A graceful labelling of an undirected graph G with n edges is a one-to-one function from the set of vertices of G to the set {0, 1, 2, . . . , n} such that the induced edge labels are all distinct. An induced edge label is the absolute value of the difference between the two end-vertex labels. The Graceful Tree Conjecture states that all trees have a graceful labelling. In this survey we presen...

We introduce new labeling called m-bonacci graceful labeling. A graph G on n edges is if the vertices can be labeled with distinct integers from set {0,1,2,…,Zn,m} such that derived edge labels are first numbers. show complete graphs, bipartite gear triangular grid and wheel graphs not graceful. Almost all trees give to cycles, friendship polygonal snake double graphs.

7 Partial cubes are graphs that allow isometric embeddings into hypercubes. -graceful labelings of partial cubes are introduced as a natural extension of graceful labelings of trees. It is shown that several classes of partial cubes are -graceful, for instance even 9 cycles, Fibonacci cubes, and (newly introduced) lexicographic subcubes. The Cartesian product of -graceful partial cubes is again...

Graceful labeling is one of the best known labeling methods of graphs. Despite the large number of papers published on the subject of graph labeling, there are few particular techniques to be used by researchers to gracefully label graphs. In this paper, first a new approach based on the mathematical programming technique is presented to model the graceful labeling problem. Then a “branching me...

Tree is one of the most studied and practically useful classes of graphs and is the attention of a great number of studies. There is absence of generalized results for tree as a class and even for one kind of labeling as whole. Only specialized results exist limited to specific types of trees only. A number of conjectures stand being unsolved. Graham and Sloane (1980) conjectured trees to be Ha...

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...

A (simple undirected) graph G = (V,E) with m edges is graceful if it has a distinct vertex labeling f : V −→ {0, 1, 2, 3, . . . ,m} which induces a set of distinct edge labels {|f(u)− f(v)| | uv ∈ E, u, v ∈ V }. The famous Ringel-Kotzig conjecture [9, 15] is that all trees are graceful. The base of a tree T is obtained from T by deleting its one-degree vertices. A caterpillar is a tree whose ba...