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Symmetry 2019, 11(1), 38; https://doi.org/10.3390/sym11010038

Article
Edge Even Graceful Labeling of Polar Grid Graphs
1
Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah 41411, Saudi Arabia
2
Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El Kom 32511, Egypt
Received: 6 December 2018 / Accepted: 23 December 2018 / Published: 2 January 2019

Abstract

:
Edge Even Graceful Labelingwas first defined byElsonbaty and Daoud in 2017. An edge even graceful labeling of a simple graph G with p vertices and q edges is a bijection f from the edges of the graph to the set { 2 , 4 , , 2 q } such that, when each vertex is assigned the sum of all edges incident to it mod 2 r where r = max { p , q } , the resulting vertex labels are distinct. In this paper we proved necessary and sufficient conditions for the polar grid graph to be edge even graceful graph.
Keywords:
graceful labeling; edge graceful labeling; edge even graceful labeling; polar grid graph
Mathematics Subject Classification:
05C78

1. Introduction

The field of Graph Theory plays an important role in various areas of pure and applied sciences. One of the important areas in graph theory is Graph Labeling of a graph G which is an assignment of integers either to the vertices or edges or both subject to certain conditions. Graph labeling is a very powerful tool that eventually makes things in different fields very ease to be handled in mathematical way. Nowadays graph labeling has much attention from different brilliant researches ingraph theory which has rigorous applications in many disciplines, e.g., communication networks, coding theory, x-raycrystallography, radar, astronomy, circuit design, communication network addressing, data base management and graph decomposition problems. More interesting applications of graph labeling can be found in [1,2,3,4,5,6,7,8,9,10].
Let G = ( V ( G ) , E ( G ) ) with p = | V ( G ) | and q = | E ( G ) | be a simple, connected, finite, undirected graph.
A function f is called a graceful labeling of a graph G if f : V ( G ) { 0 , 1 , 2 , , q } is injective and the induced function f : E ( G ) { 1 , 2 , , q } defined as f ( e = u v ) = | f ( u ) f ( v ) | is bijective. This type of graph labeling first introducedby Rosa in 1967 [11] as a β valuation, later on Solomon W. Golomb [12] called as graceful labeling.
A function f is called an odd graceful labeling of a graph G if f : V ( G ) { 0 , 1 , 2 , , 2 q 1 } is injective and the induced function f : E ( G ) { 1 , 3 , , 2 q 1 } defined as f ( e = u v ) = | f ( u ) f ( v ) | is bijective. This type of graph labeling first introducedby Gnanajothi in 1991 [13]. For more results on this type of labeling see [14,15].
A function f is called an edge graceful labeling of a graph G if f : E ( G ) { 1 , 2 , , q } is bijective and the induced function f : V ( G ) { 0 , 1 , 2 , , p 1 } defined as f ( u ) = e = u v E ( G ) f ( e ) mod p is bijective. This type of graph labeling first introducedby Lo in 1985 [16]. For more results on this labeling see [17,18].
A function f is called an edge odd graceful labeling of a graph G if f : E ( G ) { 1 , 3 , , 2 q 1 } is bijective and the induced function f : V ( G ) { 0 , 1 , 2 , , 2 q 1 } defined as f ( u ) = e = u v E ( G ) f ( e ) mod 2 q is injective. This type of graph labeling first introducedby Solairaju and Chithra in 2009 [19]. See also Daoud [20].
A function f is called an edge even graceful labeling of a graph G if f : E ( G ) { 2 , 4 , , 2 q } is bijective and the induced function f : V ( G ) { 0 , 2 , 4 , , 2 q 2 } defined as f ( u ) = e = u v E ( G ) f ( e ) mod 2 r , where r = max { p , q } is injective. This type of graph labeling first introduced by Elsonbaty and Daoud in 2017 [21].
For a summary of the results on these five types of graceful labels as well as all known labels so far, see [22].

2. Polar Grid Graph P m , n

The polargrid graph P m , n is the graph consists of n copies of circles C m which will be numbered from the inner most circle to the outer circle as C m ( 1 ) , C m ( 2 ) , , C m ( n 1 ) , C m ( n ) and m copies of paths P n + 1 intersected at the center vertex v 0 which will be numbered as P n + 1 ( 1 ) , P n + 1 ( 2 ) , , P n + 1 ( m 1 ) , P n + 1 ( m ) . See Figure 1.
Theorem 1.
If m and n are even positive integes such that m 4 and n 2 , then the polar grid graph P m , n is an edge even graceful graph.
Proof. 
Using standard notation p = | V ( P m , n ) | = m n + 1 , q = | E ( P m , n ) | = 2 m n and r = max { p , q } = 2 m n Let the polar grid graph P m , n be labeled as in Figure 2. Let f : E ( G ) { 2 , 4 , , 2 q } .
First we label the edges of paths P n + 1 ( k ) , k m begin with the edges of the path P n + 1 ( 1 ) to the edges of the path P n + 1 ( m ) as follows: Move clockwise to label the edges v 0 v 1 , v 0 v 2 , , v 0 v m 1 , v 0 v m by 2 , 4 , , 2 m 2 , 2 m , then move anticlockwise to label the edges v 1 v m + 1 , v m v 2 m , v m 1 v 2 m 1 , , v 3 v m + 3 , v 2 v m + 2 by 2 m + 2 , 2 m + 4 , 2 m + 6 , , 4 m 2 , 4 m , then move clockwise to label the edges v m + 1 v 2 m + 1 , v m + 2 v 2 m + 2 , v m + 3 v 2 m + 3 , , v 2 m 1 v 3 m 1 , v m v 3 m by 4 m + 2 , 4 m + 4 , 4 m + 6 , , 6 m 2 , 6 m and so on. Finally move anticlockwise to label the edges v m ( n 2 ) + 1 v m ( n 1 ) + 1 , v m ( n 1 ) 1 v m n , v m ( n 1 ) 1 v m n 1 , , v m ( n 2 ) + 2 v m ( n 1 ) + 2 by 2 m ( n 1 ) + 2 , 2 m ( n 1 ) + 4 , 2 m ( n 1 ) + 6 , , 2 m n 2 , 2 m n . Second we label the edges of the circles C m ( k ) , 1 k n begin with the edges of the inner most circle C m ( 1 ) to the edges of the circle C m ( n 2 ) , then the edges of the outer circle C m ( n ) . Finally the edges of circles C m ( n 2 + 1 ) , C m ( n 2 + 2 ) , , C m ( n 1 ) respectively as follows: f ( v m ( k 1 ) + i v m ( k 1 ) + i + 1 ) = 2 m n + 2 m ( k 1 ) + 2 i , f ( v k m v m ( k 1 ) + 1 ) = 2 m n + 2 m k , 1 i m 1 , 1 k n 2 ; f ( v m ( n 1 ) + i v m ( n 1 ) + i + 1 ) = 3 m n + 2 i , 1 i m 1 ; f ( v k m v m ( k 1 ) + 1 ) = 2 m n + ( 2 k + 1 ) m , n 2 + 1 k n 1 .
Now the corresponding labels of vertices mod 4 m n are assigned as follows:
Case (1) m 4 k mod 4 n , 1 k n 1 and m = 2 mod 4 n .
The labels of the vertices of the inner most circle C m ( 1 ) to the circle C m ( n 2 ) are given by f ( v ( k 1 ) m + i ) 4 m ( 2 k 1 ) + 4 i , 1 k n 2 , 1 i m , the labels of the vertices of the outre circle C m ( n ) are given by f ( v ( n 1 ) m + i ) 2 i + 2 , 1 i m and the labels of the vertices of the circles C m ( n 2 + 1 ) , , C m ( n 1 ) are given by f ( v ( k 1 ) m + i ) 8 m ( k n 2 ) + 4 i + 2 , 1 i m , n 2 + 1 k n 1 .
The label of the center vertex v 0 is assigned as follows: when m = 4 k mod 4 n , 1 k n 1 , f ( v 0 ) m 2 ( 2 m + 2 ) = m 2 + m , since m = 4 k mod 4 n then m = 4 n h + 4 k , thus f ( v 0 ) = m ( 4 k + 1 ) and when m = 2 mod 4 n , we have f ( v 0 ) = 3 m .
Case (2) m ( 8 k 2 ) mod 4 n , 1 k n 2 . In this case the vertex v k m ( m + 2 4 ) in the circle C m ( k ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v k m ( m + 2 4 ) v k m ( m 2 4 ) and v k m ( m 2 4 ) v k m ( m 6 4 ) . That is f ( v k m ( m + 2 4 ) v k m ( m 2 4 ) ) = 2 m n + m ( 2 k 1 ) + m + 2 2 and f ( v k m ( m 2 4 ) v k m ( m 6 4 ) ) = 2 m n + m ( 2 k 1 ) + m 2 2 and we obtain the labels of the corresponding vertices as follows f ( v k m ( m + 2 4 ) ) m ( 8 k 1 ) + 2 , f ( v k m ( m 2 4 ) ) m ( 8 k 1 ) + 4 , f ( v k m ( m 6 4 ) ) m ( 8 k 1 ) + 6 and the label of the center vertex v 0 is assigened as f ( v 0 ) m ( 8 k 1 ) .The rest vertices are labeled as in case(1).
Case (3) m ( 8 k + 2 ) mod 4 n , 1 k n 2 1 . In this case the vertex v k m ( m + 2 2 ) in the circle C m ( n 2 + k ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v k m ( m + 2 2 ) v k m ( m 2 ) and v k m ( m 2 ) v k m ( m 2 2 ) . That is f ( v k m ( m + 2 2 ) v k m ( m 2 ) ) = 2 m n + m ( 2 k + 1 ) + m + 2 2 and f ( v k m ( m 2 ) v k m ( m 2 2 ) ) = 2 m n + m ( 2 k + 1 ) + m 2 2 and we obtain the labels of the corresponding vertices as follows f ( v k m ( m + 2 2 ) ) m ( 8 k + 3 ) + 2 , f ( v k m ( m 2 ) ) m ( 8 k + 3 ) + 4 , f ( v k m ( m 2 2 ) ) m ( 8 k + 3 ) + 6 and the label of the center vertex v 0 is assigened f ( v 0 ) m ( 8 k 29 ) as. The rest vertices are labeled as in case(1).
Case (4) m 0 mod 4 n . In this case the vertex v m n ( m + 2 2 ) in the outer circle will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m n ( m + 4 2 ) v m n ( m + 2 2 ) and v m n v m ( n 1 ) + 1 . That is f ( v m n ( m + 4 2 ) v m n ( m + 2 2 ) ) = m ( 3 n + 2 ) and f ( v m n v m ( n 1 ) + 1 ) = 3 m n + m 4 and we obtain the labels of the corresponding vertices as follows f ( v m n ( m + 4 2 ) ) 2 m + 2 , f ( v m n ( m + 2 2 ) ) 2 m + 4 , f ( v m n ) m 2 and f ( v m ( n 1 ) + 1 ) 4 m n m and the label of the center vertex v 0 is assigened as f ( v 0 ) m . The rest vertices are labeled as in case (1).
Illustration. 
The edge even graceful labeling of the polar grid graphs P 14 , 6 , P 16 , 6 , P 18 , 6 , P 24 , 6 and P 26 , 6 respectively are shown in Figure 3.
Remark 1.
In case m = 2 and n is even, n > 2 .
Let the label of edges of the polar grid graph be as in Figure 4. Thus we have the label of the corresponding vertices are as follows:
  • f ( v 1 ) 4 n + 12 ; f ( v i ) 16 i 2 , 2 i n 2 ; f ( v n 2 + i ) 16 i + 6 , 1 i n 2 1 ;
  • f ( v n ) 4 ; f ( v i ) 16 i + 2 , 1 i n 2 1 ; f ( v n 2 + i ) 2 ; f ( v n 2 + i ) 16 i + 10 , 1 i n 2 2 ;
  • f ( v n 1 ) 4 n 4 ; f ( v n ) 4 n + 8 and f ( v 0 ) 4 n + 4 .
Note that P 2 , 2 is an edge even graceful graph but not follow this rule. See Figure 5.
Theorem 2.
If m is an odd positive integer greater than 1 and n an is even positive integer greater than or equal 2 , then the polar grid graph P m , n is an edge even graceful graph.
Proof. 
Let the edges of the polar grid graph P m , n be labeled as in Figure 2. □
Now the corresponding labels of vertices mod 4 m n are assigned as follows: There are four cases
Case (1): m ( 4 k 1 ) mod 4 n , 1 k n
The labels of the vertices of the inner most circle C m ( 1 ) to the circle C m ( n 2 ) are given by f ( v ( k 1 ) m + i ) 4 m ( 2 k 1 ) + 4 i + 2 , 1 k n 2 , 1 i m , the labels of the vertices of the outer circle C m ( n ) are given by f ( v ( n 1 ) m + i ) 2 i + 2 , 1 i m and the labels of the vertices of the circles C m ( n 2 + 1 ) , , C m ( n 1 ) are given by f ( v ( k 1 ) m + i ) 8 m ( k n 2 ) + 4 i + 2 , 1 i m , n 2 + 1 k n 1 .
The center vertex v 0 is labeled as f ( v 0 ) 4 m k , and if k = n , we have f ( v 0 ) 0 .
Case (2): m ( 8 k 3 ) mod 4 n , 1 k n 2 .
In this case the vertex v k m ( m + 1 2 ) in the circle C m ( k ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v k m ( m + 1 2 ) v k m ( m 1 2 ) and v k m ( m 1 2 ) v k m ( m 3 2 ) . That is f ( v k m ( m + 1 2 ) v k m ( m 1 2 ) ) = 2 m n + m ( 2 k 1 ) + 1 and f ( v k m ( m 1 2 ) v k m ( m 3 2 ) ) = 2 m n + m ( 2 k 1 ) 1 and we obtain the labels of the corresponding vertices as follows f ( v k m ( m + 1 2 ) ) 2 m ( 4 k 1 ) + 2 , f ( v k m ( m 1 2 ) ) 2 m ( 4 k 1 ) + 4 and f ( v k m ( m 3 2 ) ) 2 m ( 4 k 1 ) + 6 . The center vertex v 0 is labeled as f ( v 0 ) = 2 m ( 4 k 1 ) . The rest vertices are labeled as in case (1).
Case (3): m ( 8 k + 1 ) mod 4 n , 1 k n 2 1 .
In this case the vertex v m ( n 2 + k ) ( m + 1 2 ) in the circle C m ( n 2 + k ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m ( n 2 + k ) ( m + 1 2 ) v m ( n 2 + k ) ( m 1 2 ) and v m ( n 2 + k ) ( m 1 2 ) v m ( n 2 + k ) ( m 3 2 ) . That is f ( v m ( n 2 + k ) ( m + 1 2 ) v m ( n 2 + k ) ( m 1 2 ) ) = 3 m n + m ( 2 k 1 ) + 1 and f ( v m ( n 2 + k ) ( m 1 2 ) v m ( n 2 + k ) ( m 3 2 ) ) = 3 m n + m ( 2 k + 1 ) 1 and we obtain the labels of the corresponding vertices as follows f ( v m ( n 2 + k ) ( m + 1 2 ) ) 2 m ( 4 k + 1 ) + 2 , f ( v m ( n 2 + k ) ( m 1 2 ) ) 2 m ( 4 k + 1 ) + 4 , and f ( v m ( n 2 + k ) ( m 3 2 ) ) 2 m ( 4 k + 1 ) + 6 and in this case the center vertex v 0 is labeled as f ( v 0 ) = 2 m ( 4 k + 1 ) . The rest vertices are labeled as in case (1).
Case (4): m 1 mod 4 n
In this case the vertex v m n 1 in the outer circle C m ( n ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m n 2 v m n 1 and v m n v m ( n 1 ) + 1 . That is f ( v m n 2 v m n 1 ) = m ( 3 n + 2 ) and f ( v m n v m ( n 1 ) + 1 ) = m ( 3 n + 2 ) 4 and we obtain the labels of the corresponding vertices as follows f ( v m n 2 ) 2 m + 2 , f ( v m n 1 ) 2 m + 4 , f ( v m n ) 2 m 2 and f ( v m ( n 1 ) + 1 ) 0 , the center vertex v 0 is labeled as f ( v 0 ) 2 m . The rest vertices are labeled as in case (1).
Illustration. 
The edge even graceful labeling of the polar grid graphs P 13 , 6 , P 15 , 6 , P 17 , 6 and P 25 , 6 respectively are shown in Figure 6.
Theorem 3.
If m is an even positive integer greater than or equal 4 and n is an odd positive integer greater than or equal 3 . Then the polar grid graph P m , n is an edge even graceful graph.
Proof. 
Let the polar grid graph P m , n be labeled as in Figure 7. Let f : E ( G ) { 2 , 4 , , 2 q } .
First we label the edges of the circles C m ( k ) , 1 k n begin with the edges of the inner most circle C m ( 1 ) to edges of the outer circle C m ( n ) as follows:
f ( v m ( k 1 ) + i v m ( k 1 ) + i + 1 ) = 2 m ( k 1 ) + 2 i , f ( v k m v ( k 1 ) m + 1 ) = 2 k m , 1 i m 1 , 1 k n .
Second we label the edges of paths P n + 1 ( k ) , 1 k m begin with the edges of the path P n + 1 ( 1 ) as follows: Move anticlockwise to label the edges v 0 v 1 , v 0 v m , v 0 v m 1 , , v 0 v 3 , v 0 v 2 by 2 m n + 2 , 2 m n + 4 , 2 m n + 6 , , 2 m ( n + 1 ) 2 , 2 m ( n + 1 ) , then move clockwise to label the edges v 1 v m + 1 , v 2 v m + 2 , v 3 v m + 3 , , v m 1 v 2 m 1 , v m v 2 m by 2 m ( n + 1 ) + 2 , 2 m ( n + 1 ) + 4 , 2 m ( n + 1 ) + 6 , , 2 m ( n + 2 ) 2 , 2 m ( n + 2 ) , then move anticlockwise to label the edges v m + 1 v 2 m + 1 , v 2 m v 3 m , v 2 m 1 v 3 m 1 , , v m + 3 v 2 m + 3 , v m + 2 v 2 m + 2 by 2 m ( n + 2 ) + 2 , 2 m ( n + 2 ) + 4 , 2 m ( n + 2 ) + 6 , , 2 m ( n + 3 ) 2 , 2 m ( n + 3 ) and so on. Finally move anticlockwise to label the edges v m ( n 2 ) + 1 v m ( n 1 ) + 1 , v m ( n 1 ) + 1 v m n , v m ( n 1 ) 1 v m n 1 , , v m ( n 2 ) + 3 v m ( m 1 ) + 3 , v m ( n 2 ) + 2 v m ( m 1 ) + 2 by 2 m ( 2 n 1 ) + 2 , 2 m ( 2 n 1 ) + 4 , 2 m ( 2 n 1 ) + 6 , , 4 m n 2 , 4 m n .
The corresponding labels of vertices mod 4 m n are assigned as follows: There are four cases
Case (1) m 4 k mod 4 n , 1 k n 2 ; m ( 4 n 2 ) mod 4 n and m 0 mod 4 n
f ( v ( k 1 ) m + i ) 4 m ( 2 k 1 ) + 4 i + 2 , 1 i m , 1 k n 1 . That is the labels of the vertices in the most inner circle C m ( 1 ) are assigned by f ( v i ) 4 m + 4 i + 2 , 1 i m , the labels of the vertices in the circle C m ( 2 ) are assigned by f ( v m + i ) 12 m + 4 i + 2 , the labels of vertices of the circle C m ( n 1 2 ) are assigned by f ( v m ( n 3 ) 2 + i ) 4 m n 8 m + 4 i + 2 , the labels of vertices of the circle C m ( n + 1 2 ) are assigned by f ( v m ( n 1 ) 2 + i ) 4 i + 2 , the labels of vertices of the circle C m ( n + 3 2 ) are assigned by f ( v m ( n + 1 ) 2 + i ) 8 m + 4 i + 2 , , the labels of the vertices in the circle C m ( n 1 ) are assigned by f ( v m ( n 2 ) + i ) 4 m n 12 m + 4 i + 2 , 1 i m and the labels of the vertices of the outer circle C m ( n ) are assigned by f ( v m ( n 1 ) + i ) 4 m n 4 m + 2 i + 2 , 1 i m . The labels of the center vertex v 0 is assigned by f ( v 0 ) m 2 ( 2 n + 1 ) when m 4 k mod 4 n , we have f ( v 0 ) m ( 4 k + 1 ) , when m ( 4 n 2 ) mod 4 n , f ( v 0 ) 4 m n m and when m 0 mod 4 n , f ( v 0 ) m .
Case (2) m ( 8 k 2 ) mod 4 n , 1 k n 1 2 .
In this case the vertex v k m ( m + 2 4 ) in the circle C m ( k ) will repeat with the center vertex v 0 . To avoid this problem we replace the label of two edges v k m ( m + 2 4 ) v k m ( m 2 4 ) and v k m ( m 2 4 ) v k m ( m 6 4 ) . That is f ( v k m ( m + 2 4 ) v k m ( m 2 4 ) ) = m ( 2 k 1 ) + m + 2 2 and f ( v k m ( m 2 4 ) v k m ( m 6 2 ) ) = m ( 2 k 1 ) + m 2 2 and we obtain the labels of the corresponding vertices as follows f ( v k m ( m + 2 4 ) ) m ( 8 k 1 ) + 2 , f ( v k m ( m 2 4 ) ) m ( 8 k 1 ) + 4 and f ( v k m ( m 6 2 ) ) m ( 8 k 1 ) + 6 . In this case the center vertex v 0 is labeled as f ( v 0 ) m ( 2 m n + m + 1 ) m ( 8 k 1 ) . The rest vertices are labeled as in case (1).
Case (3) m ( 8 k 6 ) mod 4 n , 1 k n 1 2 and m 2 . In this case the vertex v m ( n + 2 k 1 2 ) ( m + 2 4 ) in the circle C m ( n + 2 k 1 2 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m ( n + 2 k 1 2 ) ( m + 2 4 ) v m ( n + 2 k 1 2 ) ( m 2 4 ) and v m ( n + 2 k 1 2 ) ( m 2 4 ) v m ( n + 2 k 1 2 ) ( m 6 4 ) . That is f ( v m ( n + 2 k 1 2 ) ( m + 2 4 ) v m ( n + 2 k 1 2 ) ( m 2 4 ) ) = m n + 2 m ( k 1 ) + m + 2 2 and f ( v m ( n + 2 k 1 2 ) ( m 2 4 ) v m ( n + 2 k 1 2 ) ( m 6 4 ) ) = m n + 2 m ( k 1 ) + m 2 2 and we obtain the labels of the corresponding vertices as follows f ( v m ( n + 2 k 1 2 ) ( m + 2 4 ) ) m ( 8 k 5 ) + 2 , f ( v m ( n + 2 k 1 2 ) ( m 2 4 ) ) m ( 8 k 5 ) + 4 and f ( v m ( n + 2 k 1 2 ) ( m 6 4 ) ) m ( 8 k 5 ) + 6 and in this case the center vertex v 0 is labeled as f ( v 0 ) m ( 8 k 5 ) . The rest vertices are labeled as in case (1).
Case (4) m ( 4 n 4 ) mod 4 n . In this case the vertex v m n ( m + 2 2 ) in the outer circle C m ( n ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m n ( m + 6 4 ) v m n ( m + 2 4 ) and v m n v m ( n 1 ) + 1 . That is f ( v m n ( m + 6 4 ) v m n ( m + 2 4 ) ) = 2 m n and f ( v m n v m ( n 1 ) + 1 ) = m ( 2 n 1 ) 4 and we obtain the labels of the corresponding vertices as follows f ( v m n ( m + 6 4 ) ) 4 m n 2 m + 2 , f ( v m n ( m + 2 4 ) ) 4 m n 2 m + 8 , f ( v m n ) 4 m n 3 m 2 and f ( v m ( n 1 ) + 1 ) 4 m n 5 m and in this case the center vertex v 0 is labeled as f ( v 0 ) m ( 4 n 3 ) . The rest vertices are labeled as in case (1).
Illustration. 
The edge even graceful labeling of the polar grid graphs P 10 , 5 , P 12 , 5 , P 14 , 5 , P 16 , 5 , P 18 , 5 and P 20 , 5 respectively are shown in Figure 8.
Remark 2.
In case m = 2 , n is odd, n 3 .
Let the label of edges of the polar grid graph P 2 , n be as in Figure 9. Thus we have the labels of the corresponding vertices as follows: f ( v 1 ) 12 ; f ( v i ) 16 i 2 , 2 i n 1 2 ; f ( v n + 2 i 1 2 ) 16 i 10 , 1 i n 1 2 ; f ( v n ) 8 n 2 ; f ( v i ) 16 i + 2 , 1 i n 1 2 ; f ( v n + 1 2 ) 2 ; f ( v n + 2 i 1 2 ) 16 i 6 , 2 i n 1 2 ; f ( v n ) 0 and f ( v 0 ) 4 .
Illustration. 
The edge even graceful labeling of the polar grid graphs P 2 , 5 is shown in Figure 10.
Theorem 4.
If m and n are odd positive integers greater than 1 . Then the polar grid graph P m , n is an edge even graceful graph.
Proof. 
Let the polar grid graph P m , n be labeled as in Figure 7. Let f : E ( G ) { 2 , 4 , , 2 q } .
The corresponding labels of vertices mod 4 m n are assigned as follows: There are two cases:
Case (1) n 1 mod 4 , this case contains five subcases as follows:
SubCase (i) m ( 4 k 3 ) mod 4 n , 2 k n
f ( v ( k 1 ) m + i ) 4 m ( 2 k 1 ) + 4 i + 2 , 1 k n 1 , 1 i m . That is the labels of vertices of the most inner circle C m ( 1 ) are assigned by f ( v i ) = 4 m + 4 i + 2 , the labels of vertices of the circle C m ( 2 ) are assigned by f ( v m + i ) 12 m + 4 i + 2 , the labels of the vertices of the circle C m ( n 1 2 ) are assigned by f ( v m ( n 3 ) 2 + i ) 4 m n 8 m + 4 i + 2 , the labels of the vertices of the circle C m ( n + 1 2 ) are assigned by f ( v m ( n 1 ) 2 + i ) 4 i + 2 , the labels of the vertices of the circle C m ( n + 3 2 ) are assigned by f ( v m ( n + 1 ) 2 + i ) 8 m + 4 i + 2 , , the labels of the vertices of the circle C m ( n 1 ) are assigned by f ( v m ( n 2 ) + i ) 4 m n 12 m + 4 i + 2 , 1 i m and the labels of the vertices of the outer circle C m ( n ) are assigned by f ( v m ( n 1 ) + i ) 4 m n 12 m + 2 i + 2 , 1 i m . The label of the center vertex v 0 is assigned by f ( v 0 ) 2 m n + 2 m ( 2 k 1 ) , when k = n + 1 2 , we have f ( v 0 ) 0 .
SubCase (ii) m ( 8 k 5 ) mod 4 n , 1 k n + 1 2 , m 3 . In this subcase the vertex v m ( n + 4 k 1 4 ) ( m + 1 2 ) in the circle C m ( n + 4 k 1 4 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m ( n + 4 k 1 4 ) ( m + 1 2 ) v m ( n + 4 k 1 4 ) ( m 1 2 ) and v m ( n + 4 k 1 4 ) ( m 1 2 ) v m ( n + 4 k 1 4 ) ( m 3 2 ) . That is f ( v m ( n + 4 k 1 4 ) ( m + 1 2 ) v m ( n + 4 k 1 4 ) ( m 1 2 ) ) = m [ 2 k + n 3 2 ] + 1 and f ( v m ( n + 4 k 1 4 ) ( m 1 2 ) v m ( n + 4 k 1 4 ) ( m 3 2 ) ) = m [ 2 k + n 3 2 ] 1 and we obtain the labels of the corresponding vertices as follows f ( v m ( n + 4 k 1 4 ) ( m + 1 2 ) ) 2 m n + 4 m ( 2 k 1 ) + 2 , f ( v m ( n + 4 k 1 4 ) ( m 1 2 ) ) 2 m n + 4 m ( 2 k 1 ) + 4 , f ( v m ( n + 4 k 1 4 ) ( m 3 2 ) ) 2 m n + 4 m ( 2 k 1 ) + 6 , and in this case the center vertex v 0 is labeled as f ( v 0 ) 2 m n + 4 m ( 2 k 1 ) . The rest vertices will be labeled as in subCase (i).
Remark 3.
When n 1 mod 4 and m = 3 , in this case the vertex v 3 ( n 1 4 ) + 1 in the circle C 3 ( n 1 4 + 1 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v 3 ( n 1 4 ) + 2 v 3 ( n 1 4 ) + 3 and v 3 ( n 1 4 ) + 1 v 3 ( n 1 4 ) + 3 . That is f ( v 3 ( n 1 4 ) + 2 v 3 ( n 1 4 ) + 3 ) = 3 ( n 1 2 ) + 6 and f ( v 3 ( n 1 4 ) + 1 v 3 ( n 1 4 ) + 3 ) = 3 ( n 1 2 ) + 4 and we obtain the labels of the corresponding vertices as follows f ( v 3 ( n 1 4 ) + 1 ) 6 n + 10 , f ( v 3 ( n 1 4 ) + 2 ) 6 n + 18 , f ( v 3 ( n 1 4 ) + 3 ) 6 n + 16 and the center vertex v 0 is labeld as f ( v 0 ) 6 n + 12 .
SubCase (iii) m ( 8 k 1 ) mod 4 n , 1 k n 5 4 . In this subcase the vertex v m ( 3 n + 4 k + 1 4 ) ( m + 1 2 ) in the circle C m ( 3 n + 4 k 1 4 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m ( 3 n + 4 k + 1 4 ) ( m + 1 2 ) v m ( 3 n + 4 k + 1 4 ) ( m 1 2 ) and v m ( 3 n + 4 k + 1 4 ) ( m 1 2 ) v m ( 3 n + 4 k + 1 4 ) ( m 3 2 ) . That is f ( v m ( 3 n + 4 k + 1 4 ) ( m + 1 2 ) v m ( 3 n + 4 k + 1 4 ) ( m 1 2 ) ) = 2 m n + m [ 2 k n + 1 2 ] + 1 and f ( v m ( 3 n + 4 k 1 4 ) ( m 1 2 ) v m ( 3 n + 4 k 1 4 ) ( m 3 2 ) ) = 2 m n + m [ 2 k n + 1 2 ] 1 and we obtain the labels of the corresponding vertices as follows f ( v m ( 3 n + 4 k + 1 4 ) ( m + 1 2 ) ) 2 m n + 8 k m + 2 , f ( v m ( 3 n + 4 k 1 4 ) ( m 1 2 ) ) 2 m n + 8 k m + 4 , f ( v m ( 3 n + 4 k + 1 4 ) ( m 3 2 ) ) 2 m n + 8 k m + 6 , and in this case the center vertex v 0 is labeled as f ( v 0 ) 2 m n + 8 k m . The rest vertices will be labeled as in subCase (i).
SubCase (iv) m ( 8 k 1 ) mod 4 n , n + 3 4 k n 1 2 . In this case the vertex v m ( 4 k n + 1 4 ) ( m + 1 2 ) in the circle C m ( 4 n k + 1 4 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m ( 4 k n + 1 4 ) ( m + 1 2 ) v m ( 4 k n + 1 4 ) ( m 1 2 ) and v m ( 4 k n + 1 4 ) ( m 1 2 ) v m ( 4 k n + 1 4 ) ( m 3 2 ) . That is f ( v m ( 4 k n + 1 4 ) ( m + 1 2 ) v m ( 4 k n + 1 4 ) ( m 1 2 ) ) = m [ 2 k n + 1 2 ] + 1 and f ( v m ( 4 k n + 1 4 ) ( m 1 2 ) v m ( 4 k n + 1 4 ) ( m 3 2 ) ) = m [ 2 k n + 1 2 ] 1 and we obtain the labels of the corresponding vertices as follows f ( v m ( 4 k n + 1 4 ) ( m + 1 2 ) ) 2 m n + 8 k m + 2 , f ( v m ( 4 k n + 1 4 ) ( m 1 2 ) ) 8 k m 2 m n + 4 , f ( v m ( 4 k n + 1 4 ) ( m 3 2 ) ) 8 k m 2 m n + 6 , and in this case the center vertex v 0 is labeled as f ( v 0 ) 8 k m 2 m n . The rest vertices will be labeled as in subCase (i).
SubCase (v) m ( 2 n 3 ) mod 4 n . In this case the vertex v m n 1 in the outer circle C m ( n ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m n 2 v m n 1 and v m n v m ( n 1 ) + 1 . That is f ( v m n 2 v m n 1 ) = 2 m n 4 , f ( v m n v m ( n 1 ) + 1 ) = 2 m n and we obtain the labels of the corresponding vertices are as follows f ( v m n 2 ) 4 m n 2 m + 2 , f ( v m n 1 ) 4 m n 2 m + 4 , f ( v m n ) 4 m n 2 m 2 and f ( v m ( n 1 ) + 1 ) 4 m n 4 m , and in this case the center vertex v 0 is labeled as f ( v 0 ) 4 m n 2 m . The rest vertices will be labeled as in subCase (i).
Illustration. 
The edge odd graceful labeling of the polar grid graphs P 3 , 5 , P 13 , 5 , P 11 , 5 , P 7 , 9 , P 15 , 5 and P 7 , 5 respectively are shown in Figure 11.
Case (2) n 3 mod 4 . This case contains also five subcases as follows:
SubCase (i) m ( 4 k 3 ) mod 4 n , 2 k n
f ( v ( k 1 ) m + i ) 4 m ( 2 k 1 ) + 4 i + 2 , 1 k n 1 , 1 i m . That is the labels of vertices of the most inner circle C m ( 1 ) are assigned by f ( v i ) 4 m + 4 i + 2 , the label of vertices of the circle C m ( 2 ) are assigned by f ( v m + i ) 12 m + 4 i + 2 , the labels of vertices of the circle C m ( n 1 2 ) are assigned by f ( v m ( n 1 ) 2 + i ) 4 m n 8 m + 4 i + 2 , the labels of vertices of the circle C m ( n + 1 2 ) are assigned by f ( v m ( n 3 ) 2 + i ) 4 i + 2 , the labels of vertices of the circle C m ( n + 3 2 ) are assigned by f ( v m ( n + 1 ) 2 + i ) 8 m + 4 i + 2 , , the labels of vertices of the circle C m ( n 1 ) are assigned by f ( v m ( n 2 ) + i ) 4 m n 12 m + 4 i + 2 , 1 i m and the labels of the vertices of the outer circle C m ( n ) are assigned by f ( v m ( n 1 ) + i ) 4 m n 12 m + 2 i + 2 , 1 i m . The label of the center vertex v 0 is assigned by f ( v 0 ) 2 m n + 2 m ( 2 k 1 ) , when k = n + 1 2 , we have f ( v 0 ) 0 .
SubCase (ii) m ( 8 k 5 ) mod 4 n , 1 k n 3 4 , m 3
In this subcase the vertex v m ( 3 n + 4 k 1 4 ) ( m + 1 2 ) in the circle C m ( 3 n + 4 k 1 4 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m ( 3 n + 4 k 1 4 ) ( m + 1 2 ) v m ( 3 n + 4 k 1 4 ) ( m 1 2 ) and v m ( 3 n + 4 k 1 4 ) ( m 1 2 ) v m ( 3 n + 4 k 1 4 ) ( m 3 2 ) . That is f ( v m ( 3 n + 4 k 1 4 ) ( m + 1 2 ) v m ( 3 n + 4 k 1 4 ) ( m 1 2 ) ) = 2 m n + m [ 2 k n + 3 2 ] + 1 and f ( v m ( 3 n + 4 k 1 4 ) ( m 1 2 ) v m ( 3 n + 4 k 1 4 ) ( m 3 2 ) ) = 2 m n + m [ 2 k n + 3 2 ] 1 and we obtain the labels of the corresponding vertices as follows f ( v m ( 3 n + 4 k 1 4 ) ( m + 1 2 ) ) 2 m n + 4 m ( 2 k 1 ) + 2 , f ( v m ( 3 n + 4 k 1 4 ) ( m 1 2 ) ) 2 m n + 4 m ( 2 k 1 ) + 4 , f ( v m ( 3 n + 4 k 1 4 ) ( m 3 2 ) ) 2 m n + 4 m ( 2 k 1 ) + 6 , and the label of the center vertex v 0 is assigned by f ( v 0 ) 2 m n + 4 m ( 2 k 1 ) . That rest vertices will be labeled as in subcase (i).
Remark 4.
When n 3 mod 4 and m = 3 , we have the vertex v 3 ( 3 n 1 4 ) + 1 in the circle C m ( 3 n 1 4 + 1 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v 3 ( 3 n 1 4 ) + 2 v 3 ( 3 n 1 4 ) + 3 and v 3 ( 3 n 1 4 ) + 1 v 3 ( 3 n 1 4 ) + 3 . That is f ( v 3 ( 3 n 1 4 ) + 2 v 3 ( 3 n 1 4 ) + 3 ) = 3 ( 3 n 1 2 ) + 6 and f ( v 3 ( 3 n 1 4 ) + 1 v 3 ( 3 n 1 4 ) + 3 ) = 3 ( 3 n 1 2 ) + 4 and we obtain the labes of the corresponding vertices mod 4 m n are as follows: f ( v 3 ( 3 n 1 4 ) + 1 ) 6 n + 10 , f ( v 3 ( 3 n 1 4 ) + 2 ) 6 n + 18 , f ( v 3 ( 3 n 1 4 ) + 3 ) 6 n + 20 and the label of the center vertex v 0 is assigned by f ( v 0 ) 6 n + 12 .
Note that P 3 , 3 is an edge even graceful grapg but not follow this rule. See Figure 12.
SubCase (iii) m ( 8 k 1 ) mod 4 n , n + 5 4 k n + 1 2 ,
In this subcase the vertex v m ( 4 k n 1 4 ) ( m + 1 2 ) in the circle C m ( 4 k n 1 4 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m ( 4 k n 1 4 ) ( m + 1 2 ) v m ( 4 k n 1 4 ) ( m 1 2 ) and v m ( 4 k n 1 4 ) ( m 1 2 ) v m ( 4 k n 1 4 ) ( m 3 2 ) That is f ( v m ( 4 k n 1 4 ) ( m + 1 2 ) v m ( 4 k n 1 4 ) ( m 1 2 ) ) = ( 2 k 1 ) m + 1 and f ( v m ( 4 k n 1 4 ) ( m 1 2 ) v m ( 4 k n 1 4 ) ( m 3 2 ) ) = ( 2 k 1 ) m 1 and we obtain the labels of the corresponding vertices as follows: f ( v m ( 4 k n 1 4 ) ( m + 1 2 ) ) 2 m ( 4 k 1 ) + 2 , f ( v m ( 4 k n 1 4 ) ( m 1 2 ) ) 2 m ( 4 k 1 ) + 4 and f ( v m ( 4 k n 1 4 ) ( m 3 2 ) ) 2 m ( 4 k 1 ) + 6 . The label of the center vertex v 0 is assigned by f ( v 0 ) 2 m ( 4 k 1 ) . The rest vertices will be labeled as in subCase (i).
SubCase (iv) m ( 8 k 1 ) mod 4 n , 1 k n 1 2
In this subcase the vertex v m ( n + 4 k + 1 4 ) ( m + 1 2 ) in the circle C m ( n + 4 k + 1 4 ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m ( n + 4 k + 1 4 ) ( m + 1 2 ) v m ( n + 4 k + 1 4 ) ( m 1 2 ) and v m ( n + 4 k + 1 4 ) ( m 1 2 ) v m ( n + 4 k + 1 4 ) ( m 3 2 ) That is f ( v m ( n + 4 k + 1 4 ) ( m + 1 2 ) v m ( n + 4 k + 1 4 ) ( m 1 2 ) ) = ( 2 k + n 1 2 ) m + 1 and f ( v m ( n + 4 k + 1 4 ) ( m 1 2 ) v m ( n + 4 k + 1 4 ) ( m 3 2 ) ) = ( 2 k + n 1 2 ) m 1 and we obtain the labels of the corresponding vertices as follows f ( v m ( n + 4 k + 1 4 ) ( m + 1 2 ) ) = 2 m n + 8 k m + 2 , f ( v m ( n + 4 k + 1 4 ) ( m 1 2 ) ) = 2 m n + 8 k m + 4 , f ( v m ( n + 4 k + 1 4 ) ( m 3 2 ) ) = 2 m n + 8 k m + 6 and the label of the center vertex v 0 is labeled as f ( v 0 ) 2 m n + 8 k m . The rest vertices will be labeled as in subCase (i).
Remark 5.
If k = n + 1 4 we have
f ( v m ( n + 4 k + 1 4 ) ( m + 1 2 ) ) = 8 k m 2 m n + 2 , f ( v m ( n + 4 k + 1 4 ) ( m 1 2 ) ) = 8 k m 2 m n + 4 , f ( v m ( n + 4 k + 1 4 ) ( m 3 2 ) ) = 8 k m 2 m n + 6
and the center vertex v 0 is labeled as f ( v 0 ) 8 k m 2 m n .
SubCase (v) m ( 2 n 3 ) mod 4 n
In this subcase the vertex v m n 1 in the outer circle C m ( n ) will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m n 2 v m n 1 and v m n v m ( n 1 ) + 1 . That is f ( v m n 2 v m n 1 ) = 2 m n 4 , f ( v m n v m ( n 1 ) + 1 ) = 2 m n and we obtain the labes of the corresponding vertices as follows:
f ( v m n 2 ) = 4 m n 2 m + 2 , f ( v m n 1 ) = 4 m n 2 m + 4 , f ( v m n ) = 4 m n 2 m 2 and f ( v m ( n 1 ) + 1 ) = 4 m ( n 1 ) and the label of the center vertex v 0 is assigned by f ( v 0 ) 2 m ( 2 n 1 ) .
The rest vertices will be labeled as in subCase (i).
Illustration. 
The edge odd graceful labeling of the polar grid graphs P 3 , 7 , P 13 , 7 , P 19 , 7 , P 11 , 7 and P 15 , 7 respectively are shown in Figure 13.

3. Conclusions

This paper gives some basic knowledge about the application of Graph labeling and Graph Theory in real life which is the one branch of mathematics. It is designed for the researcher who research in graph labeling and graph Theory. In this paper, we give necessary and sufficient conditions for a polar grid graph to admit edge even labeling. In future work we will study the necessary and sufficient conditions for the cylinder P m × C n , torus C m × C n and rectangular P m × P n grid graphs to be edge even graceful.

Funding

This work was supported by the deanship of Scientific Research, Taibah University, Al-Madinah Al-Munawwarah, Saudi Arabia.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Figure 1. Polar grid graph P m , n .
Figure 1. Polar grid graph P m , n .
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Figure 2. Labeling ofthe polar grid graph P m , n when n is even, n 2 .
Figure 2. Labeling ofthe polar grid graph P m , n when n is even, n 2 .
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Figure 3. The edge even graceful labeling of the polar grid graphs P 14 , 6 , P 16 , 6 , P 18 , 6 , P 24 , 6 and P 26 , 6 .
Figure 3. The edge even graceful labeling of the polar grid graphs P 14 , 6 , P 16 , 6 , P 18 , 6 , P 24 , 6 and P 26 , 6 .
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Figure 4. Labeling of the polar grid graph P 2 , n , n is even integer greater than 2 .
Figure 4. Labeling of the polar grid graph P 2 , n , n is even integer greater than 2 .
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Figure 5. The polar grid graph P 2 , 2 .
Figure 5. The polar grid graph P 2 , 2 .
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Figure 6. The edge even graceful labeling of the polar grid graphs P 13 , 6 , P 15 , 6 , P 17 , 6 and P 25 , 6 .
Figure 6. The edge even graceful labeling of the polar grid graphs P 13 , 6 , P 15 , 6 , P 17 , 6 and P 25 , 6 .
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Figure 7. Labeling of the polar grid graph P m , n when n is odd and n 3 .
Figure 7. Labeling of the polar grid graph P m , n when n is odd and n 3 .
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Figure 8. The edge even graceful labeling of the polar grid graphs P 10 , 5 , P 12 , 5 , P 14 , 5 , P 16 , 5 , P 18 , 5 and P 20 , 5 .
Figure 8. The edge even graceful labeling of the polar grid graphs P 10 , 5 , P 12 , 5 , P 14 , 5 , P 16 , 5 , P 18 , 5 and P 20 , 5 .
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Figure 9. The labeling of the polar grid graph P 2 , n , n 3 .
Figure 9. The labeling of the polar grid graph P 2 , n , n 3 .
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Figure 10. The labeling of the polar grid graph P 2 , 5 .
Figure 10. The labeling of the polar grid graph P 2 , 5 .
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Figure 11. The polar grid graphs P 3 , 5 , P 13 , 5 , P 11 , 5 , P 7 , 9 , P 15 , 5 and P 7 , 5 .
Figure 11. The polar grid graphs P 3 , 5 , P 13 , 5 , P 11 , 5 , P 7 , 9 , P 15 , 5 and P 7 , 5 .
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Figure 12. The polar grid graphs P 3 , 3 .
Figure 12. The polar grid graphs P 3 , 3 .
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Figure 13. The polar grid graphs P 3 , 7 , P 13 , 7 , P 19 , 7 , P 11 , 7 P 15 , 7 .
Figure 13. The polar grid graphs P 3 , 7 , P 13 , 7 , P 19 , 7 , P 11 , 7 P 15 , 7 .
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