Edge Even Graceful Labeling of Cylinder Grid Graph

Edge even graceful labeling (e.e.g., l.) of graphs is a modular technique of edge labeling of graphs, introduced in 2017. An e.e.g., l. of simple finite undirected graph G = ( V ( G ) , E ( G ) ) of order P = | ( V ( G ) | and size q = | E ( G ) | is a bijection f : E ( G ) → { 2 , 4 , … , 2 q } , such that when each vertex v ∈ V ( G ) is assigned the modular sum of the labels (images of f ) of the edges incident to v , the resulting vertex labels are distinct mod 2 r , where r = max ( p , q ) . In this work, the family of cylinder grid graphs are studied. Explicit formulas of e.e.g., l. for all of the cases of each member of this family have been proven.

graph labeling was first introduced by Solairaju and Chithra in 2009 [20]. For more results on this labeling, see References [21][22][23]. A function f is called an edge even graceful labeling of a graph G if f : E(G) → 2, 4, . . . , 2q − 2 is bijective and the induced function f * : V(G) → 0, 2, 4, . . . , 2q − 2 , defined as f * (u) = e=uv∈E (G) f (e)(mod2r) where r = max p, q , is injective. This type of graph labeling was first introduced by Elsonbaty and Daoud in 2017 [24,25]. For a summary of the results on these five types of graceful labels as well as all known labeling techniques, see Reference [26].

Cylinder Grid Graph
The Cartesian product G 1 × G 2 of two graphs G 1 and G 2 , is the graph with vertex set V(G 1 ) × V(G 2 ), and any two vertices (u 1 , v 1 ) and (u 2 , v 2 ) are adjacent in G 1 × G 2 whenever u 1 = u 2 and v 1 v 2 ∈ E(G 2 ) or v 1 = v 2 and u 1 u 2 ∈ E(G 1 ). The cylinder grid graph C m,n is the graph formed from the Cartesian product P m × C n of the path graph P m and the cycle graph C n . That is, the cylinder grid graph consists of m copies of C n represented by circles, and will be numbered from the innermost circle to the outer circle as C (1) n , C (2) n , C   [20]. For more results on this labeling, see References [21][22][23]. A function f is called an edge even graceful labeling of a graph  [24,25]. For a summary of the results on these five types of graceful labels as well as all known labeling techniques, see Reference [26].

Cylinder Grid Graph
The Cartesian product 12 GG  of two graphs 1 G and 2 G , is the graph with vertex set   Theorem 1. If m is an even positive integer greater than or equal 2 and n ≥ 2, then the cylinder grid graph C m,n , is an edge even graceful graph. Proof. Using standard notation p = V(C m,n ) = mn, q = E(C m,n ) = 2mn − n and r = max(p, q) = 2mn − n and f : E(C m,n ) → {2, 4, 6, . . . , 4mn − 2n − 2}. Let the cylinder grid graph C m,n be as in Figure 2.     First, we label the edges of the paths P , v 2n+2 v 3n+2 by 4n + 2, 4n + 4, 4n + 6, . . . , 6n − 2, 6n and so on. Finally, move anticlockwise to label the edges v Secondly, we label the edges of the circles C n . Finally, we label the edges of the circles C n as follows: Thus, the labels of corresponding vertices mod(4mn − 2n) will be: Illustration: An e.e.g., l, of the cylinder grid graphs C 8,11 and C 8,12 are shown in Figure 3.
Theorem 2. If m = 3 and n is an odd positive integer greater than 3, then the cylinder grid graph C 3,n , is an edge even graceful graph.
Thus, the labels of corresponding vertices mod (4 2 ) m n n  will be: f Illustration: An e.e.g., l, of the cylinder grid graphs 8 ,11 C and 8 ,12 C are shown in Figure 3. Proof. Using standard notation 3, , let the cylinder grid graph 3, n C be as in Figure 4.   f Then, move anticlockwise to label the edges    First, we label the edges of the paths P Secondly, we label the edges of the circles C   n as follows:    Label the edges of the circle ( 2) n C as follows: Label the edges of the circle (3) n C as follows: The labels of corresponding of vertices mod10n are as follows: The labels of vertices of the circle  Label the edges of the circle C n as follows: The labels of corresponding of vertices mod10n are as follows: The labels of vertices of the circle C (1) n are as follows: The labels of vertices of the circle C Case (2): If n ≡ 3mod6, let the cylinder grid graph C 3,n be as in Figure 5.
First, we label the edges of the paths P , let the cylinder grid graph 3, n C be as in Figure 6.
The labels of vertices of the circles ( 2) n C and (3) n C are the same as in case (1). Illustration: An e.e.g., l. of the cylinder grid graphs 3, 25 3, 27 , CC and 3, 29 C are shown in Figure 7. First, we label the edges of the paths P n as follows: The labels of corresponding vertices mod10n are as follows: The labels of vertices of the circle C (1) The labels of vertices of the circles C (2) n and C (3) n are the same as in case (1). Illustration: An e.e.g., l. of the cylinder grid graphs C 3,25 , C 3,27 and C 3,29 are shown in Figure 7. Remark 1. Note that 3,5 C is an edge even graceful graph but it does not follow the pervious rule (see Figure 8).  Remark 1. Note that C 3,5 is an edge even graceful graph but it does not follow the pervious rule (see Figure 8).   Theorem 3. If m is an odd positive integer greater than 3 and n is an even positive integer, n ≥ 2, then the cylinder grid graph C m,n , is an edge even graceful graph. Proof. Using standard notation p = V(C m,n ) = mn, q = E(C m,n ) = 2mn − n and r = max(p, q) = 2mn − n and f : E(C m,n ) → {2, 4, 6, . . . , 4mn − 2n − 2}.
Label the edges of the circle C (4) n as follows: Label the edges of C (2) n as follows: Thus, the labels of corresponding vertices mod(4mn − 2n) will be: The label the vertices of C (1) n are: The label the vertices of C respectively are: Case (2): n ≡ 2mod12, n 2. First, we label the edges of the paths P   Label the edges of the circle C (2) n as follows:  (1). Thus, the labels of corresponding vertices mod(4mn − 2n) will be: The label the vertices of C The label the vertices of the circle C n are: The label the vertices of C n respectively are as the same as in case (1).

Remark 2.
In case n = 2. Let the edges of the cylinder grid graph C m,2 are labeled as shown in Figure 10.
The corresponding labels of vertices mod(8m − 4) are as follows: Symmetry 2019, 10, x FOR PEER REVIEW 15 of 35 n as follows: Label the edges of C n as in case (1). Thus we have the labels of corresponding vertices of the circle C (1) n mod(4mn − 2n) will be: The label the vertices of C n respectively are as same in case (1) Remark 3. In case n = 4. Let the the edges of the cylinder grid graph C m,4 are labeled as shown in Figure 11. The corresponding labels of vertices mod(16m − 8) are as follows: Case (4): n ≡ 6mod12. First, we label the edges of the paths P n as follows: Label the edges of C n , C n as in case (1). Label the edges of the circle C (m−2) n as follows: Label the edges of the circle C (m) n as follows:   Figure 11. An e.e.g., l. of the cylinder grid graph Thus we have the labels of corresponding vertices mod(4mn − 2n) will be: The labels the vertices of the circle C (1) n are: The labels the vertices of the circle C The labels the vertices of C First, we label the edges of the paths P n as follows: Label the edges of the circle C Label the edges of C n as the same in case (1).
Thus we labels of corresponding vertices of the circle C n mod(4mn − 2n) will be: The labels the vertices of the circle C n as follows: Label the edges of the circle C (2) n as follows: Label the edges of C  (1). Thus we have the labels of corresponding vertices mod(4mn − 2n) will be: The labels the vertices of the circle C (1) n are as follows: The labels the vertices of the circle C (2) n are as follows: n respectively are as the same as in case (1).

Theorem 4.
If m is an odd positive integer greater than 3 and n is an odd positive integer, n ≥ 3, then the cylinder grid graph C m,n , is an edge even graceful graph.
Finally, move anticlockwise to label the edges v Second, we label the edges of the circles C n as follows: Then, label the edges of C as follows: Label the edges of the circle C (m−2) n as follows: Label the edges of the circle C n as follows: Label the edges of the circle C (m−1) n as follows: Label the edges of the circle C Label the edges of the circle C (4) n as follows: Label the edges of C (2) n as follows: Thus, the labels of corresponding vertices of the circle C n mod(4mn − 2n) will be: The labels of the vertices of C n , respectively, are as follows: Case (2): n ≡ 3mod12. First, we label the edges of the paths P n as follows: Label the edges of the circle C n as follows: Label the edges of C n as in case (1). Thus, the labels of corresponding vertices mod(4mn − 2n) will be: The labels of the vertices of C (1) n are as follows: The labels of the vertices of the circle C (2) n are as follows: The labels of the vertices of C n , respectively, are the same as in case (1).

Remark 4.
In case n = 3 and m is odd, m ≥ 3.

Conclusions
In this paper, using the connection of labeling of graphs with modular arithmetic and theory of numbers in general, we give a detailed study for e.e.g., l. of all cases of members of the cylinder grid graphs. The study of necessary and sufficient conditions for e.e.g., l. of other important families including torus m n C C × and rectangular m n P P × grid graphs should be taken into consideration in future studies of e.e.g., l.
Author Contributions: All authors contributed equally to this work.
Funding: This work was supported by the deanship of Scientific Research, Taibah University, Al-Madinah Al-Munawwarah, Saudi Arabia.