A shell graph is the join of a path P k of 'k' vertices and K 1. A subdivided shell graph can be constructed by subdividing the edges in the path of the shell graph. In this paper we prove that the disjoint union of two subdivided shell graphs is odd graceful and also one modulo three graceful. 1. Introduction A graph labeling is an assignment of integers to the vertices or edges, or both, subj...

Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q− 1} such that, when each edge xy is assigned the label |f (x)− f (y)| , the resulting edge labels are {1, 3, 5, . . . , 2q− 1} and f is called an odd graceful labeling of G. Motivated by the work of Z. Gao [6] in which he studied th...

The Graceful Tree Conjecture stated by Rosa in the mid 1960s says that every tree can be gracefully labelled. It is one of best known open problems Graph Theory. conjecture has caused a great interest study gracefulness simple graphs and led to many new contributions list graceful graphs. However, it acknowledged not much about structure after 55 years.

We studied the Einstein-Brans-Dicke cosmology in detail. The difference of the evolution of the universe is significant between Einstein-Brans-Dicke cosmology and standard big-bang model during the radiation-dominated era. The power-law evolution of the scale factor is fast enough to solve the cosmological puzzles and slow enough to avoid the graceful exit problem. However, the constraints from...

String cosmology solutions are examined in a generalized phase-space including sources representing arbitrary corrections to lowest order string-dilatongravity effective action. We find a set of necessary conditions for a graceful exit transition from a dilaton-driven inflationary phase to a radiation dominated era. We show that sources allowing such a transition have to violate energy conditio...

If the vertex set V of > =< V,E G can be divided into two un empty sets X and Y , ∅ = = Y V,X Y X ∩ ∪ , but also two nodes of every edge belong to X and Y separately, the G is called bipartite graph. If E ) ,y Y,(x X,y x i i i i ∈ ∈ ∈ ∀ then G is called complete bipartite graph. if n Y m, X = = , the G is marked m,n K . In this paper the graceful labeling, k-graceful labeling, odd graceful labe...

A graph G on m edges is considered graceful if there is a labelling f of the vertices of G with distinct integers in the set {0, 1, . . . ,m} such that the induced edge labelling g defined by g(uv) = |f(u) − f(v)| is a bijection to {1, . . . ,m}. We here consider some relaxations of these conditions as applied to tree labellings: 1. Edge-relaxed graceful labellings, in which repeated edge label...

Let G be a (p,q)-graph and k~O. A graph G is said to be k-edge­ graceful if the edges can be labeled by k,k+I, ... ,k+q-I so that the vertex sums are distinct, modulo p. We denote the set of all k such that G is k-edge graceful by egS(G). The set is called the edge-graceful spectrum ofG. In this paper, we are concerned with the problem of exhibiting sets of natural numbers which are the edge-gr...