All graphs in this paper are finite, simple and undirected. Terms not defined here are used in the sense of Harary [1]. In 1966, Rosa [3] introduced
A graph
then the resulting edges get distinct labels from the set {1,3,5, ⋅⋅⋅, 2
(c.f. [4]). A wedge is defined as an edge connecting two components of a graph, denoted as ∧,
Primarily, assume that |
To prove that
Without loss of generality, we assume that
Consider the graph
The corresponding edge labels are as follows: The edge label of
Hence the vertex labels and the induced edge labels of G are distinct.
Thus
Consider the graph
The required vertex labeling
The corresponding edge labels are as follows: The edge label of
Hence the vertex labels and the induced edge labels of G are distinct.
Thus
Consider the graph
The required vertex labeling
The corresponding edge labels are as follows: The edge label of
Hence the vertex labels and the induced edge labels of G are distinct.
Thus
Consider the graph
The required vertex labeling
The corresponding edge labels are as follows: The edge label of
Hence the vertex labels and the induced edge labels of G are distinct.
Thus
Therefore,
Now we shall assume that
That is to prove the graph
Let us prove this by the method of induction. Consider the graph when
Now we shall prove that distinct vertex labels cannot be allotted to the vertices, on considering all the possible value for the vertex v (without loss of generality). The vertex labeling,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 0 or 1, 4 or 5, 8 or 9 and 12.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 1 or 2, 5 or 6, 9 or 10 and 13.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 2 or 3, 6 or 7 and 10.
There are only three possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 0, 3 or 4, 7 or 8 and 11 or 12.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 0 or 1, 4 or 5, 8 and 12 or 13.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 1 or 2, 5 or 6, 9 or 10 and 13.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 2 or 3, 6 or 7 and 10 or 11.
There are only three possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 0, 3 or 4, 7 or 8 and 11 or 12.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 0 or 1, 4 or 5, 8 or 9 and 12 or 13.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 1 or 2, 5 or 6, 9 or 10 and 13.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 2, 6 or 7 and 10 or 11.
There are only three possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 0, 3 or 4, 7 or 8 and 11 or 12.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 0, 4 or 5, 8 or 9 and 12 or 13.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Let us first consider
The vertices should be labeled as such all the edges receives distinct odd label.
We should get 5 distinct odd labels for the edges of
The possibilities of the vertices are 1 or 2, 5 or 6, 9 or 10 and 13.
There are only four possibilities to label five vertices, which is not sufficient.
Therefore,
Therefore,
By the method of induction assume that the result is true for all
The vertex labeling,
By the induction method we know that when
We have proved that
Therefore
Hence the theorem.
The odd mean labeling is applied on a graph (network) in order to enhance fastness, efficient communication and various issues,
A protocol, with secured communication can be achieved, provided the graph (network) is sufficiently connected.
To find an efficient way for safer transmissions in areas such as Cellular telephony, Wi-Fi, Security systems and many more.
Channel labeling can be used to determine the time at which sensor communicate.
Researchers may get the use of odd mean labeling in their research concerned with the above discussed issues.