2 Face Irregular strength of grid graph
Let , where ,
and
Theorem 5.
Let be a positive integers and be grid graph, then
Proof.
According to (1) it is sufficient to show that . So we define the face irregular k-labeling ρ of type (1, 0, 0) of in the following way
The weight of the face f under labeling ρ of type (1, 0, 0) is defined as
The differences between the weights of the horizontal faces will be calculated as follows:
Let, and be two horizontal consecutive faces. Then,
Now we compute differences between the weights of the vertical faces will be calculated as follows:
Let, and be two vertical consecutive faces. Then,
Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □
Theorem 6.
Let be a positive integers and be grid graph, then
Proof.
We define a face irregular k-labeling ρ of type (0, 1, 0) of G in the following way
The weight of the face f under the labeling ρ of type (0, 1, 0) is
Now we calculate differences between the weights of the horizontal faces as follows:
Let, and be two horizontal consecutive faces.
Then for
Now we calculate the differences between the weights of the vertically adjacent faces as follows:
Let, and be two vertical consecutive faces.
Then for
For,
For,
For,
For,
Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □
Theorem 7.
Let be a positive integers and be grid graph, then
Proof.
We define vertex-face labeling ρ of type (1, 0, 1) of G in the following way
The weight under labeling ρ of type (1, 0, 1) can be defined as
The differences between the weights of every two horizontally adjacent faces will be calculated as follows:
Let, and be two horizontal consecutive faces.
Then for and
For and
For and
Now we measure the differences between the weights for every two vertically adjacent faces:
Let, and be two vertical consecutive faces.
Then for and
For and
For and
For and
For and
Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □
Theorem 8.
Let be a positive integers and be grid graph, then
Proof.
We define a k-labeling ρ of type (0, 1, 1) of G in the following way
The face labels are defined as
The weights can be defined as
The differences between the horizontal weights for every two adjacent faces are as follows: Let, and be two horizontal consecutive faces.
Then for and
Now the differences between the weights of every two adjacent vertical faces of type (0, 1, 1) are as follows:
Let, and be two vertical consecutive faces. Then,
For and
For and
For and
For and
For and
For and
For and
For and
Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □
Theorem 9.
Let be a positive integers and be grid graph, then
Proof.
Consider, an integer such that . Then the entire face irregular labeling is defined as follow
For the face labels are as follows
For the horizontal edges labels are as follow
The weight of a face is defined as
Now we calculate the difference between the weights for every two adjacent horizontal faces in each row as follow.
Let, and be two horizontal consecutive faces. Then,
For
For
For
For
For
For
For
Now the calculate the difference between the weights for every two adjacent vertical faces as follow.
Let, and be two vertical consecutive faces.
For
Since , for every value of i, so we have
For
For
For
For
For
Hence it follows that the weights of any two faces f and g are distinct. Complete the proof by clearly giving the two required inequalities. □