Some methods for constructing some classes of graceful uniform trees

A tree T (V,E) is graceful if there exists an injective function f from the vertex set V (T ) into the set {0, 1, 2, ..., |V | − 1} which induces a bijective function f ′ from the edge set E(T ) onto the set {1, 2, ..., |E|}, with f ′(uv) = |f(u) − f(v)| for every edge {u, v} ∈ E. Motivated by the conjecture of Alexander Rosa (see [1]) saying that all trees are graceful, a lot of works have addressed gracefulness of some trees. In this paper we show that some uniform trees are graceful. This results will extend the list of graceful trees.


Introduction
Let G(V, E) be a graph of |V (G)| vertices and |E(G)| edges.Graphs labelling is a notorious topic in graph theory.This is not because of its interesting combinatorial property of graphs labelling but also because of applications side such as X-ray crystallography, coding theory, circuit design, or astronomy (see [5]).
In year 1967, Alexander Rosa introduced β labelling or β−valuations which was popularized by Solomon W. Golomb as graceful labelling.A graceful labelling for the graph G(V, E) is an injective function f from the vertex set V (G) into the set {0, 1, 2, ..., |E(G)|} which induces edge labels |f (u) − f (v)| for all {u, v} ∈ E(G) which are all distinct.A graph which admits a graceful labelling is said to be graceful.Not too many classes of trees are yet known to be graceful (see [1]).Erdös in an unpublished paper said that most graphs are not graceful, but graphs with some regularity are graceful (see [1]).Especially with trees, the conjecture of Alexander Rosa (see [1]) saying that all trees are graceful, has motivated many published papers regarding graceful trees.There are many classes of trees known to be graceful.Here we construct some classes of trees which are then proven to be graceful.

Results
In this section we will observe some classes of trees which then are known to be graceful.

Uniform Caterpillars
Denote by P x the path of x vertices.One end vertex of a path will be called the head and the other one end will be called the back of the path.A uniform (q, r)-caterpillar or shortly (q, r)caterpillar is a tree which is obtained by appending r leaves to each vertex of P q except the back which is appended by r + 1 leaves.The path P q will be called the backbone of (q, r)-caterpillar.It was proved in [2] that for any positive integers q and r, (q, r)-caterpillar is graceful.Furthermore, it is clear that the number of vertices in a uniform (q, r)-caterpillar is equal to q(r + 1) + 1.A tree which is constructed from p copies of (q, r)-caterpillars and a path P p by identifying the head of each (q, r)-caterpillar to exactly one vertex of P p is called a uniform super (p, q, r)-caterpillar, and is denoted by SC(p, q, r).It is easy to check that the number of vertices of SC(p, q, r) is equal to p(1 + q + qr).
The path P p is called the spine of SC(p, q, r) (see [6]).For r = 0, SC(p, q, 0) is proved to be graceful in [6].
Without loss of generality, we count the vertices of the spine P p from one end point to the other one (from left to right) from 1 until p.Furthermore, we assign a number to each (q, r)-caterpillar corresponds to the number of vertex of P p on which the head of the caterpillar is identified.Denote each vertex of SC(p, q, r) by v i,j,k with i := 1, 2, ..., r, r + 1, j := 0, 1, 2, ..., q, k := 1, 2, ..., p, where v r+1,j,k are for the vertices in the backbone of the k th (q, r)-caterpillar which is at distant j from the spine P p , and v i,j,k , with j > 0, is the leaf of the k th (q, r)-caterpillar which is adjacent with vertex v r+1,j−1,k .
Theorem 2.1.For every positive integers p, q, and r, the uniform super caterpillar SC(p, q, r) is graceful.
for j = 0 even, k even, 1 2 [ks − (j − 1)(r + 1)] − i for, j odd, k even. ( This function induces the function f as follows. for j > 0 and k even (4) Now we want to show that f is injective by showing that the range of f is a subset of {0, 1, 2, . . ., ps− 1}, and the cardinality of the range of f is equal to ps − 1.
Moreover, we will show that the range of f is equal to the set of {1, 2, . . ., ps − 1}, and therefore f is bijective.
Let us denote the ranges of f in Eq. ( 1) as V 1 , V 2 , V 3 , and V 4 , for the conditions of the pairs (j, k): (even, odd),(odd, odd),(even, even) and (odd, even), respectively.Thus for the whole range of f is To this end, we will see each of the four cases of the pairs of (p, q).
For the sake of space, we will present the proof only for the case (p, q)=(even, even).The proof for other cases is similar.For case (p, q) = (even, even) we have Based on the above observation we get For this case we conclude that f is an injection.By doing a similar way, we may conclude that f is an injective function.
For p odd we have For case p odd we get that For p even we have So, for case p even, again we obtain that

Uniform Super Lobster Tree
Here we will observe another class of trees which are graceful.These trees are called super uniform lobsters.
Consider a (q, r)-caterpillar.If we append s leaves to each leaf of this (q, r)-caterpillar, except one leaf which is adjacent to the back, then we obtain a new tree which is called a uniform (q, r, s)lobster.The leaf of the (q, r)-caterpillar which is not appended by s leaves will be called the tail of the uniform (q, r, s)-lobster.The uniform (q, r, s)-lobster is called 2-distant tree in [2].The number of vertices in a uniform (q, r, s)-lobster is equal to q(rs + r + 1) + 1.We will call again the path P q as the backbone of the uniform (q, r, s)-lobster.For a positif integer p > 1, take p copies of uniform (q, r, s)-lobster and then identify their all tails.The resulting tree is called a uniform (p, q, r, s)-super lobster or shortly a (p, q, r, s)-super lobster, which is denoted by SL(p, q, r, s).For r = 0, (p, q, r, 0)-super lobster in [4] is called super caterpillar, and was proved to be graceful.Later we will show that SL(p, q, r, s) is graceful.
First, we name the vertices of SL(p, q, r, s) with v i,j,k and with v h i,j,k where i = 1, 2, ..., q, j = 0, 1, 2, . . ., r, k = 1, 2, . . ., p, h = 1, 2, . . ., s, with the following details: v i,0,k for the vertex which is on the k th backbone and at distant i from the center v 0,0,0 ; v i,j,k , j = 0, for the vertex which is adjacent to v i,0,k ; and v h i,j,k for the leaf which is adjacent to v i,j,k .Let G(V, E) := SL(p, q, r, s) and m = r(s + 1) + 1.It is easy to see that |V | = pqm + 1.Now we define a function f from V into {0, 1, 2, . . ., pqm} as the following.For 1 ≤ k ≤ p, we define qm, if j = 0 and i even, and for 1 ≤ k ≤ p and 0 < j ≤ r we define Theorem 2.2.For all positive integers p, q, r, and s, super lobster graph SL(p, q, r, s) is graceful.
Proof.Consider the function defined in Eq. ( 7) and Eq.(8 for the ranges of f for conditions (i, j, k) = (0, 0, 0); (i odd, j = 0, k); (i even, j = 0, k); (i odd, j > 0, k); (i even, j > 0, k); (i odd, j, k); and (i even, j, k); respectively.Furthermore, let First we will prove that f is an injection.We partition into two cases on q: (i) q is even, and (ii) q is odd.(i) Case q is even.Here we have In this case we will calculate A into two phases with respect to q: for 0 < i < q, and then for i = q.
Here we have that Furthermore, let B = V 2 ∪ V 5 ∪ V 6 .As we did above, here again we will calculate B into two phases with respect to q: for 0 < i < q, and then for i = q.Let B 1 = V 2 ∪ V 5 ∪ V 6 for 0 < i < q, and B 2 = V 2 ∪ V 5 ∪ V 6 for i = q.Thus the total We obtain the following result.

Thus we have
We see that for every case we obtain that V 1 ∪V 2 ∪V 3 ∪V 4 ∪V 5 ∪V 6 ∪V 7 = {0, 1, 2, . . ., pqm}.Thus the function f is not only injective but also bijective.We will leave the proof that the function f is bijective from E onto {1, 2, . . ., pqm}.

Uniform Distant Trees
The third class of trees we want to show to be graceful is the class of uniform distant trees.A tree in which all internal vertices have degrees r + 1 except one internal vertex, is called an full r-ary tree.the vertex of degrees r will be called the root of the tree.A uniform full r-ary tree is a full r-ary tree in which all of its leaves are at the same level, say q, and is denoted by T r q .A tree which is obtained from p copies of T r q and a path of p vertices, P p , by identifying each vertex of P p with exactly the root of each T r q is called a uniform (p, q, r)-distant tree and will be denoted by D(p, q, r).In [6] it was proved that D(p, q, 1) is graceful.Here we extend this result for r > 1.
It is easy to see that the order of T r q is r q+1 −1 r−1 , and hence the order of D(p, q, r) is ps with s = r q+1 −1 r−1 .Now we want to name vertices of D(p, q, r) to facilitate the definition of a function from the set of vertices to the set {0, 1, 2, ..., ps − 1}.First we name p vertices of the path P p from one end to the other by 1 up until p.If the root of the tree T r q is identified with the k th vertex of the path P p ., then we call the tree as the k th tree in P p .We name vertices as v i j ,j,k which has the following details: i) j = 0, 1, . . ., q for referring the distance of the vertex from the path P p .ii) i = 1, 2, . . ., r j for referring the order of vertices at level j in each T r q .iii) k = 1, 2, . . ., p indicates that the vertex is in the k th tree T r q .f 1 (v 1,0,1 ) = 0, then let n = 1, 2, . . ., q.For k = 1 we will see two cases: 1 ≤ j < n and j = n.For case 1 ≤ j < n, we define r j , if j even and i = 1, 2, . . ., r j , f n−1 (v i,j,k ) + r n (p − i−1 r j ), if j odd and i = 1, 2, . . ., r j . (9) For j = n let m = 1, 2, . . ., r n−1 and i = 1 + r(m − 1), . . ., r + r(m − 1).Here we define with s n = r n+1 −1 r−1 .For 1 < k ≤ p, 0 ≤ j ≤ q, and 1 ≤ i ≤ r j , we define Theorem 2.3.For all positive integers p, q, and r ≥ 2, the graph D(p, q, r) is graceful.
Proof.Let G(V, E) be the graph D(p, q, r).Consider the function f as defined in Eqs. 9, 10, and 11 above.We define a function f from E onto {1, 2, . . ., ps − 1} such that for every edge uv ∈ E, f (uv) = |f (u) − f (v)|.Next we will show that f and f are injective and bijective, respectively.First we want to show that the function f is injective.Let the ranges of f eqs.9, 10, and 11 be V 1 , V 2 , and V 3 respectively.Thus the total range of f is Consider now the cases of p. First, assume that p is even.For the case k odd, j even, or the case k even, j odd, let Secondly, consider the case p odd.Here, we will partition into case 1 ≤ k < p and case k For the case k odd, j even, or the case k even, j odd, let Then, for the case k odd, j odd, or the case k even, j even, let 2 )s − 1}.Thus, by some simple algebraic calculations we obtain V = A ∪ B ∪ C ={0, 1, 2, . . ., ps − 1}.Since for every case we have that the total image of f is V = {0, 1, 2, . . ., ps − 1}, and hence |V | = |V |, then we may conclude that f is an injection (even a bijection).Next, it remains to show that the induced function f is a bijection.a.First let us consider edges {v 1,0,k , v 1,0,k+1 }.For n = 1, 2, . . ., q let s n = r

Conclusion
We should mention here that all uniform trees in this paper can indeed be proven by the method introduced in [3], but in this paper we label the whole graph directly, whereas in [3] the labeling proses is carried out part by part.