On the Graceful Cartesian Product of Alpha-Trees

A graceful labeling of a graph G of size n is an injective assignment of integers from the set {0, 1, . . . , n} to the vertices of G such that when each edge has assigned a weight, given by the absolute value of the difference of the labels of its end vertices, all the weights are distinct. A graceful labeling is called an α-labeling when the graph G is bipartite, with stable sets A and B, and the labels assigned to the vertices in A are smaller than the labels assigned to the vertices in B. In this work we study graceful and α-labelings of graphs. We prove that the Cartesian product of two α-trees results in an α-tree when both trees admit α-labelings and their stable sets are balanced. In addition, we present a tree that has the property that when any number of pendant vertices are attached to the vertices of any subset of its smaller stable set, the resulting graph is an α-tree. We also prove the existence of an α-labeling of three types of graphs obtained by connecting, sequentially, any number of paths of equal size.


Introduction
A difference vertex labeling of a graph G of size n is an injective mapping f from V (G) into a set N of nonnegative integers, such that every edge uv of G has assigned a weight defined by |f (u) − f (v)|. All labelings considered in this work are difference vertex labelings. A labeling is called graceful when N = {0, 1, . . . , n} and the induced weights are 1, 2, . . . , n. If G admits such a labeling, then it is called a graceful graph.
Let G be a bipartite graph where {A, B} is the natural bipartition of V (G), we refer to A and B as the stable sets of V (G). A bipartite labeling of G is an injection f : V (G) → {0, 1, . . . , t} for which there is an integer λ, named the boundary value of f , such that f (u) ≤ λ < f (v) for every (u, v) ∈ A × B, that induces n different weights. This is an extension of the definition given by Rosa andŠiráň in [11]; there, they focused on bipartite labelings of trees. From the definition we may conclude that t ≥ |E(G)| , furthermore, the labels assigned by f on the vertices of A and B are in the integer intervals [0, λ] and [λ + 1, t], respectively. If t = n, the bipartite labeling f is an α-labeling. By an α-graph we mean a graph that admits an α-labeling. If G is an α-graph, λ is the smaller of the two vertex labels of the edge of weight 1. If G is an α-tree, then λ = |A| − 1.
Let f : V (G) → {0, 1, ..., t} be a labeling of a graph G of size n : , is the complementary labeling of f. If f is graceful, f is also graceful. Moreover, if f is an α-labeling with boundary value λ, then f is an α-labeling with boundary value n − λ − 1.
• g : V (G) → {c, c + 1, ..., c + t}, defined for every v ∈ V (G) and c ∈ Z as g(v) = c + f (v), is the shifting of f in c units. Note that this labeling preserves the weights induced by f.
Note that if f is an α-labeling, then f r is also an α-labeling with boundary value λ. Let f be an α-labeling of a tree G of size n with boundary value λ. Suppose that f is transformed into a d-graceful labeling shifted c units. Then the stable set A receives the labels c, c + 1, ..., c + λ and the stable set B receives the labels c + λ + d, c + λ + d + 1, ..., c + n + d − 1.
Several authors have studied graceful and α-labelings of graphs that are the Cartesian product of two graphs. Maheo [9] proved that if G is strongly graceful (a special type of graceful labeling with some specific restrictions), then G × K 2 is strongly graceful. Fu and Wu [5] showed, in one way, a stronger result. They proved that if T is a tree with an αlabeling and the sizes of the two stable sets of T differ by at most one, then T × P m has an α-labeling. In [13], Wu went even further, proving that if G has size n, order n + 1, and admits an α-labeling with boundary value λ, where |n − 2λ − 1| ≤ 1, then G × P m is graceful for all m. Jungreis and Read [8] proved, among other results, that P m × P n is graceful. This result is a special case of the main result of Fu and Wu [5]. El-Zanati and Vanden Eynden [3] used a different hypothesis to prove that G × P m is graceful. They proved that if G has a strongly graceful labeling, then G × P m has an α-labeling. Note that they used the name strong α-labeling instead of strongly graceful labeling.
In Section 2 we consider the family R of all trees of size n such that when T ∈ R, the cardinality of the two stable sets of T differ by at most one. We prove that for any two α-trees, T 1 and T 2 , in R, the Cartesian product T 1 × T 2 results in an α-graph.
Frucht and Harary [4] defined the corona of two graphs as the graph obtained by taking one copy of a graph G of order n, and n copies of a graph H, and then joining, by an edge, the kth vertex of G to every vertex in the kth copy of H. The corona of G and H is denoted by G H. As a consequence of the results of Stanton and Zarnke [12], we know that if G is a graceful tree, then G rK 1 is graceful for every r ≥ 1; in other terms, when r pendant vertices are attached to every vertex of a graceful tree G, the resulting tree, G rK 1 is also graceful. In Section 3 we study a related problem, presenting a tree of size 12 and stable sets with cardinalities 5 and 8. This tree has the property that we can attach any number of pendant vertices to the vertices of any nonempty subset of the stable set with five elements and the resulting tree can be α-labeled. The path P n is the only graph, that we know, with the same property; that is, we can attach any number of vertices to any subset of vertices of P n .
In Section 4, we give α-labelings for three families of trees obtained by connecting, sequentially, any number of copies of P n with paths of length 2. We discuss here the connection between these trees and other types of combinatorial structures.
The reader interested in graph labeling is referred to Gallian' survey [6] for more information about the subject. In this paper we follow the notation and terminology used in [2] and [6].

Cartesian Product of Regular α-Trees
Jungreis and Read [8] proved that the Cartesian product of the paths P m and P n is an α-graph. For any given path P n , the difference of the cardinalities of its stable sets is at most one and P n is an α-tree. These facts motivate the study of the Cartesian product of other trees satisfying these conditions. A tree of order n is said to be regular when the cardinalities of its stable sets are equal or differ by one. We claim that the Cartesian product of two regular α-trees is an α-graph.
Theorem 2.1. If S and T are regular α-trees, then S × T is an α-graph.
Proof. Suppose that S and T are regular α-trees of order m and n, respectively. Thus, S × T is a graph of order µ = mn and size ξ = (m − 1)n + (n − 1)m. Let S 1 , S 2 , ..., S n be the n copies of S in S ×T. Assume that f is an α-labeling of S with boundary value λ such that the label λ is assigned to a vertex of the largest stable set of V (S). Recall that f corresponds to the complementary labeling of f. For all odd values of i, the labeling Hence, the weights obtained on the lth copy of S form the set V n+1−l = {j + (2m − 1)(l − 1) : 1 ≤ j ≤ m − 1 and 1 ≤ l ≤ n}. The smallest label used is 0 and the largest one is m−1+(2m−1)(n−1) = m(n−1)+n(m−1) = ξ. In addition, the shiftings guarantee that every label is used exactly once.
Fu and Wu (see [5], Proposition 2.6) proved that if S is a regular α-tree of order m, then S × P 2 is an α-graph. In their labeling of S × P 2 , the first copy of S has an α-labeling f transformed into a 2m-graceful labeling; the second copy of S is labeled using f shifted m units. The edges connecting both copies of S have weights m, m + 1, ..., 2m − 1. In our case, for every edge uv of T, we have a graph of the form S × P 2 .
Applying the result of Fu and Wu to our case, we have that for every edge uv of T, there is a subgraph S × P 2 of S × T , such that the weights of the edges, connecting the two copies of S, are consecutive integers. Hence, the weights of the edges connecting the copies of S l and S l+1 in S × T form the set Therefore, for a fixed value of l ∈ {1, 2, . . . , n − 1}, In addition, and Since any pair of "adjacent" copies of S are labeled using f and f , the final labeling of S × T is an α-labeling with boundary value m(n − 1) − 1.
In Figure 1 we show an example of the α-labeling of S × T where S and T are caterpillars of order m = 7 and m = 8, respectively; the original α-labelings of the graphs are exhibited on the left and on the top of S × T . In addition we show the labelings f 1 and f 2 of the first two copies of S. A graph G of size n is harmonious if there exists an injection f : V (G) → Z n such that when each edge uv is assigned the weight f (u) + f (v) (mod n), the resulting weights are distinct. When each edge uv is assigned the weight f (u) + f (v), we say that f is sequential if the weights are consecutive integers. Thus, it is possible to transform a sequential labeling into a harmonious labeling by reducing the weights f (u) + f (v) modulo n. Grace [7] has shown how to transform an α-labeling f of a graph G of size n with boundary value λ, into a labeling g such that the weights g(u) + g(v) are λ + 1, λ + 2, ..., λ + n. The labeling g is defined for all the vertices in V (G) by: Jungreis and Read [8] used Grace's result to prove that the Cartesian product of two paths is sequential when one of the paths has even order. In the case P 2m × P 2n+1 , they started with an α-labeling f of P 2m × P 2n+1 , which is transformed into the labeling h which is defined as: The case P 2m × P 2n is a little more complicated. Start with an α-labeling f of P 2m × P 2n and subtract 1 from every vertex label of the 2nth copy of P 2m . The new labeling of P 2m ×P 2n , denoted by f * , is also an α-labeling with boundary value λ * . They followed the same steps, used in the previous case, replacing f by f * .
Suppose that S is a regular tree of even order; following the same steps of Jungreis and Read, we can convert the α-labeling of S × T into a sequential labeling, which can be transformed into a harmonious labeling. Thus, as a consequence of Theorem 1, and the results in [7] and [8], we have the following two corollaries.
Corollary 2.2. If S and T are regular α-trees and S has even order, then S × T admits a sequential labeling. Corollary 2.3. If S and T are regular α-trees and S has even order, then S × T admits a harmonious labeling.

α-Labelings of a Special Tetrapod
A tetrapod is a rooted tree whose root r has degree four and all the internal vertices have degree two, in other terms, the one-point union of two paths, where the amalgamated vertices are internal vertices. Within this class of trees, we distinguish the tetrapod obtained by amalgamating the central vertices of two copies of the path P 7 . Let S denote this special tetrapod; suppose that A and B are its stable sets, where |A| = 5 and |B| = 8. A picture of S is shown in Figure 2. We claim that any tree obtained from S, by attaching any number of pendant vertices to the elements of A, is an α-tree. (Note that the number of vertices attached may vary from vertex to vertex in A.) This property is interesting to us, because it resembles the fact that caterpillars of diameter d can be obtained in the same form, by attaching pendant vertices to the internal vertices of P d+1 . We are not aware of any other tree with the same property. This raises the question about the existence of other trees with the same property.
Theorem 3.1. Any tree T of size n > 12, obtained from S by attaching any number of pendant vertices to the elements of A, is an α-tree.
Proof. Recall that the star with m ≥ 1 leaves is the graph S m ∼ = K 1,m . For practical reasons, we allow here m ≥ 0, that is, an isolated vertex is considered a star with 0 leaves.
The tree T can be constructed by connecting a leaf, from each of four stars, to the central vertex of a new star. For each 0 ≤ i ≤ 4, let S i be the star of size n i , where n i ≥ 2 except for i = 2, where n 2 ≥ 0. Let f i be the α-labeling of S i that places the label 0 at the center of the star. The labeling f i is transformed into a k -graceful labeling shifted i units, where k = 1 when i = 4 and k = 5 − i + 4 j=i+1 n j , otherwise. Suppose that n = 4 + 4 j=0 n j , so n is the size of T. Consequently, the labels assigned to the vertices of 4 i=0 S i form the set {0, 1, ..., n}, and the induced weights form the set {1, 2, ..., n} − {n 4 + 1, n 4 + n 3 + 2, n 4 + n 3 + n 2 + 3, n 4 + n 3 + n 2 + n 1 + 4}. Since the centers of the stars S i are labeled 0, 1, 2, 3, and 4, respectively, and S 4 has a leaf labeled n 4 + 3, S 3 has a leaf labeled n 4 + n 3 + 4, S 1 has a leaf labeled n 4 + n 3 + n 2 + 5, and S 0 has a leaf labeled n 4 + n 3 + n 2 + n 1 + 6, we can connect these leaves to the central vertex of S 2 , labeled 2, to obtain four edges whose weights correspond to the missing weights mentioned before. Summarizing, the new edges connect four leaves to a central vertex, therefore the resulting labeling is a bipartite labeling of T . Moreover, the labels assigned to the vertices of T are 0, 1, . . . , n, the induced weights are 1, 2, . . . , n, and 4 is the largest label assigned to a central vertex of the starts S i . Hence, we have obtained an α-labeling of T with boundary value λ = 4.
In Figure 2 we show a possible tree T with the α-labeling described in the proof of Theorem 4. In this example, n 0 = 5, n 1 = 7, n 2 = 4, n 3 = 4, and n 4 = 8.

α-Labelings of Some Fences
Suppose that for every 1 ≤ i ≤ t, P n i is a path of order n i ≥ 3. For each 1 ≤ i ≤ t − 1, a fence F is a tree obtained by connecting, with a path of length l i , an internal vertex of P n i with an internal vertex of P n i+1 . The length of the fence is given by l = In [10], Rosa proved that for any n and any vertex v of P n+1 , there exists an α-labeling f of P n+1 such that f (v) = 0, if and only if, v is not the central vertex of P 5 . If λ is the boundary value of f, then f r (v) = λ. In [1], Barrientos proved that given two α-labeled graphs, G 1 and G 2 , the vertex amalgamation of the vertices labeled 0 in G 1 and G 2 , produces a new α-graph. Thus, this argument can be used to prove that all trees with up to four leaves are graceful; some special cases need ad hoc arguments. Extending the result in [1], it is possible to prove that any fence, constructed connecting, sequentially, the vertices labeled 0 of the P n i , is an α-tree. This fact motivates our study of graceful and α-labelings of other types of fences, that is, where the vertices labeled 0 are not the ones connected by a path of length l i .
In this subsection, we study a type of fence where all the P n i are isomorphic to the path P 2n+1 . We also show an α-labeling for a fence constructed using the path P 2n . Let P 1 , P 2 , ..., P t be copies of the path P 2n+1 and V ( We denote by R n,t the family of all fences of length 2(t − 1) obtained by connecting, with a path of length 2, the .., 2n − 2}. Note that the family R n,t is quite robust; for instance, when n = 4 and t = 5, there are 25 nonisomorphic fences in R 4,5 . There is a bijection between R 4,t and the set of equivalence classes of (t − 1)-tuples of elements of Z t−1 3 , where two (t − 1)-tuples are equivalent if one can be obtained from the other by a sequence of operations R and C, where R denotes reversal and C denotes the 2's complements (i.e., C(x) = 2 − x). More information about the number of (t − 1)-tuples with elements in Z 3 can be found in the Online Encyclopedia of Integer Sequences, sequence A001998 [14]. It is mentioned there that a(19) = 290, 585, 050; that is, the number of elements in R 4,20 is 290, 585, 050.
Theorem 4.1. If F ∈ R n,t , then F is an α-tree for all positive values of n and t.
Proof. Let C 1 = P 1 and for every 2 ≤ i ≤ t, C i be the caterpillar obtained from P i , by attaching a pendant vertex to the vertex v i q i , where q i is any number in {2, 4, ..., 2n − 2}. We denote the new vertex by u i q i . Suppose that f is an α-labeling of C i such that f (v i 0 ) = 0 and f r is its reverse α-labeling, so f r (v i 0 ) = n. For each 1 ≤ i ≤ t, we assume that the initial labeling f i of C i is the labeling f when i is odd, and the labeling f r when i is even. Thus, f i is transformed into a k i -graceful labeling g i shifted d i units, where k i = 2(n + 1)(t − i) + 1 and d i = (n + 1)(i − 1).
The use of f and f r in alternated copies of C i implies that the vertices connected belong to different stable sets of F ; therefore, the final labeling of F is a bipartite labeling that uses the labels 0, 1, . . . , 2(nt + t − 1) to induce the weights 1, 2, . . . , 2(nt + t − 1); the boundary value of this labeling is λ = t(n + 1) − 1.
In Figure 3 we show the α-labeling of a fence in R 5,4 , constructed using Theorem 4.1, with q 1 = 4, q 2 = 6, and q 3 = 2. Suppose now that P 1 , P 2 , . . . , P t are copies of the path P 2n . Let S n,t be the family of all fences of length 2(t − 1) obtained by connecting the vertex v i n−1 of P i to the vertex v i+1 n−1 of P i+1 , for all 1 ≤ i ≤ t − 1. We claim that if F ∈ S n,t , then F is an α-tree. Proposition 4.1. If F ∈ S n,t , then F is an α-tree.
Up to this point, we have that the vertices of F are labeled with the integers 0, 1, . . . , t(2n+ 1) + 2(t − 1), and the weights induced on the edges of t i=1 C i form the set {1, 2, . . . , 2t(n + 1) Note that for each 1 ≤ i ≤ t − 1, the vertex v i n−1 has label n−1 2 + n(i − 1) and the vertex u i+1 n−1 has label t(2n + 1) − (n + 1)i − n+1 2 . So, the edge v i n−1 u i+1 n−1 has weight (2n + 1)(t − i). Since the labelings of the C i are α-labelings shifted conveniently and the edges v i n−1 u i+1 n−1 connect vertices in different stable sets, we have that the final labeling of F is an α-labeling with boundary value λ = nt − 1.

α-Labelings of 2-Link Fences
Another type of α-graph that can be constructed using α-labeled paths are the 2-link fences. Let P 1 , P 2 , . . . , P r be disjoint copies of P n . For each and v i+1 k i of P i+1 , respectively, for all 1 ≤ i ≤ r − 1, where j i , k i ∈ {1, 2, . . . , n} and |j i − k i | is odd. Let F r,n be the family of all the 2-link fences constructed using r copies of P n . Note that F r,n is a robust family, for instance, when r = 3 and n = 7, we have counted 42 non-isomorphic elements in F 3,7 .
The main result of this subsection is associated to a property of an α-labeling of the path P n . Suppose that v 1 , v 2 , . . . , v n are the consecutive vertices of P n . Let f : V (P n ) → {0, 1, . . . , n − 1} be the α-labeling of P n , given by Rosa [10]; thus, Then, the reverse of the complementary labeling is defined as: when n is odd, and when n is even. We claim that all the elements of F r,n are α-graphs. Before proving this result, we prove that all the members of F 2,n are α-graphs.
Proof. Suppose that G ∈ F 2,n , then there exist j, k ∈ {1, 2, . . . , n} such that |j − k| is odd and j 1 = j, k 1 = k. The vertices of P 1 are labeled using the α-labeling f , described above, the vertices of P 2 are labeled using f r . Once these labelings are in place, f is transformed into a (n + 2)-graceful labeling and f r is shifted n 2 units. In this way, when i is odd, f (v 1 i ) = i−1 2 and f r (v 2 i ) = n + i−1 2 . Hence, the edge v 1 i v 2 i has weight n. When i is even, , and the edge v 1 i v 2 i has weight n + 1. Since j and k have different parity, the edges v 1 j v 2 j and v 1 k v 2 k have weights n and n + 1. Recall that the induced weights on P 1 are n+2, n+3, . . . , 2n, and on P 2 are 1, 2, . . . , n−1. So, the weights of the edges of G are 1, 2, . . . , 2n. The use of f and f r guarantees that the final labeling of G is an α-labeling with boundary value λ = n − 1.
Theorem 4.3. Let n ≥ 4; if G ∈ F r,n , then G is an α-graph.
Proof. Let G ∈ F r,n and P 1 , P 2 , . . . , P r be the disjoint copies of P n used to construct G. We label the vertices of P i using the labeling f when i is odd, and f r when i is even. The labeling of P i is transformed into a ((n+1)(r−i)+1)-graceful labeling shifted (n+1)(i−2)/2 units except when both, n and i are even, where the labeling is shifted n/2 + (n + 1)(i − 2)/2 units.
Using the lemma, it is straight forward to prove that the edges connecting P i and P i+1 , 1 ≤ i ≤ r − 1, have weights (n + 1)(r − i) and (n + 1)(r − i) − 1. Since consecutive copies of P n use the labelings f and f r , the resulting labeling of G is an α-labeling. Consider the 2-link fence F formed by using t copies of P 2n , where the vertices v i 1 and v i 2n of P i are connected to the vertices v i+1 1 and v i+1 2n , respectively when 1 ≤ i ≤ t − 1.
Since v i 1 and v i 2n are in different stable sets of V (P i ), F is an α-graph. This 2-link fence can also be seen as a cycle C 2(t+2n−2) with t − 2 parallel P 2n -chords.
Corollary 4.4. For all t ≥ 2 and n ≥ 1, the cycle with t − 2 parallel P 2n -chords is an α-graph.
Remark 1. Note that the path P n , used in Theorem 4.3, can be replaced by any α-tree of size n and the result still holds.