Since f is also an α-labeling of G, the m-edge consecutive edge-magic labeling obtained using f instead of f , has valence k = 2m + 1 + s 2 .
Several years after the publication of his seminal paper on difference vertex labeling, Rosa [11] introduced the following definition. Let f be an α-labeling of a graph G of size n that assigns the label 1 to a vertex of S 1 ; the α-labeling f r of G given by is called inverse labeling of f . This labeling is also called reverse labeling of f . In general, f = f r but they have the same boundary value. Consequently, if f is an α-labeling of G, then f , f r , and f r are also α-labelings of G, where f and f r have boundary value s 1 , while f and f r have boundary value s 2 . With all these facts, the proof of the following result is straightforward. Theorem 1.3. Let G be an α-graph of order m and size m − 1, and let f be an α-labeling of G such that the vertex labeled 1 by f is in S 1 . If f = f r , then there are four m-edge consecutive edge-magic labelings of G, two with valence 2m + 1 + s 1 and two with valence 2m + 1 + s 2 .
In Figure 2 we show the labelings f , f r , f , and f r for a tree of order 11 with stable sets of cardinalities 5 and 6; in addition, we show the corresponding 11-edge consecutive edge-magic labelings.
In this work, we continue the study of b-edge consecutive edge-magic labelings initiated by

Main Results
In the introduction we mentioned that a super edge-magic labeling of a graph of size m is an m-edge consecutive edge-magic labeling. Figueroa-Centeno et al. [6] proved that if G is an αgraph of order m and size m − 1, then G is super edge-magic. Formulating this result in terms of b-edge consecutive edge-magic labelings we have the following.  Proof. Suppose that f is an α-labeling of G. By Theorem 2.1 we know that f can be transformed into a m-edge consecutive edge-magic labeling of G. Let g denote the m-edge consecutive labeling of G obtained using f . Then, the set of edge labels is {m + 1, m + 2, . . . , 2m − 1}. Therefore, the set of labels assigned by g to the edges of G is {1, 2, . . . , m − 1}. Consequently, g is a 0-edge consecutive edge-magic labeling of G.
In the next result we prove that for any α-graph of order m and size m − 1, with stable sets of cardinalities s 1 and s 2 , there exists a b-edge consecutive edge-magic labeling, where b is either s 1 or s 2 . Theorem 2.3. If G is an α-graph of order m and size m − 1, then there exists a b-edge consecutive edge-magic labeling of G where b is the cardinality of any of its stable sets.
Proof. Suppose that G is an α-graph of order m and size m − 1, with stable sets S 1 and S 2 , where |S 1 | = s 1 and |S 2 | = s 2 . Since G is an α-graph, there is an α-labeling f of G that assigns the label 1 to a vertex of S 1 . Consider the following labeling of the vertices of G: Thus, the labels assigned by g to the elements of S 1 form the set {1, 2, . . . , s 1 }, while the elements of S 2 receive the labels in Since 1 ≤ w ≤ m − 1, we get that m + s 1 + 1 ≤ g(u) + g(v) ≤ 2m + s 1 − 1. In other terms, {g(u) + g(v) : uv ∈ E(G)} is a set of m − 1 consecutive integers. We extend the labeling g to include the edges of G. Let uv ∈ E(G) such that g(u) + g(v) = 2m + s 1 − i for some i ∈ {1, 2, . . . , m − 1}, then g(uv) = m + s 1 + i. Hence, the labels assigned to the edges of G form the set {s 1 + 1, s 1 + 2, . . . , m + s 1 − 1}. Consequently, the labels assigned by g to the edges of G are m − 1 consecutive integers; in addition, the labels on V (G) ∪ E(G) form the set {1, 2, . . . , 2m − 1}. Therefore, g is a s 1 -edge consecutive edge-magic labeling. Since Thus, the valence of g is k = 2(m + s 1 ). Then, g is a s 1 -edge consecutive edge-magic labeling of G with valence k = 2(m + s 1 ).
If we use f instead of f , the resulting labeling is s 2 -edge consecutive edge-magic, and its valence is k = 2(m + s 2 ).  In Figure 3 we show the two b-edge consecutive edge-magic labelings obtained using f and f for a disconnected graph of order 22, size 21, with stable sets of cardinalities 10 and 12.
Since any tree of order m has size m−1, we deduce that any α-tree admits a b-edge consecutive edge-magic labeling where b is the cardinality of any of its stable sets.
Corollary 2.1. Let T be an α-tree with a stable set of cardinality b, then T admits a b-edge consecutive edge-magic labeling.
Some families of disconnected α-graphs, of order m and size m − 1, are known. Let x and y be positive integers such that y ≥ x + 2; in [2], Barrientos and Minion proved that if G y is a caterpillar of size y with stable sets S 1 and S 2 , such that there exist v ∈ S 1 adjacent to a leaf, and an α-labeling f of G y with the property that f (v) = s 1 − x − 1, then C 4x ∪ G y is an α-graph. The graph depicted in Figure 3 satisfies the conditions described above, where the vertex v is the vertex labeled 6 in the first representation. Barrientos and Minion [3] continued their work about disconnected graphs that admit an α-labeling; they proved that if G is an α-graph of order and size y, then tG ∪ P t is an α-graph for every positive integer t, where P t is the path of order t. Let L t−1 be any linear forest of size t − 1; Barrientos and Minion [3] proved that if G is an α-graph of order m and size n, with m < n, then tG ∪ L t−1 is an α-graph for every t ≥ 2. Therefore, all these graphs admit a b-edge consecutive edge-magic labeling for some values of b.

Conclusions
Suppose that G is a super edge-magic graph of size m. We have presented the fact that a super edge-magic labeling of G is an m-edge consecutive labeling with valence k. We proved that the existence of this m-edge consecutive labeling implies the existence of a 0-edge consecutive edge-magic labeling as well; the valence of this new labeling is 6m − k. In the case where G is a bipartite graph, with stable sets S 1 and S 2 , where |S 1 | = s 1 and |S 2 | = s 2 , an m-edge consecutive edge-magic labeling exists, provided that G is an α-graph; the valence of the m-edge consecutive labeling is k = 2m + 1 + s 1 or k = 2m + 1 + s 2 , depending on whether the vertex labeled 1 belongs to S 1 or S 2 , respectively.
We know now that when G is an α-graph, it also admits a b-edge consecutive edge-magic labeling, where b = s 1 or b = s 2 , whose valences are k = 2m + 1 + s 1 or k = 2m + 1 + s 2 , respectively. Furthermore, if we use the reverse labeling of the α-labeling f , we obtain a different b-edge consecutive edge-magic labeling, where the parameters b and k are the same for the labelings f and f r .
Since in the definition of a b-edge consecutive edge-magic labeling of a graph of size n, we have that b could be any element of {0, 1, . . . , n}, we may ask for which values of b such a labeling exists when the graph is an α-graph.