G be a graph with p = | V ( G ) | vertices and q = | E ( G ) | edges. Graph labeling, where the vertices and edges are assigned real values or subsets of a set are subject to certain conditions. A detailed survey of graph labeling can be found in [2].Terms not defined here are used in the sense of Harary in [5].The concept of **adjacent** **edge** **graceful** labeling was first introduced in [18].**Some** results on **adjacent** **edge** **graceful** labeling of **graphs** and **some** non- **adjacent** **edge** **graceful** **graphs** are discussed in [18].

Show more
10 Read more

All **graphs** in this paper are finite, simple and undirected **graphs**. Let ( p , q ) be a graph with p = | V ( G ) | vertices and q = | E ( G ) | edges. Graph labeling, where the vertices and edges are assigned real values or subsets of a set are subject to certain conditions. A detailed survey of graph labeling can be found in [1].Terms not defined here are used in the sense of Harary in [3].The concept of **edge** **graceful** labeling was first introduced in [2] and the concept of strong **edge** **graceful** labeling was introduced in [4]. **Some** results on strong **edge** **graceful** labeling of **graphs** are discussed in [4]. In this paper, we introduced a new **edge** **graceful** labeling. We use the following definitions in the subsequent sections.

Show more
11 Read more

the avd-colorings of **graphs**, see [3–5, 7–9, 11, 13–16, 18]. A bicyclic graph is a connected graph in which the num- ber of edges equals the number of vertices plus one. In this paper, we investigate the avd-coloring of bicyclic **graphs** and show that χ a ′ (G) ≤ ∆(G) + 1 for bicyclic **graphs** G. This implies that Conjecture 1 holds for all bicyclic **graphs**.

11 Read more

chromatic number of coloring, For special H , the exact value of the **adjacent** vertex distinguishing **edge**-coloring of G [ H ] is obtained. In this paper, we prove that the chromatic number of **adjacent** vertex distinguishing **edge**-coloring of lexicographic product G [ H ] for any two **graphs** G and H is equal to the graph class

m may change from **edge** to **edge**; they called this replacement, supersubdivision. More general results about super subdivisions can be found in [2], [3], and [18]. In Section 3 we analyze this concept in more detail. There, we introduce a replacement theorem that allows us to replace, within a **graceful** labeled graph, **some** specific labeled subgraphs by **some** analogous **graphs**.

11 Read more

The tables which are constructed in the theorem1 and theorem2 are not unique. There are many ways to construct the tables but in every way the required minimum no. of colours for avd **edge** colouring in the table is same as the theorems. Also the tables are constructed in the theorem 2 are symmetric. Any one can work on algorithm of avd **edge** colouring of any arbitrary permutation **graphs**.

15 Read more

again have **some** more cases to consider, depending on the values of i, and j. When 1 6 i 6 r, or p(n − 1) + r + 1 6 i 6 min{(p + 1)(n − 1) + r, m}, then similar to Case(1), and Case(2), we can easily observe that q is not in A 1 , A 2 . The remaining case that we need to consider

12 Read more

Abstract - A graph G = (V, E) with p vertices and q edges is said to admit pentagonal **graceful** labeling if its vertices can be labeled by non negative integers such that the induced **edge** labels obtained by the absolute difference of the labels of end vertices are the first 𝒒 pentagonal numbers. A graph 𝑮 which admits pentagonal **graceful** labeling is called a pentagonal **graceful** graph. In this paper, we prove that Caterpillar is a pentagonal **graceful** graph.

A graph G of size q is odd-**graceful**, if there is an injection from V(G) to {0, 1, 2, …, 2q-1} such that, when each **edge** xy is assigned the label or weight | (x) - (y)|, the resulting **edge** labels are {1, 3, 5, …, 2q-1}. This definition was introduced in 1991 by Gnanajothi [1] who proved that the class of odd **graceful** **graphs** lies between the class of **graphs** with α-labelings and the class of bipartite **graphs**. Gnanajothi [1] proved that every cycle C n is odd **graceful** if n is even. It is

Show more
Proof: Let H be the kC 4 -snake with k blocks, so the number of vertices of H is 3 k + 1 and the number of edges of H is 4 . Let G be the super graph of H such that G is the k complete m-points projection on the disjoint vertices of kC 4 -snake. Let N = m + 3 k + 1 be the number of vertices of G and M = 2 mk + 4 k be the number of edges of G. [Refer Figure 2.5]. To prove G is **graceful** it is enough to prove that the M edges o f G having the **edge** values as { M , M − 1 , M − 2 , ..., 3 , 2 , 1 } . Name the k + 1 adjoint vertices by

Show more
(**graphs** obtained by joining a single pendent **edge** to each vertex of C n ) if and only if n is even ; the disjoint union of copies of C 4 ; the one-point union of copies of C 4 ; C n × K 2 if and only if n is even; caterpillars; rooted trees of height 2; the **graphs** obtained from P n (n > 3) by adding exactly two leaves at each vertex of degree 2 of P n ; the **graphs** obtained from P n × P 2 by deleting an **edge** that joins to end points of the P n paths ; the **graphs** obtained from a star by adjoining to each end vertex the path P 3 or by adjoining to each end vertex the path P 4 . She conjectures that all trees are odd-**graceful** and proves the conjecture for all trees with order up to 10. Barrientos [2] has extended this to trees of order up to 12. For details in the progress made so far in this area one can refer to the latest Survey on graph labeling problems due to Gallian [3]. Here we are inspired by the works of Solairaju [4] and Moussa and Badr [5] and give odd **graceful** labeling to the **graphs** : Bistar, K 1 , n : 2 ,

Show more
, Then Rosa [10] called the mapping the β-labeling (valuation) of a graph G, Golomb [4] subsequently called such labeling to be **graceful** labeling and the graph is called a **graceful** graph, while is called an induced edge’s **graceful** labeling. -**graceful** labeling is the generalization of **graceful** labeling that introduced by Slater [11] in 1982 and by Maheo and Thuillier [8] in 1982.

By a graph G = (V, E) we mean a finite undirected graph without loops and multiple edges. A subset S of V is called a dominating set of G if every vertex not in S is **adjacent** to **some** vertex in S. The domination number of G denoted by γ(G) is the minimal cardinality taken over all dominating sets of G. A subset F of E is called an **edge** dominating set if each **edge** in E is either in F or is **adjacent** to an **edge** in F . An **edge** dominating set F is called minimal if no proper subset of F is an **edge** dominating set. The **edge** domination number of G denoted by γ 0 (G) is the minimum cardinality taken over all **edge** dominating sets of G.

Show more
In [5], the **edge** version of geometric-arithmetic index introduced based on the end-vertex degrees of edges in a line graph of G which is a graph such that each vertex of L(G ) represents an **edge** of G; and two vertices of L(G) are **adjacent** if and only if their corresponding edges share a common endpoint in G, as follows

and means f (e) = 0 and since f : E(G) → {0,1,2} is –function, then any **edge** must be **adjacent** to **edge** h , i.e.,f (h) = 2 and in the function g : E(G) → {1,2}, if then g( h ) = {1,2}. Hence g : E(G) → {1,2} is a 2-rainbow **edge** domination function in G with the weight W(g) that means

11 Read more

A **graceful** labeling of a graph G of size n is an injective assignment of integers from {0, 1, . . . , n} to the vertices of G, such that when each **edge** of G has assigned a weight, given by the absolute difference of the labels of its end vertices, the set of weights is {1, 2, . . . , n}. If a **graceful** labeling f of a bipartite graph G assigns the smaller labels to one of the two stable sets of G, then f is called an α-labeling and G is said to be an α-graph. A tree is a caterpillar if the deletion of all its leaves results in a path. In this work we study **graceful** labelings of the disjoint union of a cycle and a caterpillar. We present necessary conditions for this union to be **graceful** and, in the case where the cycle has even size, to be an α- graph. In addition, we present a new family of **graceful** trees constructed using α-labeled caterpillars.

Show more
The study of graph labeling and **graceful** **graphs** was introduced by Rosa[4].(Gnanajothi R.B,1991)[5 ] introduced the concept of odd **graceful** **graphs** and she has proved many results. Kathiresan K.M.,2008[6] has discussed odd gracefulness of ladders and **graphs** obtained fromthem by subdividing each step exactly once. Vaidya.S.K. e-tal 2010[7 ]proved the odd gracefulness of joining of even cycle with path and cycle sharing a common **edge**. Vaidya S.K. e-tal 2013[ 8] proved the odd gracefulness of splitting graph and the shadow graph of bister. Barrientos Christian 2009[ 9] discussed the odd gracefulness of Trees of Diameter 5.For detailed survey on graph labeling and related results we refer to Gallian J.A.,2015[10 ]

Show more
Rosa [2] introduced the notion of **graceful** labeling. Bloom and Hsu [3],[4],[5] extended the concept of **graceful** labeling for digraphs also. Lo [6] introduced the idea of **edge** **graceful** **graphs**. Lee [7] further developed the concept for more **graphs**. Lee et al [8] defined super **edge** **graceful** labeling for **some** undirected **graphs**. Also Gayathri et al [9] discussed the even **edge** **graceful** labeling for undirected **graphs**. The Cayley graph that presents the exact structure of a group and this definition is introduced by Cayley [10]. Thirusangu et al [11] defined labeling concepts for **some** classes of Cayley digraphs. Further many authors started various labeling technologies on the Cayley digraphs [12],[13]. We now extended super **edge** **graceful** and the even **edge** **graceful** labeling to directed **graphs** and applied both the methodologies on the Cayley digraphs.

Show more
We look at a computer network as a connected undirected graph. A network designer may want to know which edges in the network are most important. If these edges are removed from the network, there will be a great decrease in its performance. Such edges are called the most vital edges in a network [5, 6, 12]. However, they are only concerned with the effect of the maximum flow or the shortest path in the network. We can consider the effect of a minimum spanning tree in the network. Suppose that G = (V, E) is a weighted graph with a weight w(e) assigned to every **edge** e in G. In the weighted graph G, the weight of a spanning tree T, w(T ) is defined to be P w(e)

Show more
mod (2k), is an injective function, where k = max(p, q). The graph that admits an **edge** even **graceful** labeling is called an **edge** even **graceful** graph. In Fig. 1 , we present an **edge** even **graceful** labeling of the Peterson graph and the complete graph K 5 .

15 Read more