Series-Parallel Operations with Alpha-Graphs

Among difference vertex labelings of graphs, α-labelings are the most restrictive one. A graph is an α-graph if it admits an α-labeling. In this work, we study a new alternative to construct α-graphs using, the well-known, series-parallel operations on smaller α-graphs. As an application of the series operation, we show that all members of a subfamily of all trees with maximum degree 4, obtained using vertex amalgamation of copies of the path P11, are α-graphs. We also show that the one-point union of up to four copies of Kn,n is an α-graph. In addition we prove that any α-graph of order m and size n is an induced subgraph of a graph of order m+ 2 and size m+ n. Furthermore, we prove that the Cartesian product of the bipartite graph K2,n and the path Pm is an α-graph.


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)|. The labeling f is called graceful when N = {0, 1, . . . , n} and the set of induced weights is {1, 2, . . . , n}. If this is the case, G is called a graceful graph. Let G be a bipartite graph and {A, B} be the natural bipartition of V (G), we refer to A and B as the stable sets of V (G) and assume that |A| = a and |B| = b. A bipartite labeling of G is an injection f : V (G) → {0, 1, . . . , s} 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 of bipartite labeling given by Rosa andŠiráň [14]. From the definition we may conclude that s ≥ |E(G)|; furthermore, the labels assigned by f on the vertices of A and B are in the sets {0, 1, . . . , λ} and {λ + 1, λ + 2, . . . , s}, respectively. If s = n, the function f is an α-labeling and G is an α-graph. If f is an α-labeling of a tree and f −1 (0) ∈ A, then its boundary value is λ = a − 1.
Suppose that f : V (G) → {0, 1, . . . , n} is a graceful labeling of a graph G of size n: • f : V (G) → {0, 1, . . . , n}, defined for every v ∈ V (G) as f (v) = n − f (v), is the complementary labeling of f . Note that f preserves the weights induced by f .
• g : V (G) → {c, c+1, . . . , c+n}, defined for every v ∈ V (G) and c ∈ N as g(v) = c+f (v), is the shifting of f in c units. Note that this labeling preserves the weights induced by f .
Suppose now that f is an α-labeling of G with boundary value λ.
is the reverse labeling of f . Note thatf is also an α-labeling with boundary value λ. For example: let f be an α-labeling of a tree T of size n with boundary value λ. Suppose that f is transformed into a d-graceful labeling shifted c units. Then the elements of A are labeled with the integers in [c, λ + c], the elements of B are labeled with the integers in [c + λ + d, c + n + d − 1], and the induced weights form the interval [d, n + d − 1].
Several graph operations involving graceful and/or α-graphs have been studied in the last fifty years. The Cartesian product has been extensively investigated for several families of graphs, as well as the union and the one-point union, the corona product, the join, the tensor product, and many other ways to combine graceful graphs to obtain new greaceful and α-graphs. A good account of the newest techniques can be found in [11].
In Section 2 we perform series-parallel operations on α-graph to create new α-graphs; in addition, we prove that the one-point union of up to four copies of K n,n results in a new αgraph. In addition we present another example of a family of α-trees that can be constructed using the series operation. In Section 3 we show that any α-graph of order m and size n is an induced subgraph of an α-graph of order m + 2 and size m + n. We close this section showing that the Cartesian Product K 2,n × P m is an α-graph for all positive values of m and n. The reader interested in graph labelings is refered to Gallian's survey [9] for more information about the subject. In this paper we follow the notation and terminology used in [8] and [9].

Series-Parallel Operations with α-Graphs
In this section we investigate how to operate α-graphs to produce larger α-graphs using the well-known series-parallel operations. We start analyzing the series operations, showing that we can always combine smaller α-graphs to produce new and larger α-graphs. As a consequence of this result we prove that the one-point union, of up-to four copies of K n,n , is an α-graph. We also prove here that all trees with maximum degree 4, obtained applying the series operation to a collection of α-labeled copies of P 11 , are α-graphs when the distance between any pair of consecutive vertices of degree 4 is even. The last part of the section is devoted to the study of the parallel operation on a family of α-graphs.

The Series Operation
A series-parallel graph with distinguished terminals l and r, denoted (G, l, r), is defined recursively as follows: • The graph consisting of a single edge v 1 v 2 is a series-parallel graph (G, l, r) with l = v 1 , and r = v 2 .
• A parallel operation (G 1 , l 1 , r 1 ) p (G 2 , l 2 , r 2 ) forms a series-parallel graph by identifying l 1 with l 2 and r 1 with r 2 .
Let f be an α-labeling of a graph G of size n with boundary value λ. Let u, v ∈ V (G) such that f (u) = λ + 1 and f (v) = n. In the following result, these vertices correspond to the vertices l and r in the definition of series-parallel graph.
As a consequence of this result we can prove that the one-point union of up to four complete bipartite graphs K n,n is an α-graph. This result is related to some other problems that we can find in the literarture. For example, in [15], Selvaraju worked with α-labelings of the one point union of complete bipartite graphs, showing that there is an α-labeling for the one-point union of the following graphs: K m,n 1 and K m,n 2 ; K m 1 ,n 1 , K m 2 ,n 2 , K m 3 ,n 3 when m 1 ≤ m 2 ≤ m 3 and n 1 < n 2 < n 3 ; and K m 1 ,n , K m 2 ,n , K m 3 ,n where m 1 < m 2 < m 3 ≤ 2n. In a related line, Sethuraman and Selvaraju [16], proved that the one-point union of any number of non-isomorphic complete bipartite graphs is a graceful graph. Sudha [17] showed that the graph formed with any number of complete bipartite graphs that share one stable set, is graceful. Barrientos [5] proved that the graph obtained as the one-point union of K m 1 ,n 1 , K m 2 ,n 2 , . . . K mt,nt , where each K m i ,n i appears at most twice in that list and gcd(n 1 , n 2 , . . . , n t ) = 1, is graceful. Proposition 2.1. For the one-point union of two copies of K n,n , there exists an α-labeling that assigns the label 2n 2 to the vertex of degree 2n.
Since the intersection between all these label sets is n 2 + n, we can identify the two vertices with this label to produce an α-labeled version of the one-point union of two copies of K n,n , where n 2 +n−1 is the boundary value of the associated α-labeling. Thus, the reverse labeling places the label 2n 2 on the vertex of degree 2n that results of the amalgamation.
Consider now K n,n with the α-labeling described within the proof of Proposition 2.1, so, one of the stable sets of K n,n has a vertex labeled n and this is the smallest label in that stable set. By Proposition 2.1, we know that there is an α-labeling, of the one-point union of two copies of K n,n , that places the highest label, 2n 2 , on the vertex of maximum degree. Therefore we can apply Theorem 2.1 to prove that the one-point union of three copies of K n,n is an α-graph. On the case of the one-point union of four copies of K n,n , we use Theorem 2.1 on two copies of the one-point union of two K n,n . We start with two copies of the one-point union of two K n,n labeled using Proposition 2.1; once this is done, one of these labelings is transformed into its reverse labeling. In this way, the vertex of highest degree in the first copy is labeled 2n 2 and in the second copy is labeled n 2 + n − 1. Hence Theorem 2.1 can be applied to prove the following theorem.
Theorem 2.2. The one-point union of three or four copies of K n,n is an α-graph.
In Figure 1 we show an example of these results for the case of K 3,3 .

A Chain of Paths Crossing: An Application
Suppose that for every 1 ≤ i ≤ t, P i is a path of length at least four with distinguished vertices v i 1 and v i 2 such that they are not leaves. A tree T is said to be a chain of paths crossing if for every Since the distinguished vertices, used in the amalgamation of P i and P i+1 , are interior vertices, we may calculate the number of leaves in T (vertices of degree 1) to be 2t. Therefore, the number of vertices of degree 3 is t i=1 |V (P i )| − 4t + 2. When the distinguished vertices are taken in an ad hoc manner, we can apply the series operation to produce an α-labeling of this type of tree. Rosa [12], proved that for every v ∈ V (P n ), n ≥ 1, there exists an α-labeling f of P n such that f (v) = 0, except when v is the central vertex of P 5 . Thus, for a given path P n , n ≥ 4, with an α-labeling f , the distinguished vertices v 1 and v 2 are those where f (v 1 ) = 0 and f (v 2 ) = λ, where λ is the boundary value of f . We want to show how powerful is the series operation, to do that, we work here with the case where the P i are copies of P 11 ; certainly the next result can be modified to include paths of other lengths.
Before presenting the theorem, let us introduce the α-labelings of P 11 that are used in its proof. We show these labelings in Figure 2. On each path, we have highlighted the vertices labeled 0 and λ, which are crucial to apply the series operation. Theorem 2.3. If T is a chain of paths crossing where every path is a copy of P 11 and the distance between any pair of consecutive vertices of degree four is even, then T is an α-tree.
Proof. Suppose that T is a chain of paths crossing formed with t copies of P 11 in such a way that the distance between the distinguished vertices, v i 1 and v i 2 , of the ith copy of P 11 is even, that is, dist(v i 1 , v i 2 ) ∈ {2, 4, 6, 8}. Note that the labelings of P 11 in Figure 2 can be used here because, for every j ∈ {1, 2, 3, 4}, under the labeling φ j , dist(v i 1 , v i 2 ) = 2j. The first and the last copy of P 11 can be labeled with any of the labelings in Figure 2. For every 1 < i < t, the labeling of the ith copy of P 11 is determined by the distance between the distinguished vertices of P 11 ; thus, if dist(v i 1 , v i 2 ) = 2j, then the selected labeling is φ j . Assuming that the ith copy of P 11 has been labeled, its initial α-labeling is amplified to produce a (10i + 1)-graceful labeling; finally this transitory labeling must be shifted conveniently to produce the desired α-labeling of T .
In Figure 3 we show an example of this construction for a tree T of size 60 where the sequence of distances, in between vertices of degree four, is 6, 4, 2, 8.
It may seem that this result does not contribute that much in the discovery of new graceful trees, however, it produces a large amount of them. In fact, for the case that we have under consideration, that is, for P 11 , the number of non-isomorphic trees constructed with t copies of P 11 is of the order of 2 2t−1 .
Theorem 2.4. Let t ≥ 3 be an integer, the number of trees formed as a chain of t paths crossing, where every path used is isomorphic to P 11 , and the distance between distinguished vertices belong to {2, 4, 6, 8}, is given by (i) 2 2t−1 + 2 t when t is odd, Proof. Let T be a tree that is a chain of paths crossing, formed using t ≥ 3 copies of P 11 such that the distance between consecutive vertices of degree four belongs to D = {2, 4, 6, 8}. Every copy of P 11 is labeled with one of the labelings in Figure 2. Since we have four different α-labelings of P 11 , there are 4 t different posibilities for T . Each of these α-labeled trees has associated a string of length t − 2, where every number on this string comes from D. But the string and its reverse represent the same tree and some strings are reversible, that is, they are the same when read backwards. So, in order to determine the number of different strings (or non-isomorphic trees formed in the prescribed way) we need to calculate the number of reversible strings.
When t is odd, there is an integer s such that t − 2 = 2s + 1. Thus, the number of reversible strings of length t − 2 is given by (4 s )(4)(1 s ), where the first factor is the number of strings of length s, the second factor is the amount of elements of D that can be placed in the median position of the string. The third factor is the number of options for the last s positions in the string; recall that these integers are determined by the selection made for the first s entries in the string. Thus, we have 4 s+1 reversible strings of length t − 2. But this number does not consider the first and the last copy of P 11 in T . So, the number of symmetric chains of t paths crossing is 4 · 4 s+1 = 4 s+2 . Hence, the number of non-isomorphic chains of t paths crossing is When t is even, there is an integer r such that t−2 = 2r. Hence, the number of reversible strings of length t − 2 is given by (4 r )(1 r ) = 4 r . As in the odd case, the first factor is the number of strings of length r and the second factor is the number of options for the last r entries of the string. The number 4 r does not consider the first and the last copy of P 11 in T . So, the number of symmetric chains of t paths crossing is 4 · 4 r = 4 r+1 . Now the number of non-isomorphic chains of t paths crossing is This concludes the proof.
Let a(t) be the number of non-isomorphic trees T we can form in this way. In Table 1 we show the first values of a(t), showing how fast this number grows. The sequence formed by the consecutive values of a(t) corresponds to OEIS sequence A032121 [6].

The Parallel Operation
Now we turn our attention to graphs that can be constructed using the parallel operation. If G is an α-graph of size n, then G is bipartite. Let A and B be the stable sets of G, we assume that |A| ≤ |B|. For i = 1, 2 let G i be an α-graph of size n i and let f i be an α-labeling of G i , with boundary value λ i , such that the label λ i is assigned to a vertex of A i . We say that G 1 and G 2 are compatible if (i) the vertices x i and y i of G i labeled 0 and λ i are leaves and (ii) n 1 = λ 1 + λ 2 .
We claim that the graph G obtained, using the parallel operation on G 1 and G 2 , is an α-graph.
Theorem 2.5. If G 1 and G 2 are compatible graphs, then there exist vertices x 1 , y 1 ∈ G 1 and x 2 , y 2 ∈ G 2 , such that G = G 1 p G 2 is an α-graph.
Now we present a construction that produces compatible α-graphs. Suppose that G is an α-labeled graph of size n with an α-labeling f . If the vertex labeled 0 by f is not a leaf, then we create a graph G by attaching a leaf to the vertex labeled n in G, in such a way that the new vertex is labeled 0 and all the other labels are increased by one unit, we obtain an α-labeling of G where the vertex labeled 0 is a leaf. Something similar can be done to G if its labeling f does not assign the label λ to a leaf. In this case the new graph G , is obtained by attaching a leaf to the vertex labeled λ + 1. The α-labeling of G is obtained assigning the label λ + 1 to this leaf and adding one unit to each original label greater than λ. Clearly, if this does not assign the labels 0 and λ to leaves of G, a graph G can be formed applying these modifications. As an example, in Figure 4 we show, step by step, how the standard α-labeling of G = C 8 can be used to transform G into a self-compatible graph. In Figure 5 we show some examples of the graphs obtained using the parallel operation on the cycles C 16 , C 12 , C 8 , and C 4 .
The family E , of all caterpillars of even diameter, provides infinitely many pairs of compatible graphs. Rosa [13] proved that for any given caterpillar (or path), there exists an α-labeling f that assigns the label 0 to a vertex of maximum eccentricity, and in the case where the diameter is even, the vertex labeled λ (where λ is the boundary value of f ) is the other extreme of the path of maximum length that has one extreme at the vertex labeled 0. So, for any given caterpillar X of size n in E with stable sets of cardinalities a and b, any caterpillar X in E having one stable set of cardinality n − a is compatible with X. Therefore, X p X is an α-graph. Note that the graph X p X is a type of unicyclic graph named hairy cycle; that is, a cycle with pendant vertices attached. In [4], Barrientos proved that all hairy cycles are graceful; when the girth of the cycle is even, the labeling used to proved that result is in fact an α-labeling.

α-Graphs Inside α-Graphs
In [2], it was proved that given a graceful graph G and an α-graph H, there is a vertex amalgamation of G and H that results in a graceful graph. This idea is a generalization of the quite natural construction of newer graceful graphs by attaching pendant vertices to the vertex labeled zero. In this section, we explore how to extend an α-labeled graph to a larger α-graph. Let G be an α-graph of order n + 1 and size n. As usual, we are assuming that there exists an α-labeling of G that assigns its boundary value to a vertex of the stable set A. We claim that there exists an α-graph H of size 2n + 1 and order n + 3 that contains G as an induced subgraph.
Theorem 3.1. For each α-graph G of order n + 1 and size n, there exists an α-graph H of order n + 3 and size 2n + 1, such that G is an induced subgraph of H.
In Figure 6 we show two examples of this procedure. In the first case G is a caterpillar of size 8; in the second case, G is a disconnected α-graph of size 7. : α-labelings of graphs containing smaller α-graphs Using the construction of α-graphs just presented, many other graphs of this type can be obtained. To explain this statement we can analyze the first graph in Figure 6. There, the edges are located in three levels: level 1 contains the edges incident to the vertex labeled 0, level 2 contains the edges of G, and level 3 contains the edges incident to the vertex labeled 9 (or λ + 1 in a more general version). We can delete some of the edges in levels 1 and 3 and still have an α-graph; if we delete, from left to right, any number of consecutive edges of level 1, and from right to left, any number of consecutive edges on level 3, then the labeling of the resulting graph can be transformed into an α-labeling. We illustrate this fact in Figure  7. In this section we present a new family of α-graphs that are the result of the Cartesian product of K 2,n and P m . The graph K 2,n × P m has order m(n + 2) and size 3mn + 2m − n − 2.
In the next theorem we use the following well-known α-labelings of K 2,n . First, suppose that A = {u 1 , u 2 } and B = {v 1 , v 2 , . . . , v n } are the stable sets of K 2,n : • f (u 1 ) = 0, f (u 2 ) = n, and f (v i ) = n + i for all 1 ≤ i ≤ n.
Essentially these labelings are adaptations of the α-labeling of K m,n introduced by Rosa [13]. Proof. Let R 1 , R 2 , . . . , R m be the m copies of K 2,n in K 2,n × P m . The stable sets of R j , Suppose that the inital labeling of R j is f when j is odd and g when j is even. Regardless of the parity of j, this initial labeling is transformed into a δ j -graceful labeling shifted i units, where δ j = (3n + 2)(m − j) + 1 and Assume that j is even. The new labels of the vertices in A j and B j form the sets

respectively.
For j − 1 and j + 1, if R j+1 exists, we get the following sets: , we conclude that we have an injective assignment of labels. Furthermore, the smallest label assigned is minL A 2−1 = 0 and the largest one is maxL B 2−1 = (3n + 2)(2m − 2)/2 + n + n = 3mn + 2m − n − 2. Now we turn our attention to the weights induced by this labeling on the edges of K 2,n × P m . Let h denote the labeling of our graph. As we said before, when h is restricted to R j , 1 ≤ j ≤ m, it is an amplification of an α-labeling, so the weights on the edges of K 2,n × P m form the set Note that for every 2 ≤ j ≤ m, minW R j−1 − maxW R j = n + 3.
One more time, suppose that j is even. Then, for every 1 ≤ i ≤ n, Hence, the weights of the edges connecting R j−1 and R j form the set and On the other side, and assuming that j is even, for every 1 ≤ i ≤ n, we get The weights of the edges connecting R j and R j+1 form the set Since min(W R j−1 ∪W j−1,j )−max(W R j ∪W j,j+1 ) = ((3n+2)(m−j +1)−n−1)−((3n+2)(m− j) + 2n) = 1, we can see that every weight appears exactly once. In addition, the largest weight, obtained on R 1 , equals (3n + 2)(m − 2 + 1) + 2n = 3mn + 2m − n − 2, which is the size of the graph. Recall that the weight 1 is obtained on R m because when h is restricted to R m , h is just a shifting of an α-labeling.
Since the bipartite nature of the inital α-labelings of the R j is not affected by any of the transformations applied to them and the fact that when one stable set of R j has the largest labels, the corresponding stable set on R j+1 has the smallest labels assigned to the (j + 1)th copy of K 2,n . Hence, the labeling h of K 2,n × P m is bipartite with boundary value λ = n + (3n + 2)(m − 1)/2 if m is odd, (3n + 2)m/2 − 2 if m is even.
In Figure 8 we show an example for the case K 2,7 × P 4 .

Conclusions
The two operations introduced in this work can be used to find new families of graceful or α-graphs. The series operation was used here to prove that any tree that is a chain of t paths crossing, where the paths are isomorphic to P 11 and the distance between any pair of vertices of degree 4 is even. In this context we can ask the following: if T is any chain of t paths crossing, is T an α-tree? In other terms, can we obtain an α-tree independently of the parity of the distance between any pair of vertices of degree 4? Also, if not all the paths used in T are isomorphic, is the result still valid? Aldred,Širáň, andŠiráň [1] proved that the number of graceful labelings of P n grows at least as fast as ( 5 3 ) n ; Cattell [7] proved that in the majority of the cases, for any vertex v in P n and any label r ∈ {0, 1, . . . , n − 1}, there exists a graceful labeling f of P n such that f (v) = r. These two results together with Rosa's result [12], provide support to our idea that all trees that are a chain of paths crossing, are in fact α-trees, regardless of the paths used and the parity between consecutive crossings (vertices of degree 4).
In order to apply the parallel operation, we just need two compatible graphs. In the example given in Figure 5, the cycles can be replaced by any graph obtained identifying the corresponding end-vertices of any number of paths of even size, the labeling of the new parallel graph follows the same pattern that the one exhibited in Figure 5. We can also apply the parallel operation to any pair of compatible path-like trees, defined by Barrientos in [3] (these trees are called T P by Hegde and Shetty [10]), because their α-labelings are based on Rosa's α-labelings of caterpillars [13]. Thus, when the diameter of these trees is even, the labels 0 and λ are placed in the extreme vertices of a path of maximum length; the resulting graph is, as in the case of the caterpillars mentioned in Section 2, a unicyclic α-graph. In this way we obtain more evidence that support Truszczyński's conjecture [18] that all unicyclic graphs are graceful.
Finally, we have three questions related to Theorem 4.1, where we proved that the Cartesian product of K 2,n and P m is an α-graph. For which values of n 1 , is the graph K n 1 ,n 2 × P m an α-graph? Let T m be any tree of order m. Is K 2,n × T m an α-graph? Is K n 1 ,n 2 × T m an α-graph?