We prove that all uniform
Labeling of a graph is an assignment of labels (numbers) to its vertices or/and edges or faces, which satisfy some conditions. These are different from coloring problems since some properties and structures of numbers such as ordering, addition, and subtraction used here are not properties of colors. Graph labelings have several applications in many fields. They have found usage in various coding theory problems, including the design of good radar-type codes, synch-set codes, and convolutional codes with optimal autocorrelation properties. They facilitate the optimal nonstandard encodings of integers. They have also been applied to determine ambiguities in X-ray crystallographic analysis, to design of a communication network addressing system, to determine optimal circuit layouts, and to problems in additive number theory.
Graham and Sloane [
A
A uniform
To prove our results, we name the vertices of any
Order to name the vertices.
Uniform
In this section, we list a few existing labelings which are useful for the development of this paper. Here, we consider a graph
An injective function
This was introduced by Graham and Sloane [
A graph which has a harmonious labeling is called a harmonious graph.
An injective function
This was introduced by Chang et al. [
A graph which has a strongly
An injective function
This was introduced by Grace [
A graph which has a sequential labeling is called a sequential graph.
Every uniform
Let
To prove
Define a labeling
That is, the edges receive the labels
That is, the edges receive the labels
That is, the edges receive the labels
That is, the edges receive the labels
That is, the edges receive the labels
That is, the edges receive the labels
Hence,
Hence,
Every uniform
Let
Define a labeling
The odd vertices
That is, the edge labels are
That is,
Every uniform
Let
Define a labeling
The odd vertices
That is,
Every uniform
Let
Define a labeling
The author would like to thank the editors for their valuable comments.