Edge Even Graceful Labeling of Polar Grid Graphs

: 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 mod2 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.


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].
A function f is called an odd graceful labeling of a graph G if f : V(G) → {0, 1, 2, . . . , 2q − 1} is injective and the induced function f * : E(G) → {1, 3, . . . , 2q − 1} defined as f * (e = uv) = | 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]. 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]. f (e)mod2q is injective. This type of graph labeling first introducedby Solairaju and Chithra in 2009 [19]. See also Daoud [20]. f (e)mod2r, 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].

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 and m copies of paths P n+1 intersected at the center vertex v 0 which will be numbered as P (1) n+1 , P and Chithra in 2009 [19]. See also Daoud [20]. A function f is called an edge even graceful labeling of a graph G if :  [21]. For a summary of the results on these five types of graceful labels as well as all known labels so far, see [22].   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.
In this case the vertex v km−( m+2 4 ) in the circle C (k) m will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v km−( m+2 and we obtain the labels of the corresponding vertices as follows 4 ) ) ≡ m(8k − 1) + 6 and the label of the center vertex v 0 is assigened as f * (v 0 ) ≡ m(8k − 1).The rest vertices are labeled as in case (1).
In this case the vertex v km−( m+2 2 ) in the circle C ( n 2 +k) m will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v km−( m+2 and we obtain the labels of the corresponding vertices as follows ) ≡ m(8k + 3) + 6 and the label of the center vertex v 0 is assigened f * (v 0 ) ≡ m(8k − 29) as. The rest vertices are labeled as in case (1).
Case (4) m ≡ 0mod4n. In this case the vertex v mn−( 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 mn−( m+4 and v mn v m(n−1)+1 . That is f (v mn−( m+4 2 ) v mn−( m+2 2 ) ) = m(3n + 2) and f (v mn v m(n−1)+1 ) = 3mn + m − 4 and we obtain the labels of the corresponding vertices as follows The rest vertices are labeled as in case (1).

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: P .

Remark 1.
In case 2 m = and is even, 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: 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 .  Note that 2, 2 P is an edge even graceful graph but not follow this rule. See Figure 5. Note that P 2,2 is an edge even graceful graph but not follow this rule. See Figure 5.  Note that 2, 2 P 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 mod4mn are assigned as follows: There are four cases The labels of the vertices of the inner most circle C In this case the vertex v km( m+1 2 ) in the circle C (k) m will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v km−( m+1 2 ) ) = 2mn + m(2k − 1) − 1 and we obtain the labels of the corresponding vertices as follows . The rest vertices are labeled as in case (1).
Case (3): will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m( n 2 ) ) = 3mn + m(2k + 1) − 1 and we obtain the labels of the corresponding vertices as follows f * (v m( n 2 ) ) ≡ 2m(4k + 1) + 6 and in this case the center vertex v 0 is labeled as f * (v 0 ) = 2m(4k + 1). The rest vertices are labeled as in case (1).
Case (4): m ≡ 1mod4n In this case the vertex v mn−1 in the outer circle C (n) m will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v mn−2 v mn−1 and v mn v m(n−1)+1 . That is f (v mn−2 v mn−1 ) = m(3n + 2) and f (v mn v m(n−1)+1 ) = m(3n + 2) − 4 and we obtain the labels of the corresponding vertices as follows 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 Second we label the edges of paths P Finally move anticlockwise to label the edges         The corresponding labels of vertices mod4mn are assigned as follows: There are four cases That is the labels of the vertices in the most inner circle C (1) m are assigned by f * (v i ) ≡ 4m + 4i + 2, 1 ≤ i ≤ m, the labels of the vertices in the circle C (2) m are assigned by f * (v m+i ) ≡ 12m + 4i + 2, the labels of vertices of the circle C are assigned by f * (v m(n−2)+i ) ≡ 4mn − 12m + 4i + 2, 1 ≤ i ≤ m and the labels of the vertices of the outer circle Case (2) m ≡ (8k − 2)mod 4n, 1 ≤ k ≤ n−1 2 . In this case the vertex v km−( m+2 4 ) in the circle C (k) m will repeat with the center vertex v 0 . To avoid this problem we replace the label of two edges v km−( m+2 and we obtain the labels of the corresponding vertices as follows In this case the center vertex v 0 is labeled as f * (v 0 ) ≡ m(2mn + m + 1) ≡ m(8k − 1). The rest vertices are labeled as in case (1).
Case ( and we obtain the labels of the corresponding vertices as follows 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.
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:  . 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 . Let the label of edges of the polar grid graph 2, n P be as in Figure (9). Thus we have the labels of the corresponding vertices as follows:   Figure 9. The labeling of the polar grid graph 2, , 3 n P n ≥ .
Illustration.The edge even graceful labeling of the polar grid graphs 2 , 5 P is shown in Figure 10.  Illustration. The edge even graceful labeling of the polar grid graphs P 2,5 is shown in Figure 10.   Figure 9. The labeling of the polar grid graph 2, , 3 n P n ≥ .
Illustration.The edge even graceful labeling of the polar grid graphs 2 , 5 P is shown in Figure 10.   The corresponding labels of vertices mod4mn are assigned as follows: There are two cases: Case (1) n ≡ 1 mod4, this case contains five subcases as follows: That is the labels of vertices of the most inner circle C (1) m are assigned by f * (v i ) = 4m + 4i + 2, the labels of vertices of the will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m( n+4k−1 2 ] − 1 and we obtain the labels of the corresponding vertices as follows 2 ) ) ≡ 2mn + 4m(2k − 1) + 6, and in this case the center vertex v 0 is labeled as f * (v 0 ) ≡ 2mn + 4m(2k − 1). The rest vertices will be labeled as in subCase (i).

Remark 3.
When n ≡ 1 mod4 and m = 3, in this case the vertex v 3( n−1 4 )+1 in the circle C will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v 3( n−1 2 ) + 4 and we obtain the labels of the corresponding vertices as follows f SubCase (iii) m ≡ (8k − 1)mod4n, 1 ≤ k ≤ n−5 4 . In this subcase the vertex v m( 3n+4k+1 in the circle C 2 ) ) = 2mn + m[2k − n+1 2 ] − 1 and we obtain the labels of the corresponding vertices as follows f * (v m( 3n+4k+1 2 ) ) ≡ 2mn + 8km + 6, and in this case the center vertex v 0 is labeled as f * (v 0 ) ≡ 2mn + 8km. The rest vertices will be labeled as in subCase (i).
) m will repeat with the center vertex v 0 .
To avoid this problem we replace the labels of the two edges v m( 4k−n+1 2 ] − 1 and we obtain the labels of the corresponding vertices as follows 2 ) ) ≡ 8km − 2mn + 6, and in this case the center vertex v 0 is labeled as f * (v 0 ) ≡ 8km − 2mn. The rest vertices will be labeled as in subCase (i).
Case (2) n ≡ 3 mod4. This case contains also five subcases as follows: That is the labels of vertices of the most inner circle C (1) m are assigned by f * (v i ) ≡ 4m + 4i + 2, the label of vertices of the circle C (2) m are assigned by f * (v m+i ) ≡ 12m + 4i + 2, the labels of vertices of the circle C will repeat with the center vertex v 0 . To avoid this problem we replace the labels of the two edges v m( 3n+4k−1 and v m( 3n+4k−1 ] − 1 and we obtain the labels of the corresponding vertices as follows 2 ) ) ≡ 2mn + 4m(2k − 1) + 6, and the label of the center vertex v 0 is assigned by f * (v 0 ) ≡ 2mn + 4m(2k − 1). That rest vertices will be labeled as in subcase (i). vertices will be labeled as in subCase (i).
Note that P 3,3 is an edge even graceful grapg but not follow this rule. See Figure 12. That rest vertices will be labeled as in subcase (i). Note that 3, 3 P is an edge even graceful grapg but not follow this rule. See Figure 12.

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 m