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Symmetry 2019, 11(4), 584; https://doi.org/10.3390/sym11040584

Article
Edge Even Graceful Labeling of Cylinder Grid Graph
1
Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah 41411, Saudi Arabia
2
Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt
3
Department of Mathematics and Compuer Science, Faculty of Science, Menoufia University, Shebin El Kom 32511, Egypt
*
Author to whom correspondence should be addressed.
Received: 26 March 2019 / Accepted: 14 April 2019 / Published: 22 April 2019

Abstract

:
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.
Keywords:
graceful labeling; edge even graceful labeling; cylinder grid graph

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 began nearly 50 years ago. Over these decades, more than 200 methods of labeling techniques were invented, and more than 2500 papers were published. In spite of this huge literature, just few general results were discovered. Nowadays, graph labeling has much attention from different brilliant researchers in graph theory, which has rigorous applications in many disciplines, e.g., communication networks, coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing, database management, and graph decomposition problems. More interesting applications of graph labeling can be found in References [1,2,3,4,5,6,7,8,9,10,11]. 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 was first introduced by Rosa in 1967 [12] as a β valuation, and later, Solomon W. Golomb [13] termed it 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 introduced by Gnanajothi in 1991 [14]. For more results on this type of labeling, see References [15,16]. 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 was first introduced by Lo in 1985 [17]. For more results on this labeling see [18,19]. 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 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 , , 2 q 2 } 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 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].

2. 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 n ( 1 ) , C n ( 2 ) , C n ( 3 ) , , C n ( m 1 ) , C n ( m ) and we call them simply circles; n copies of P m represented by paths transverse the m circles and will be numbered clockwise as P m ( 1 ) , P m ( 2 ) , P m ( 3 ) , , P m ( n 1 ) , P m ( n ) and we call them paths (see Figure 1).
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 ) | = m n , q = | E ( C m , n ) | = 2 m n n and r = max ( p , q ) = 2 m n n and f : E ( C m , n ) { 2 , 4 , 6 , , 4 m n 2 n 2 } . Let the cylinder grid graph C m , n be as in Figure 2. □
First, we label the edges of the paths P m ( k ) , 1 k n beginning with the edges of the path P m ( 1 ) as follows: Move anticlockwise to label the edges v 1 v n + 1 , v n v 2 n , v n 1 v 2 n 1 , , v 3 v n + 3 , v 2 v n + 2 by 2 , 4 , 6 , , 2 n 2 , 2 n , then move clockwise to label the edges v n + 1 v 2 n + 1 , v n + 2 v 2 n + 2 , v n + 3 v 2 n + 3 , , v 2 n 1 v 3 n 1 , v 2 n v 3 n by 2 n + 2 , 2 n + 4 , 2 n + 6 , , 4 n 2 , 4 n , then move anticlockwise to label the edges v 2 n + 1 v 3 n + 1 , v 3 n v 4 n , v 3 n 1 v 4 n 1 , , v 2 n + 3 v 3 n + 3 , v 2 n + 2 v 3 n + 2 by 4 n + 2 , 4 n + 4 , 4 n + 6 , , 6 n 2 , 6 n and so on. Finally, move anticlockwise to label the edges v ( m 2 ) n + 1 v ( m 1 ) n + 1 , v ( m 1 ) n v m n , v ( m 1 ) n 1 v m n 1 , , v ( m 2 ) n + 3 v ( n 1 ) m + 3 , v ( m 2 ) n + 2 v ( m 1 ) n + 2 by 2 n ( m 1 ) + 2 , 2 n ( m 2 ) + 4 , 2 n ( m 2 ) + 6 , 2 n ( m 2 ) + 8 , , 2 n ( m 1 ) 2 , 2 n ( m 1 ) .
Secondly, we label the edges of the circles C n ( k ) , 1 k m beginning with the edges of the innermost circle C n ( 1 ) then the edges of outer circle C n ( m ) , then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 2 ) .
Finally, we label the edges of the circles C n ( m 1 ) , C n ( m 3 ) , , C n ( 3 ) as follows: f ( v i v i + 1 ) = 2 n ( m 1 ) + 2 i , 1 i n 1 , f ( v n v 1 ) = 2 m n ; f ( v ( m 1 ) n + i v ( m 1 ) n + i + 1 ) = 2 m n + 2 i , 1 i n 1 , f ( v m n v ( m 1 ) n + 1 ) = 2 n ( m + 1 ) ; f ( v ( k 1 ) n + i v ( k 1 ) n + i + 1 ) = n ( 3 m k ) + 2 i , 1 i n 1 , f ( v k n v ( k 1 ) n + 1 ) = n ( 3 m k + 2 ) , 2 k m 2 ; f ( v ( k 1 ) n + i v ( k 1 ) n + i + 1 ) = n ( 4 m k 1 ) + 2 i , 1 i n 1 , f ( v k n v ( k 1 ) n + 1 ) = n ( 4 m k + 1 ) , 3 k m 1 , k is odd.
Thus, the labels of corresponding vertices mod ( 4 m n 2 n ) will be: f ( v i ) 2 i + 2 ; f ( v n + i ) 2 m n + 2 n + 4 i + 2 ; f ( v 2 n + i ) 4 n + 4 i + 2 ; f ( v 3 n + i ) 2 m n + 6 n + 4 i + 2 ; ; f ( v ( m 3 ) n + i ) 4 m n 6 n + 4 i + 2 ; f ( v ( m 2 ) n + i ) 2 m n 4 n + 4 i + 2 ; f ( v ( m 1 ) n + i ) 2 m n + 2 i + 2 , 1 i n .
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.
Proof. 
Using standard notation p = | V ( C 3 , n ) | = 3 n , q = | E ( C 3 , n ) | = 5 n , r = max ( p , q ) = 5 n , and f : E ( C 3 , n ) { 2 , 4 , 6 , , 10 n 2 } . There are three cases:
Case (1): If n 1 mod 6 , let the cylinder grid graph C 3 , n be as in Figure 4.
First, we label the edges of the paths P 3 ( k ) , 1 k n beginning with the edges of the path P 3 ( 1 ) as follows: Move clockwise to label the edges f ( v 1 v n + 1 ) = 2 , f ( v 2 v n + 2 ) = 6 , f ( v i v n + i ) = 2 i + 2 , 3 i n . Then, move anticlockwise to label the edges f ( v n + 1 v 2 n + 1 ) = 2 n + 4 , f ( v 2 n v 3 n ) = 2 n + 6 , f ( v 2 n 1 v 3 n 1 ) = 2 n + 8 , f ( v 2 n 2 v 3 n 2 ) = 2 n + 10 , , f ( v n + 3 v 2 n + 3 ) = 4 n , f ( v n + 2 v 2 n + 2 ) = 4 n + 2 .
Secondly, we label the edges of the circles C n ( k ) , 1 k 3 beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( 3 ) , and then the edges of the circle C n ( 2 ) . Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 4 n + 4 , f ( v 2 v 3 ) = 4 n + 6 , , f ( v n 1 3 v n + 2 3 ) = 14 n + 10 3 , f ( v n + 2 3 v n + 5 3 ) = 14 n + 4 3 , f ( v n + 5 3 v n + 8 3 ) = 14 n + 16 3 , f ( v n + 8 3 v n + 11 3 ) = 14 n + 22 3 , f ( v n + 11 3 v n + 14 3 ) = 14 n + 34 3 , f ( v n + 14 3 v n + 17 3 ) = 14 n + 28 3 , f ( v n + 17 3 v n + 20 3 ) = 14 n + 40 3 , f ( v n + 20 3 v n + 23 3 ) = 14 n + 46 3 , f ( v n + 23 3 v n + 26 3 ) = 14 n + 58 3 , f ( v n + 26 3 v n + 29 3 ) = 14 n + 52 3 , f ( v n + 29 3 v n + 32 3 ) = 14 n + 64 3 , f ( v n + 32 3 v n + 35 3 ) = 14 n + 70 3 , f ( v n + 35 3 v n + 38 3 ) = 14 n + 82 3 , f ( v n + 38 3 v n + 41 3 ) = 14 n + 76 3 , f ( v n + 41 3 v n + 44 3 ) = 14 n + 88 3 , f ( v n + 44 3 v n + 47 3 ) = 14 n + 94 3 , , f ( v n 13 v n 12 ) = 6 n 22 , f ( v n 12 v n 11 ) = 6 n 24 , f ( v n 11 v n 10 ) = 6 n 20 , f ( v n 10 v n 9 ) = 6 n 18 , f ( v n 9 v n 8 ) = 6 n 14 , f ( v n 8 v n 7 ) = 6 n 16 , f ( v n 7 v n 6 ) = 6 n 12 , f ( v n 6 v n 5 ) = 6 n 10 , f ( v n 5 v n 4 ) = 6 n 6 , f ( v n 4 v n 3 ) = 6 n 8 , f ( v n 3 v n 2 ) = 6 n 4 , f ( v n 2 v n 1 ) = 6 n 2 , f ( v n 1 v n ) = 6 n + 2 , f ( v n v 1 ) = 6 n .  □
Label the edges of the circle C n ( 2 ) as follows: f ( v n + i v n + i + 1 ) = 8 n + 2 i + 2 , 1 i n 1 , f ( v 2 n v n + 1 ) = 4 .
Label the edges of the circle C n ( 3 ) as follows: f ( v 2 n + i v 2 n + i + 1 ) = 6 n + 2 i + 2 , 1 i n .
The labels of corresponding of vertices mod 10 n are as follows:
The labels of vertices of the circle C n ( 1 ) are as follows: f ( v 1 ) 6 , f ( v 2 ) 8 n + 16 , f ( v 3 ) 8 n + 22 , , f ( v n 1 3 ) 4 , f ( v n + 2 3 ) 8 , f ( v n + 5 3 ) 12 , f ( v n + 8 3 ) 20 , f ( v n + 11 3 ) 28 , f ( v n + 14 3 ) 32 , f ( v n + 17 3 ) 36 , f ( v n + 20 3 ) 44 , f ( v n + 23 3 ) 52 , f ( v n + 26 3 ) 56 , f ( v n + 29 3 ) 60 , f ( v n + 32 3 ) 68 , f ( v n + 35 3 ) 76 , f ( v n + 38 3 ) 80 , f ( v n + 41 3 ) 84 , f ( v n + 44 3 ) 92 , f ( v n + 47 3 ) 100 , , f ( v n 12 ) 4 n 68 , f ( v n 11 ) 4 n 64 , f ( v n 10 ) 4 n 56 , f ( v n 9 ) 4 n 48 , f ( v n 8 ) 4 n 44 , f ( v n 7 ) 4 n 40 , f ( v n 6 ) 4 n 32 , f ( v n 5 ) 4 n 24 , f ( v n 4 ) 4 n 20 , f ( v n 3 ) 4 n 16 , f ( v n 2 ) 4 n 8 , f ( v n 1 ) 4 n , f ( v n ) 4 n + 4 .
The labels of vertices of the circle C n ( 2 ) are f ( v i + 1 ) = 4 i + 10 , 1 i n 1 , f ( v 2 n ) = 4 n + 12 .
The labels of vertices of the circle C n ( 3 ) are f ( v 2 i + 1 ) = 6 n + 2 i + 8 , 1 i n .
Case (2): If n 3 mod 6 , let the cylinder grid graph C 3 , n be as in Figure 5.
First, we label the edges of the paths P 3 ( k ) , 1 k n beginning with the edges of the path P 3 ( 1 ) as the same in case (1).
Secondly, we label the edges of the circles C n ( k ) , 1 k 3 beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( 3 ) , and then the edges of the circle C n ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 4 n + 4 , f ( v 2 v 3 ) = 4 n + 6 , , f ( v n + 3 3 v n + 6 3 ) = 14 n + 18 3 , f ( v n + 6 3 v n + 9 3 ) = 14 n + 12 3 , f ( v n + 9 3 v n + 12 3 ) = 14 n + 24 3 , f ( v n + 12 3 v n + 15 3 ) = 14 n + 30 3 , f ( v n + 15 3 v n + 18 3 ) = 14 n + 42 3 , f ( v n + 18 3 v n + 21 3 ) = 14 n + 36 3 , f ( v n + 21 3 v n + 24 3 ) = 14 n + 48 3 , f ( v n + 24 3 v n + 27 3 ) = 14 n + 54 3 , f ( v n + 27 3 v n + 30 3 ) = 14 n + 66 3 , f ( v n + 30 3 v n + 33 3 ) = 14 n + 60 3 , f ( v n + 33 3 v n + 36 3 ) = 14 n + 72 3 , f ( v n + 36 3 v n + 39 3 ) = 14 n + 78 3 , f ( v n + 39 3 v n + 42 3 ) = 14 n + 90 3 , f ( v n + 42 3 v n + 45 3 ) = 14 n + 84 3 , f ( v n + 45 3 v n + 48 3 ) = 14 n + 96 3 , f ( v n + 48 3 v n + 51 3 ) = 14 n + 102 3 , , f ( v n 13 v n 12 ) = 6 n 22 , f ( v n 12 v n 11 ) = 6 n 24 , f ( v n 11 v n 10 ) f ( v n 9 v n 8 ) = 6 n 14 , f ( v n 8 v n 7 ) = 6 n 16 , f ( v n 7 v n 6 ) = 6 n 12 , f ( v n 6 v n 5 ) = 6 n 10 , f ( v n 5 v n 4 ) = 6 n 6 , f ( v n 4 v n 3 ) = 6 n 8 , f ( v n 3 v n 2 ) = 6 n 4 , f ( v n 2 v n 1 ) = 6 n 2 , f ( v n 1 v n ) = 6 n + 2 , f ( v n v 1 ) = 6 n .
The labels of corresponding vertices mod 10 n are as follows: The label of vertices of the circle C n ( 1 ) are f ( v 1 ) 6 , f ( v 2 ) 8 n + 16 , f ( v 3 ) 8 n + 22 , , f ( v n 3 1 ) 10 n 2 , f ( v n 3 ) 4 , f ( v n 3 + 1 ) 12 , f ( v n 3 + 2 ) 16 , f ( v n 3 + 3 ) 20 , f ( v n 3 + 4 ) 28 , f ( v n 3 + 5 ) 36 , f ( v n 3 + 6 ) 40 , f ( v n 3 + 7 ) 44 , f ( v n 3 + 8 ) 52 , f ( v n 3 + 9 ) 60 , f ( v n 3 + 10 ) 64 , f ( v n 3 + 11 ) 68 , f ( v n 3 + 12 ) 76 , f ( v n 3 + 13 ) 84 , f ( v n 3 + 14 ) 88 , f ( v n 3 + 15 ) 92 , f ( v n 3 + 16 ) 100 , , f ( v n 12 ) 4 n 68 , f ( v n 11 ) 4 n 64 , f ( v n 10 ) 4 n 56 , f ( v n 9 ) 4 n 48 , f ( v n 8 ) 4 n 44 , f ( v n 7 ) 4 n 40 , f ( v n 6 ) 4 n 32 , f ( v n 5 ) 4 n 24 , f ( v n 4 ) 4 n 20 , f ( v n 3 ) 4 n 16 , f ( v n 2 ) 4 n 8 , f ( v n 1 ) 4 n , f ( v n ) 4 n + 4 .
The labels of vertices of the circles C n ( 2 ) and C n ( 3 ) are the same as in case (1).
Case (3): If n 5 mod 6 , let the cylinder grid graph C 3 , n be as in Figure 6.
First, we label the edges of the paths P 3 ( k ) , 1 k 2 beginning with the edges of the path P 3 ( 1 ) as the same in case (1). Second, we label the edges of the circles C n ( k ) , 1 k 3 beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( 3 ) , and then the edges of the circle C n ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 4 n + 4 , f ( v 2 v 3 ) = 4 n + 6 , , f ( v n 5 3 v n 2 3 ) = 14 n + 2 3 , f ( v n 2 3 v n + 1 3 ) = 14 n 4 3 , f ( v n + 1 3 v n + 4 3 ) = 14 n + 8 3 , f ( v n + 4 3 v n + 7 3 ) = 14 n + 14 3 , f ( v n + 7 3 v n + 10 3 ) = 14 n + 26 3 , f ( v n + 10 3 v n + 13 3 ) = 14 n + 20 3 , f ( v n + 13 3 v n + 16 3 ) = 14 n + 32 3 , f ( v n + 16 3 v n + 19 3 ) = 14 n + 38 3 , f ( v n + 19 3 v n + 22 3 ) = 14 n + 50 3 , f ( v n + 22 3 v n + 25 3 ) = 14 n + 44 3 , f ( v n + 25 3 v n + 28 3 ) = 14 n + 56 3 , f ( v n + 28 3 v n + 31 3 ) = 14 n + 62 3 , f ( v n + 31 3 v n + 34 3 ) = 14 n + 74 3 , f ( v n + 34 3 v n + 37 3 ) = 14 n + 68 3 , f ( v n + 37 3 v n + 40 3 ) = 14 n + 80 3 , f ( v n + 40 3 v n + 43 3 ) = 14 n + 86 3 , , f ( v n 13 v n 12 ) = 6 n 22 , f ( v n 12 v n 11 ) = 6 n 24 , f ( v n 11 v n 10 ) = 6 n 20 , f ( v n 10 v n 9 ) = 6 n 18 , f ( v n 9 v n 8 ) = 6 n 14 , f ( v n 8 v n 7 ) = 6 n 16 , f ( v n 7 v n 6 ) = 6 n 12 , f ( v n 6 v n 5 ) = 6 n 10 , f ( v n 5 v n 4 ) = 6 n 6 , f ( v n 4 v n 3 ) = 6 n 8 , f ( v n 3 v n 2 ) = 6 n 4 , f ( v n 2 v n 1 ) = 6 n 2 , f ( v n 1 v n ) = 6 n + 2 , f ( v n v 1 ) = 6 n .
The labels of corresponding vertices mod 10 n are as follows: The labels of vertices of the circle C n ( 1 ) : f ( v 1 ) 6 , f ( v 2 ) 8 n + 16 , f ( v 3 ) 8 n + 22 , , f ( v n 3 5 ) 10 n 4 , f ( v n 2 3 ) 0 , f ( v n + 1 3 ) 4 , f ( v n + 4 3 ) 12 , f ( v n + 7 3 ) 20 , f ( v n + 10 3 ) 24 , f ( v n + 13 3 ) 28 , f ( v n + 16 3 ) 36 , f ( v n 3 + 7 ) 44 , f ( v n + 19 3 ) 44 , f ( v n + 22 3 ) 48 , f ( v n + 25 3 ) 52 , f ( v n + 28 3 ) 60 , f ( v n + 31 3 ) 68 , f ( v n + 34 3 ) 72 , f ( v n + 37 3 ) 76 , f ( v n + 40 3 ) 84 , f ( v n + 43 3 ) 92 , f ( v n + 46 3 ) 96 , f ( v n + 49 3 ) 100 , , f ( v n 12 ) 4 n 68 , f ( v n 11 ) 4 n 64 , f ( v n 10 ) 4 n 56 , f ( v n 9 ) 4 n 48 , f ( v n 8 ) 4 n 44 , f ( v n 7 ) 4 n 40 , f ( v n 6 ) 4 n 32 , f ( v n 5 ) 4 n 24 , f ( v n 4 ) 4 n 20 , f ( v n 3 ) 4 n 16 , f ( v n 2 ) 4 n 8 , f ( v n 1 ) 4 n , f ( v n ) 4 n + 4 .
The labels of vertices of the circles C n ( 2 ) and C n ( 3 ) 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 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 ) | = m n , q = | E ( C m , n ) | = 2 m n n and r = max ( p , q ) = 2 m n n and f : E ( C m , n ) { 2 , 4 , 6 , , 4 m n 2 n 2 } .  □
Let the cylinder grid graph C m , n be as in Figure 9. There are six cases:
Case (1): n 0 mod 12 . First, we label the edges of the paths P m ( k ) , 1 k n beginning with the edges of the path P m ( 1 ) as follows: Move clockwise to label the edges v 1 v n + 1 , v 2 v n + 2 , v 3 v n + 3 , , v n 1 v 2 n 1 , v n v 2 n by 2 , 4 , 6 , , 2 n 2 , 2 n , then move anticlockwise to label the edges v n + 1 v n + 2 , v 2 n v 3 n , v 2 n 1 v 3 n 1 , , v n + 3 v 2 n + 3 , v n + 2 v 2 n + 2 by 2 n + 2 , 2 n + 4 , 2 n + 6 , , 4 n 2 , 4 n , then move clockwise to label the edges v 2 n + 1 v 3 n + 1 , v 2 n + 2 v 3 n + 2 , v 2 n + 3 v 3 n + 3 , , v 3 n 1 v 4 n 1 , v 3 n v 4 n by 4 n + 2 , 4 n + 4 , 4 n + 6 , , 6 n 2 , 6 n and so on.
Finally, move anticlockwise to label the edges v ( m 2 ) n + 1 v ( m 1 ) n + 1 , v ( m 1 ) n v m n , v ( m 1 ) n 1 v m n 1 , , v ( m 2 ) n + 3 v ( m 1 ) n + 3 , v ( m 2 ) n + 2 v m ( n 1 ) + 2 by 2 n ( m 2 ) + 2 , 2 n ( m 2 ) + 4 , 2 n ( m 2 ) + 6 , , 2 n ( m 1 ) 2 , 2 n ( m 1 ) .
Secondly, we label the edges of the circles C n ( k ) , 1 k m beginning with the edges of the inner most circle C n ( 1 ) , then the edges of outer circle C n ( m ) , then the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) . Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 6 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 4 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 8 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 14 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 12 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 16 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 22 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 20 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 24 , , f ( v n 7 v n 6 ) = 2 m n 14 , f ( v n 6 v n 5 ) = 2 m n 10 , f ( v n 5 v n 4 ) = 2 m n 12 , f ( v n 4 v n 3 ) = 2 m n 8 , f ( v n 3 v n 2 ) = 2 m n 2 , f ( v n 2 v n 1 ) = 2 m n 6 , f ( v n 1 v n ) = 2 m n , f ( v n v 1 ) = 2 m n 4 .
Label the edges of the circle C n ( m ) as follows: f ( v ( m 1 ) n + i v ( m 1 ) n + i + 1 ) = 2 m n + 2 i , f ( v m n v ( m 1 ) n + 1 ) = 2 n ( m + 1 ) , 1 i n 1 .
Label the edges of the circle C n ( m 2 ) as follows: f ( v ( m 3 ) n + i v ( m 3 ) n + i + 1 ) = 2 n ( m + 1 ) + 2 i , f ( v ( m 2 ) n v ( m 3 ) n + 1 ) = 2 n ( m + 2 ) , 1 i n 1 .
Label the edges of the circle C n ( m 4 ) as follows: f ( v ( m 5 ) n + i v ( m 5 ) n + i + 1 ) = 2 n ( m + 2 ) + 2 i , f ( v ( m 4 ) n v ( m 5 ) n + 1 ) = 2 n ( m + 3 ) , 1 i n 1 , and so on.
Label the edges of the circle C n ( 3 ) as follows: f ( v 2 n + i v 2 n + i + 1 ) = 3 n ( m 1 ) + 2 i , f ( v 3 n v 2 n + 1 ) = n ( 3 m 1 ) , 1 i n 1 ,
Label the edges of the circle C n ( m 1 ) as follows: f ( v ( m 2 ) n + i v ( m 2 ) n + i + 1 ) = n ( 3 m 1 ) + 2 i , f ( v ( m 1 ) n v ( m 1 ) n + 1 ) = n ( 3 m + 1 ) 1 , 1 i n 1 ,
Label the edges of the circle C n ( m 3 ) as follows: f ( v ( m 4 ) n + i v ( m 4 ) n + i + 1 ) = n ( 3 m + 2 ) + 2 i , f ( v ( m 3 ) n v ( m 4 ) n + 1 ) = 3 n ( m + 1 ) , 1 i n 1 , , and so on.
Label the edges of the circle C n ( 4 ) as follows: f ( v 3 n + i v 3 n + i + 1 ) = 2 n ( 2 m 3 ) + 2 i , f ( v 4 n v 3 n + 1 ) = 4 n ( m 1 ) , 1 i n 1 ,
Label the edges of C n ( 2 ) as follows: f ( v n + i v n + i + 1 ) = 4 n ( m 1 ) + 2 i , f ( v 2 n v 2 n + 1 ) = 2 n ( m 1 ) , 1 i n 1 ,
Thus, the labels of corresponding vertices mod ( 4 m n 2 n ) will be:
The label the vertices of C n ( 1 ) are: f ( v 1 ) 0 ; f ( v 2 ) 4 m n 4 n + 12 ; f ( v 3 ) 4 m n 4 n + 16 ; f ( v 4 ) 4 m n 4 n + 20 ; f ( v 5 ) 4 m n 4 n + 28 ; f ( v 6 ) 4 m n 4 n + 36 ; f ( v 7 ) 4 m n 4 n + 40 ; f ( v 8 ) 4 m n 4 n + 44 ; f ( v 9 ) 4 m n 4 n + 52 ; f ( v 10 ) 4 m n 4 n + 60 ; f ( v 11 ) 4 m n 4 n + 64 ; f ( v 12 ) 4 m n 4 n + 68 ; ; f ( v n 6 ) 4 n 36 ; f ( v n 5 ) 4 n 32 ; f ( v n 4 ) 4 n 28 ; f ( v n 3 ) 4 n 16 ; f ( v n 2 ) 4 n 12 ; f ( v n 1 ) 4 n 8 ; f ( v n ) 4 n 4 .
The label the vertices of C n ( 2 ) , C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) respectively are: f ( v n + i ) 4 i + 2 ; f ( v 2 n + i ) 2 m n + 4 n + 4 i + 2 ; f ( v 3 n + i ) 4 n + 4 i + 2 ; ; f ( v ( m 3 ) n + i ) 4 m n 6 n + 4 i + 2 ; f ( v ( m 2 ) n + i ) 2 m n 6 n + 4 i + 2 ; f ( v ( m 1 ) n + i ) 2 m n + 2 i + 2 , 1 i n .
Case (2): n 2 mod 12 , n 2 .
First, we label the edges of the paths P m ( k ) , 1 k n begin with the edges of the path P m ( 1 ) as the same in case (1).
Secondly, we label the edges of the circles C n ( k ) , 1 k m begin with the edges of the inner most circle C n ( 1 ) , then the edges of outer circle C n ( m ) , then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 6 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 4 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 8 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 14 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 12 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 16 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 22 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 20 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 24 , , f ( v n 9 v n 8 ) = 2 m n 18 , f ( v n 8 v n 7 ) = 2 m n 14 , f ( v n 7 v n 6 ) = 2 m n 16 , f ( v n 6 v n 5 ) = 2 m n 12 , f ( v n 5 v n 4 ) = 2 m n 10 , f ( v n 4 v n 3 ) = 2 m n 6 , f ( v n 3 v n 2 ) = 2 m n 8 , f ( v n 2 v n 1 ) = 2 m n 4 , f ( v n 1 v n ) = 2 m n 2 , f ( v n v 1 ) = 2 m n .
Label the edges of the circle C n ( 2 ) as follows: f ( v n + 1 v n + 2 ) = 4 n ( m 1 ) + 4 , f ( v n + 2 v n + 3 ) = 4 n ( m 1 ) + 2 , f ( v n + 3 v n + 4 ) = 4 n ( m 1 ) + 8 , f ( v n + 4 v n + 5 ) = 4 n ( m 1 ) + 6 , f ( v n + i v n + i + 1 ) = 4 n ( m 1 ) + 2 i , 6 i n 2 , f ( v 2 n 1 v 2 n ) = 2 n ( 2 m 1 ) , f ( v 2 n v n + 1 ) = 2 n ( 2 m 1 ) 2 . Label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) as in case (1).
Thus, the labels of corresponding vertices mod ( 4 m n 2 n ) will be:
The label the vertices of C n ( 1 ) are: f ( v 1 ) 4 , f ( v 2 ) 4 m n 4 n + 12 , f ( v 3 ) 4 m n 4 n + 16 , f ( v 4 ) 4 m n 4 n + 20 , f ( v 5 ) 4 m n 4 n + 28 , f ( v 6 ) 4 m n 4 n + 36 , f ( v 7 ) 4 m n 4 n + 40 , f ( v 8 ) 4 m n 4 n + 44 , f ( v 9 ) 4 m n 4 n + 52 , f ( v 10 ) 4 m n 4 n + 60 , f ( v 11 ) 4 m n 4 n + 64 , f ( v 12 ) 4 m n 4 n + 68 , f ( v 13 ) 4 m n 4 n + 76 , , f ( v n 8 ) 4 n 48 , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 40 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 24 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 16 , f ( v n 1 ) 4 n 8 , f ( v n ) 4 n 2 .
The label the vertices of the circle C n ( 2 ) are: f ( v n + 1 ) 6 , f ( v n + 2 ) 10 , f ( v n + 3 ) 14 , f ( v n + 4 ) 18 , f ( v n + 5 ) 20 , f ( v n + i ) 4 i + 2 , 6 i n 2 , f ( v 2 n 1 ) 4 n , f ( v 2 n ) 4 n + 2 .
The label the vertices of C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) 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 ( 8 m 4 ) are as follows: f ( v 1 ) 8 , f ( v 2 i + 1 ) 4 m + 8 i + 4 , 1 i m 3 2 , f ( v 2 i ) 8 i + 6 , 1 i m 1 2 ; f ( v 1 ) 12 , f ( v 2 ) 20 , f ( v 2 i + 1 ) 4 m + 8 i + 18 , 1 i m 3 2 , f ( v 2 i ) 8 i + 10 , 2 i m 1 2 .
Case (3): n 4 mod 12 .
First we label the edges of the paths P m ( k ) , 1 k n begin with the edges of the path P m ( 1 ) as the same in case (1).
Second we label the edges of the circles C n ( k ) , 1 k m begin with the edges of the inner most circle C n ( 1 ) , then the edges of outer circle C n ( m ) , then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 6 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 4 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 8 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 14 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 12 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 16 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 22 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 20 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 24 , , f ( v n 8 v n 7 ) = 2 m n 16 , f ( v n 7 v n 6 ) = 2 m n 14 , f ( v n 6 v n 5 ) = 2 m n 10 , f ( v n 5 v n 4 ) = 2 m n 12 , f ( v n 4 v n 3 ) = 2 m n 8 , f ( v n 3 v n 2 ) = 2 m n 2 , f ( v n 2 v n 1 ) = 2 m n 6 , f ( v n 1 v n ) = 2 m n , and f ( v n v 1 ) = 2 m n 4 .
Label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) , C n ( 2 ) as in case (1).
Thus we have the labels of corresponding vertices of the circle C n ( 1 ) mod ( 4 m n 2 n ) will be: f ( v 1 ) 0 , f ( v 2 ) 4 m n 4 n + 12 , f ( v 3 ) 4 m n 4 n + 16 , f ( v 4 ) 4 m n 4 n + 20 , f ( v 5 ) 4 m n 4 n + 28 , f ( v 6 ) 4 m n 4 n + 36 , f ( v 7 ) 4 m n 4 n + 40 , f ( v 8 ) 4 m n 4 n + 44 , f ( v 9 ) 4 m n 4 n + 52 , f ( v 10 ) 4 m n 4 n + 60 , f ( v 11 ) 4 m n 4 n + 64 , f ( v 12 ) 4 m n 4 n + 68 , f ( v 13 ) 4 m n 4 n + 76 , , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 36 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 28 , f ( v n 3 ) 4 n 16 , f ( v n 2 ) 4 n 12 , f ( v n 1 ) 4 n 8 , f ( v n ) 4 n 4 .
The label the vertices of C n ( 2 ) , C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) 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 ( 16 m 8 ) are as follows: f ( v 1 ) 6 , f ( v 2 ) 8 , f ( v 3 ) 16 , f ( v 4 ) 20 ; f ( v 4 i + 1 ) 4 i + 10 , 1 i 3 , f ( v 8 ) 28 ; f ( v 8 k + i ) 8 m + 4 i + 16 k 10 , 1 i 4 , 1 k m 5 2 ; f ( v 4 m 11 ) 0 , f ( v 4 m 10 ) 2 , f ( v 4 m 9 ) 4 , f ( v 4 m 8 ) 10 , f ( v 8 k + 4 + i ) 4 i + 16 k + 10 , 1 i 4 , 1 k m 3 2 .
Case (4): n 6 mod 12 .
First, we label the edges of the paths P m ( k ) , 1 k n begin with the edges of the path P m ( 1 ) as the same in case (1).
Secondly, we label the edges of the circles C n ( k ) , 1 k m begin with the edges of the inner most circle C n ( 1 ) , then the edges of outer circle C n ( m ) , then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 6 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 4 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 8 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 14 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 12 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 16 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 22 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 20 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 24 , , f ( v n 9 v n 8 ) = 2 m n 18 , f ( v n 8 v n 7 ) = 2 m n 14 , f ( v n 7 v n 6 ) = 2 m n 16 , f ( v n 6 v n 5 ) = 2 m n 12 , f ( v n 5 v n 4 ) = 2 m n 10 , f ( v n 4 v n 3 ) = 2 m n 6 , f ( v n 3 v n 2 ) = 2 m n 8 , f ( v n 2 v n 1 ) = 2 m n 4 , f ( v n 1 v n ) = 2 m n + 2 , f ( v n v 1 ) = 2 m n 2 .
Label the edges of C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) , C n ( 2 ) as in case (1).
Label the edges of the circle C n ( m 2 ) as follows: f ( v ( m 3 ) n + 1 v ( m 3 ) n + 2 ) = 2 n ( m + 2 ) , f ( v ( m 3 ) n + i v ( m 3 ) n + i + 1 ) = 2 n ( m + 1 ) + 2 i , 2 i n 1 , f ( v ( m 2 ) n v ( m 3 ) n + 1 ) = 2 n ( m + 2 ) + 2 .
Label the edges of the circle C n ( m ) as follows: f ( v ( m 1 ) n + 1 v ( m 1 ) n + 2 ) = 2 m n , f ( v ( m 1 ) n + i v ( m 1 ) n + i + 1 ) = 2 m n + 2 i , 2 i n 1 , f ( v ( m 2 ) n v ( m 3 ) n + 1 ) = 2 n ( m + 2 ) .
Thus we have the labels of corresponding vertices mod ( 4 m n 2 n ) will be:
The labels the vertices of the circle C n ( 1 ) are: f ( v 1 ) 2 , f ( v 2 ) 4 m n 4 n + 12 , f ( v 3 ) 4 m n 4 n + 16 , f ( v 4 ) 4 m n 4 n + 20 , f ( v 5 ) 4 m n 4 n + 28 , f ( v 6 ) 4 m n 4 n + 36 , f ( v 7 ) 4 m n 4 n + 40 , f ( v 8 ) 4 m n 4 n + 44 , f ( v 9 ) 4 m n 4 n + 52 , f ( v 10 ) 4 m n 4 n + 60 , f ( v 11 ) 4 m n 4 n + 64 , f ( v 12 ) 4 m n 4 n + 68 , , f ( v n 8 ) 4 n 48 , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 40 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 24 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 16 , f ( v n 1 ) 4 n 4 ; f ( v n ) 4 n .
The labels the vertices of the circle C n ( m 2 ) are: f ( v ( m 3 ) n + 1 ) 4 m n 6 n + 6 ; f ( v ( m 3 ) n + 2 ) 4 m n 6 n + 8 ; f ( v ( m 3 ) n + i ) 4 m n 6 n + 4 i + 2 , 3 i n 1 , f ( v ( m 2 ) n ) 4 m n 4 n + 4 .
The labels the vertices of C n ( 2 ) , C n ( 3 ) , C n ( 4 ) , , C n ( m 1 ) , C n ( m 3 ) respectively are the same as in case (1).
The labels the vertices of C n ( m ) are: f ( v ( m 1 ) n + 1 ) 2 m n + 2 , f ( v ( m 1 ) n + 2 ) 2 m n + 4 , f ( v ( m 1 ) n + i ) 2 m n + 2 i + 2 , 3 i n . Case (5): n 8 mod 12 .
First, we label the edges of the paths P m ( k ) , 1 k n begin with the edges of the path P m ( 1 ) as the same in case (1).
Secondly, we label the edges of the circles C n ( k ) , 1 k m begin with the edges of the inner most circle C n ( 1 ) , then the edges of outer circle C n ( m ) , then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 )
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 6 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 4 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 8 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 14 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 12 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 16 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 22 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 20 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 24 , f ( v 13 v 14 ) = 2 n ( m 1 ) + 26 , , f ( v n 9 v n 8 ) = 2 m n 20 , f ( v n 8 v n 7 ) = 2 m n 16 , f ( v n 7 v n 6 ) = 2 m n 14 , f ( v n 6 v n 5 ) = 2 m n 10 , f ( v n 5 v n 4 ) = 2 m n 12 , f ( v n 4 v n 3 ) = 2 m n 8 , f ( v n 3 v n 2 ) = 2 m n 6 , f ( v n 2 v n 1 ) = 2 m n 2 , f ( v n 1 v n ) = 2 m n 4 , f ( v n v 1 ) = 2 m n + 4 .
Label the edges of the circle C n ( m ) as follows f ( v ( m 1 ) n + 1 v ( m 1 ) n + 2 ) = 2 m n + 2 , f ( v ( m 1 ) n + 2 v ( m 1 ) n + 3 ) = 2 m n + 6 , f ( v ( m 1 ) n + i v ( m 1 ) n + i + 1 ) = 2 m n + 2 i + 2 , 3 i n 1 , f ( v m n v ( m 1 ) n + 1 ) = 2 m n .
Label the edges of C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) , C n ( 2 ) as the same in case (1).
Thus we labels of corresponding vertices of the circle C n ( 1 ) mod ( 4 m n 2 n ) will be: f ( v 1 ) 8 , f ( v 2 ) 4 m n 4 n + 12 , f ( v 3 ) 4 m n 4 n + 16 , f ( v 4 ) 4 m n 4 n + 20 , f ( v 5 ) 4 m n 4 n + 28 , f ( v 6 ) 4 m n 4 n + 36 , f ( v 7 ) 4 m n 4 n + 40 , f ( v 8 ) 4 m n 4 n + 44 , f ( v 9 ) 4 m n 4 n + 52 , f ( v 10 ) 4 m n 4 n + 60 , f ( v 11 ) 4 m n 4 n + 64 , f ( v 12 ) 4 m n 4 n + 68 , f ( v 13 ) 4 m n 4 n + 76 , , f ( v n 8 ) 4 n 52 , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 36 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 28 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 12 , f ( v n 1 ) 4 n 8 , f ( v n ) 4 n .
The labels the vertices of the circle C n ( m ) are: f ( v ( m 1 ) n + 1 ) 2 m n 4 n + 14 , f ( v ( m 1 ) n + 2 ) 2 m n + 8 , f ( v ( m 1 ) n + i ) 2 m n + 2 i + 6 , 3 i n 1 , f ( v m n ) 2 m n + 4 . The labels the vertices of C n ( 2 ) , C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) respectively are as the same in case (1).
Case (6): n 10 mod 12 . First we label the edges of the paths P m ( k ) , 1 k n begin with the edges of the path P m ( 1 ) as the same as in case (1).
Second we label the edges of the circles C n ( k ) , 1 k m begin with the edges of the inner most circle C n ( 1 ) , then the edges of outer circle C n ( m ) , then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 6 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 4 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 8 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 14 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 12 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 16 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 22 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 20 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 24 , f ( v 13 v 14 ) = 2 n ( m 1 ) + 26 , , f ( v n 9 v n 8 ) = 2 m n 18 , f ( v n 8 v n 7 ) = 2 m n 14 , f ( v n 7 v n 6 ) = 2 m n 16 , f ( v n 6 v n 5 ) = 2 m n 12 , f ( v n 5 v n 4 ) = 2 m n 10 , f ( v n 4 v n 3 ) = 2 m n 6 , f ( v n 3 v n 2 ) = 2 m n 8 , f ( v n 2 v n 1 ) = 2 m n 4 , f ( v n 1 v n ) = 2 m n 2 , f ( v n v 1 ) = 2 m n .
Label the edges of the circle C n ( 2 ) as follows: f ( v n + 1 v n + 2 ) = 4 n ( m 1 ) + 4 , f ( v n + 2 v n + 3 ) = 4 n ( m 1 ) + 2 , f ( v n + i v n + i + 1 ) = 4 n ( m 1 ) + 2 i , 3 i n 2 , f ( v 2 n 1 v 2 n ) = 2 n ( 2 m 1 ) , f ( v 2 n v 2 n + 1 ) = 2 n ( 2 m 1 ) 2 .
Label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) as in case (1).
Thus we have the labels of corresponding vertices mod ( 4 m n 2 n ) will be:
The labels the vertices of the circle C n ( 1 ) are as follows: f ( v 1 ) 4 , f ( v 2 ) 4 m n 4 n + 12 , f ( v 3 ) 4 m n 4 n + 16 , f ( v 4 ) 4 m n 4 n + 20 , f ( v 5 ) 4 m n 4 n + 28 , f ( v 6 ) 4 m n 4 n + 36 , f ( v 7 ) 4 m n 4 n + 40 , f ( v 8 ) 4 m n 4 n + 44 , f ( v 9 ) 4 m n 4 n + 52 , f ( v 10 ) 4 m n 4 n + 60 , f ( v 11 ) 4 m n 4 n + 64 , f ( v 12 ) 4 m n 4 n + 68 , f ( v 13 ) 4 m n 4 n + 76 , f ( v 14 ) 4 m n 4 n + 84 , , f ( v n 8 ) 4 n 48 , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 40 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 24 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 16 , f ( v n 1 ) 4 n 8 , f ( v n ) 4 n 2 .
The labels the vertices of the circle C n ( 2 ) are as follows: f ( v n + 1 ) 6 , f ( v n + 2 ) 10 , f ( v n + 3 ) 12 , f ( v n + i ) 4 i + 2 , 4 i n 2 , f ( v 2 n 1 ) 4 n , f ( v 2 n ) 4 n + 2 . Label the vertices of C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) respectively are as the same as in case (1).
Illustration: The edge even graceful labeling of the cylinder grid graphs C 9 , 2 , C 9 , 4 , C 7 , 10 , C 7 , 12 , C 7 , 14 C 7 , 16 C 7 , 18 and C 7 , 20 are shown in Figure 12.
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.
Proof. 
Using standard notation p = | V ( C m , n ) | = m n , q = | E ( C m , n ) | = 2 m n n , r = max ( p , q ) = 2 m n n , and f : E ( C m , n ) { 2 , 4 , 6 , , 4 m n 2 n 2 } .  □
Let the cylinder grid graph C m , n be as in Figure 9. There are six cases:
Case (1): n 1 mod 12 .
First, we label the edges of the paths P m ( k ) , 1 k n beginning with the edges of the path P m ( 1 ) as follows: Move clockwise to label the edges v 1 v n + 1 , v 2 v n + 2 , v 3 v n + 3 , , v n 1 v 2 n 1 , v n v 2 n by 2 , 4 , 6 , , 2 n 2 , 2 n , then move anticlockwise to label the edges v n + 1 v n + 2 , v 2 n v 3 n , v 2 n 1 v 3 n 1 , , v n + 3 v 2 n + 3 , v n + 2 v 2 n + 2 by 2 n + 2 , 2 n + 4 , 2 n + 6 , , 4 n 2 , 4 n , then move clockwise to label the edges v 2 n + 1 v 3 n + 1 , v 2 n + 2 v 3 n + 2 , v 2 n + 3 v 3 n + 3 , , v 3 n 1 v 4 n 1 , v 3 n v 4 n by 4 n + 2 , 4 n + 4 , 4 n + 6 , , 6 n 2 , 6 n , and so on.
Finally, move anticlockwise to label the edges v ( m 2 ) n + 1 v ( m 1 ) n + 1 , v ( m 1 ) n v m n , v ( m 1 ) n 1 v m n 1 , , v ( m 2 ) n + 3 v ( m 1 ) n + 3 , v ( m 2 ) n + 2 v m ( n 1 ) + 2 by 2 n ( m 2 ) + 2 , 2 n ( m 2 ) + 4 , 2 n ( m 2 ) + 6 , , 2 n ( m 1 ) 2 , 2 n ( m 1 ) .
Second, we label the edges of the circles C n ( k ) , 1 k m beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( m ) , and then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 4 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 8 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 6 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 12 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 16 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 14 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 20 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 24 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 22 , f ( v 13 v 14 ) = 2 n ( m 1 ) + 26 , f ( v 14 v 15 ) = 2 n ( m 1 ) + 28 , , f ( v n 7 v n 6 ) = 2 m n 14 , f ( v n 6 v n 5 ) = 2 m n 10 , f ( v n 5 v n 4 ) = 2 m n 12 , f ( v n 4 v n 3 ) = 2 m n 8 , f ( v n 3 v n 2 ) = 2 m n 6 , f ( v n 2 v n 1 ) = 2 m n 2 , f ( v n 1 v n ) = 2 m n 4 , f ( v n v 1 ) = 2 m n .
Then, label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) , C n ( 2 ) as follows:
Label the edges of the circle C n ( m ) as follows: f ( v ( m 1 ) n + i v ( m 1 ) n + i + 1 ) = 2 m n + 2 i , f ( v m n v ( m 1 ) n + 1 ) = 2 n ( m + 1 ) , 1 i n 1 .
Label the edges of the circle C n ( m 2 ) as follows: f ( v ( m 3 ) n + i v ( m 3 ) n + i + 1 ) = 2 n ( m + 1 ) + 2 i , f ( v ( m 2 ) n v ( m 3 ) n + 1 ) = 2 n ( m + 2 ) , 1 i n 1 .
Label the edges of the circle C n ( m 4 ) as follows: f ( v ( m 5 ) n + i v ( m 5 ) n + i + 1 ) = 2 n ( m + 2 ) + 2 i , f ( v ( m 4 ) n v ( m 5 ) n + 1 ) = 2 n ( m + 3 ) , 1 i n 1 , and so on.
Label the edges of the circle C n ( 3 ) as follows: f ( v 2 n + i v 2 n + i + 1 ) = 3 n ( m 1 ) + 2 i , f ( v 3 n v 2 n + 1 ) = n ( 3 m 1 ) , 1 i n 1 ,
Label the edges of the circle C n ( m 1 ) as follows: f ( v ( m 2 ) n + i v ( m 2 ) n + i + 1 ) = n ( 3 m 1 ) + 2 i , f ( v ( m 1 ) n v ( m 1 ) n + 1 ) = n ( 3 m + 1 ) , 1 i n 1 ,
Label the edges of the circle C n ( m 3 ) as follows: f ( v ( m 4 ) n + i v ( m 4 ) n + i + 1 ) = n ( 3 m + 2 ) + 2 i , f ( v ( m 3 ) n v ( m 4 ) n + 1 ) = 3 n ( m + 1 ) , 1 i n 1 , , and so on.
Label the edges of the circle C n ( 4 ) as follows: f ( v 3 n + i v 3 n + i + 1 ) = 2 n ( 2 m 3 ) + 2 i , f ( v 4 n v 3 n + 1 ) = 4 n ( m 1 ) , 1 i n 1 ,
Label the edges of C n ( 2 ) as follows: f ( v n + i v n + i + 1 ) = 4 n ( m 1 ) + 2 i , f ( v 2 n v 2 n + 1 ) = 2 n ( m 1 ) , 1 i n 1 ,
Thus, the labels of corresponding vertices of the circle C n ( 1 ) mod ( 4 m n 2 n ) will be: f ( v 1 ) 4 , f ( v 2 ) 4 m n 4 n + 10 , f ( v 3 ) 4 m n 4 n + 18 , f ( v 4 ) 4 m n 4 n + 22 , f ( v 5 ) 4 m n 4 n + 26 , f ( v 6 ) 4 m n 4 n + 34 , f ( v 7 ) 4 m n 4 n + 42 , f ( v 8 ) 4 m n 4 n + 46 , f ( v 9 ) 4 m n 4 n + 50 , f ( v 10 ) 4 m n 4 n + 58 , f ( v 11 ) 4 m n 4 n + 66 , f ( v 12 ) 4 m n 4 n + 70 , f ( v 13 ) 4 m n 4 n + 74 , f ( v 14 ) 4 m n 4 n + 82 , , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 36 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 28 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 12 , f ( v n 1 ) 4 n 8 , f ( v n ) 4 n 4 .
The labels of the vertices of C n ( 2 ) , C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) , respectively, are as follows: f ( v n + i ) 4 i + 2 ; f ( v 2 n + i ) 2 m n + 4 n + 4 i + 2 ; f ( v 3 n + i ) 4 n + 4 i + 2 ; ; f ( v ( m 3 ) n + i ) 4 m n 6 n + 4 i + 2 ; f ( v ( m 2 ) n + i ) 2 m n 6 n + 4 i + 2 ; f ( v ( m 1 ) n + i ) 2 m n + 2 i + 2 , 1 i n .
Case (2): n 3 mod 12 .
First, we label the edges of the paths P m ( k ) , 1 k n beginning with the edges of the path P m ( 1 ) as the same in case (1).
Second, we label the edges of the circles C n ( k ) , 1 k m beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( m ) , and then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 4 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 8 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 6 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 12 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 16 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 14 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 20 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 24 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 22 , f ( v 13 v 14 ) = 2 n ( m 1 ) + 26 , , f ( v n 9 v n 8 ) = 2 m n 18 , f ( v n 8 v n 7 ) = 2 m n 14 , f ( v n 7 v n 6 ) = 2 m n 16 , f ( v n 6 v n 5 ) = 2 m n 12 , f ( v n 5 v n 4 ) = 2 m n 10 , f ( v n 4 v n 3 ) = 2 m n 6 , f ( v n 3 v n 2 ) = 2 m n 8 , f ( v n 2 v n 1 ) = 2 m n , f ( v n 1 v n ) = 2 m n 2 , f ( v n v 1 ) = 2 m n 4 .
Label the edges of the circle C n ( 2 ) as follows: f ( v n + i v n + i + 1 ) = 4 n ( m 1 ) + 2 i , 1 i n 9 , f ( v 2 n 9 v 2 n 8 ) = 2 n ( 2 m 1 ) 18 , f ( v 2 n 8 v 2 n 7 ) = 2 n ( 2 m 1 ) 14 , f ( v 2 n 7 v 2 n 6 ) = 2 n ( 2 m 1 ) 16 , f ( v 2 n 6 v 2 n 5 ) = 2 n ( 2 m 1 ) 10 , f ( v 2 n 5 v 2 n 4 ) = 2 n ( 2 m 1 ) 12 , f ( v 2 n 4 v 2 n 3 ) = 2 n ( 2 m 1 ) 6 , f ( v 2 n 3 v 2 n 2 ) = 2 n ( 2 m 1 ) 8 , f ( v 2 n 2 v 2 n 1 ) = 2 n ( 2 m 1 ) 4 , f ( v 2 n 1 v 2 n ) = 2 n ( 2 m 1 ) 2 , f ( v 2 n v n + 1 ) = 2 n ( 2 m 1 ) .
Label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) as in case (1).
Thus, the labels of corresponding vertices mod ( 4 m n 2 n ) will be:
The labels of the vertices of C n ( 1 ) are as follows: f ( v 1 ) 0 , f ( v 2 ) 4 m n 4 n + 10 , f ( v 3 ) 4 m n 4 n + 18 , f ( v 4 ) 4 m n 4 n + 22 , f ( v 5 ) 4 m n 4 n + 26 , f ( v 6 ) 4 m n 4 n + 34 , f ( v 7 ) 4 m n 4 n + 42 , f ( v 8 ) 4 m n 4 n + 46 , f ( v 9 ) 4 m n 4 n + 50 , f ( v 10 ) 4 m n 4 n + 58 , f ( v 11 ) 4 m n 4 n + 66 , f ( v 12 ) 4 m n 4 n + 70 , f ( v 13 ) 4 m n 4 n + 74 , f ( v 14 ) 4 m n 4 n + 82 , , f ( v n 8 ) 4 n 48 , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 40 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 24 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 12 , f ( v n 1 ) 4 n 4 , f ( v n ) 4 n 6 .
The labels of the vertices of the circle C n ( 2 ) are as follows: f ( v n + i ) 4 i + 2 , 1 i n 9 , f ( v 2 n 8 ) 4 n 28 , f ( v 2 n 7 ) 4 n 26 , f ( v 2 n 6 ) 4 n 22 , f ( v 2 n 5 ) 4 n 18 , f ( v 2 n 4 ) 4 n 14 , f ( v 2 n 3 ) 4 n 10 , f ( v 2 n 2 ) 4 n 8 , f ( v 2 n 1 ) 4 n 2 , f ( v 2 n ) 4 n + 2 .
The labels of the vertices of C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) , respectively, are the same as in case (1).
Remark 4.
In case n = 3 and m is odd, m 3 .
Let the label of edges of the cylinder grid graph C m , 3 be as in Figure 13.
Thus, the labels of corresponding vertices mod ( 12 m 6 ) are as follows:
The labels of the vertices of the circle C 3 ( 1 ) are f ( v 1 ) 8 , f ( v 2 ) 12 , f ( v 3 ) 16 .
The labels of the vertices of the circle C 3 ( 3 ) are f ( v 3 m 2 ) 6 m + 10 , f ( v 3 m 1 ) 6 m + 12 , f ( v 3 m ) 6 m + 14 .
The labels of the vertices of the circles C 3 ( 2 ) , C 3 ( 4 ) , , C 3 ( m 1 ) are f ( v 3 k + i ) 4 i + 6 k + 4 , 1 i 3 , 1 k m 2 , k is odd.
The labels of the vertices of the circles C 3 ( 3 ) , C 3 ( 5 ) , , C 3 ( m 2 ) are f ( v 3 k + i ) 6 m + 4 i + 6 k + 10 , 1 i 3 , 2 k m 3 , k is even.
Case (3): n 5 mod 12 .
First, we label the edges of the paths P m ( k ) , 1 k n beginning with the edges of the path P m ( 1 ) the same as in case (1).
Second, we label the edges of the circles C n ( k ) , 1 k m beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( m ) , and then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 4 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 8 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 6 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 12 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 16 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 14 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 20 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 24 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 22 , f ( v 13 v 14 ) = 2 n ( m 1 ) + 26 , , f ( v n 8 v n 7 ) = 2 m n 16 , f ( v n 7 v n 6 ) = 2 m n 14 , f ( v n 6 v n 5 ) = 2 m n 10 , f ( v n 5 v n 4 ) = 2 m n 12 , f ( v n 4 v n 3 ) = 2 m n 8 , f ( v n 3 v n 2 ) = 2 m n 6 , f ( v n 2 v n 1 ) = 2 m n 2 , f ( v n 1 v n ) = 2 m n 4 , f ( v n v 1 ) = 2 m n . Label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) , C n ( 2 ) as in case (1).
Thus, the labels of corresponding vertices of the circle C n ( 1 ) mod ( 4 m n 2 n ) will be: f ( v 1 ) 4 , f ( v 2 ) 4 m n 4 n + 10 , f ( v 3 ) 4 m n 4 n + 18 , f ( v 4 ) 4 m n 4 n + 22 , f ( v 5 ) 4 m n 4 n + 26 , f ( v 6 ) 4 m n 4 n + 34 , f ( v 7 ) 4 m n 4 n + 42 , f ( v 8 ) 4 m n 4 n + 46 , f ( v 9 ) 4 m n 4 n + 50 , f ( v 10 ) 4 m n 4 n + 58 , f ( v 11 ) 4 m n 4 n + 66 , f ( v 12 ) 4 m n 4 n + 70 , f ( v 13 ) 4 m n 4 n + 74 , f ( v 14 ) 4 m n 4 n + 82 , , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 36 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 28 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 12 , f ( v n 1 ) 4 n 8 , f ( v n ) 4 n 4 .
The labels of the vertices of C n ( 2 ) , C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) , respectively, are the same as in case (1).
Case (4): n 7 mod 12 .
First, we label the edges of the paths P m ( k ) , 1 k n beginning with the edges of the path P m ( 1 ) the same as in case (1).
Second, we label the edges of the circles C n ( k ) , 1 k m beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( m ) , and then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 4 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 8 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 6 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 12 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 16 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 14 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 20 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 24 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 22 , f ( v 13 v 14 ) = 2 n ( m 1 ) + 26 , , f ( v n 10 v n 9 ) = 2 m n 20 , f ( v n 9 v n 8 ) = 2 m n 18 , f ( v n 8 v n 7 ) = 2 m n 14 , f ( v n 7 v n 6 ) = 2 m n 16 , f ( v n 6 v n 5 ) = 2 m n 12 , f ( v n 5 v n 4 ) = 2 m n 10 , f ( v n 4 v n 3 ) = 2 m n 6 , f ( v n 3 v n 2 ) = 2 m n 8 , f ( v n 2 v n 1 ) = 2 m n , f ( v n 1 v n ) = 2 m n 2 , f ( v n v 1 ) = 2 m n 4 .
Label the edges of the circle C n ( 2 ) as follows: f ( v n + i v n + i + 1 ) = 4 n ( m 1 ) + 2 i , 1 i n 9 , f ( v 2 n 9 v 2 n 8 ) = 2 n ( 2 m 1 ) 18 , f ( v 2 n 8 v 2 n 7 ) = 2 n ( 2 m 1 ) 14 , f ( v 2 n 7 v 2 n 6 ) = 2 n ( 2 m 1 ) 16 , f ( v 2 n 6 v 2 n 5 ) = 2 n ( 2 m 1 ) 10 , f ( v 2 n 5 v 2 n 4 ) = 2 n ( 2 m 1 ) 12 , f ( v 2 n 4 v 2 n 3 ) = 2 n ( 2 m 1 ) 6 , f ( v 2 n 3 v 2 n 2 ) = 2 n ( 2 m 1 ) 8 , f ( v 2 n 2 v 2 n 1 ) = 2 n ( 2 m 1 ) 4 , f ( v 2 n 1 v 2 n ) = 2 n ( 2 m 1 ) 2 , f ( v 2 n v n + 1 ) = 2 n ( 2 m 1 ) .
Label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) as in case (1).
Thus, the labels of corresponding vertices mod ( 4 m n 2 n ) will be:
The labels of the vertices of the circle C n ( 1 ) are as follows: f ( v 1 ) 0 , f ( v 2 ) 4 m n 4 n + 10 , f ( v 3 ) 4 m n 4 n + 18 , f ( v 4 ) 4 m n 4 n + 22 , f ( v 5 ) 4 m n 4 n + 26 , f ( v 6 ) 4 m n 4 n + 34 , f ( v 7 ) 4 m n 4 n + 42 , f ( v 8 ) 4 m n 4 n + 46 , f ( v 9 ) 4 m n 4 n + 50 , f ( v 10 ) 4 m n 4 n + 58 , f ( v 11 ) 4 m n 4 n + 66 , f ( v 12 ) 4 m n 4 n + 70 , f ( v 13 ) 4 m n 4 n + 74 , f ( v 14 ) 4 m n 4 n + 82 , , f ( v n 9 ) 4 n 56 , f ( v n 8 ) 4 n 48 , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 40 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 24 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 12 , f ( v n 1 ) 4 n 4 , f ( v n ) 4 n 6 .
The labels of the vertices of the circle C n ( 2 ) are as follows: f ( v n + i ) 4 i + 2 , 1 i n 9 , f ( v 2 n 8 ) 4 n 28 , f ( v 2 n 7 ) 4 n 26 , f ( v 2 n 6 ) 4 n 22 , f ( v 2 n 5 ) 4 n 18 , f ( v 2 n 4 ) 4 n 14 , f ( v 2 n 3 ) 4 n 10 , f ( v 2 n 2 ) 4 n 8 , f ( v 2 n 1 ) 4 n 2 , f ( v 2 n ) 4 n + 2 .
The labels of the vertices of C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) , respectively, are as in case (1).
Case (5): n 9 mod 12 .
First, we label the edges of the paths P m ( k ) , 1 k n beginning with the edges of the path P m ( 1 ) the same as in case (1).
Second, we label the edges of the circles C n ( k ) , 1 k m beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( m ) , and then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 2 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 4 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 8 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 6 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 10 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 12 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 16 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 14 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 18 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 20 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 24 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 22 , f ( v 13 v 14 ) = 2 n ( m 1 ) + 26 , , f ( v n 11 v n 10 ) = 2 m n 22 , f ( v n 10 v n 9 ) = 2 m n 18 , f ( v n 9 v n 8 ) = 2 m n 20 , f ( v n 8 v n 7 ) = 2 m n 16 , f ( v n 7 v n 6 ) = 2 m n 14 , f ( v n 6 v n 5 ) = 2 m n 10 , f ( v n 5 v n 4 ) = 2 m n 12 , f ( v n 4 v n 3 ) = 2 m n 8 , f ( v n 3 v n 2 ) = 2 m n 6 , f ( v n 2 v n 1 ) = 2 m n , f ( v n 1 v n ) = 2 m n 2 , f ( v n v 1 ) = 2 m n 4 .
Label the edges of the circle C n ( 2 ) as follows: f ( v n + i v n + i + 1 ) = 4 n ( m 1 ) + 2 i , 1 i n 8 , f ( v 2 n 7 v 2 n 6 ) = 2 n ( 2 m 1 ) 12 , f ( v 2 n 6 v 2 n 5 ) = 2 n ( 2 m 1 ) 14 , f ( v 2 n 5 v 2 n 4 ) = 2 n ( 2 m 1 ) 6 , f ( v 2 n 4 v 2 n 3 ) = 2 n ( 2 m 1 ) 10 , f ( v 2 n 3 v 2 n 2 ) = 2 n ( 2 m 1 ) 8 , f ( v 2 n 2 v 2 n 1 ) = 2 n ( 2 m 1 ) 4 , f ( v 2 n v n + 1 ) = 2 n ( 2 m 1 ) .
Label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) as in case (1).
Thus, the labels of corresponding vertices mod ( 4 m n 2 n ) will be:
The labels of the vertices of the circle C n ( 1 ) are as follows: f ( v 1 ) 0 , f ( v 2 ) 4 m n 4 n + 10 , f ( v 3 ) 4 m n 4 n + 18 , f ( v 4 ) 4 m n 4 n + 22 , f ( v 5 ) 4 m n 4 n + 26 , f ( v 6 ) 4 m n 4 n + 34 , f ( v 7 ) 4 m n 4 n + 42 , f ( v 8 ) 4 m n 4 n + 46 , f ( v 9 ) 4 m n 4 n + 50 , f ( v 10 ) 4 m n 4 n + 58 , f ( v 11 ) 4 m n 4 n + 66 , f ( v 12 ) 4 m n 4 n + 70 , f ( v 13 ) 4 m n 4 n + 74 , f ( v 14 ) 4 m n 4 n + 82 , , f ( v n 10 ) 4 n 60 , f ( v n 9 ) 4 n 56 , f ( v n 8 ) 4 n 52 , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 36 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 28 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 10 , f ( v n 1 ) 4 n 4 , f ( v n ) 4 n 6 .
The labels of the vertices of the circle C n ( 2 ) are as follows: f ( v n + i ) 4 i + 2 , 1 i n 8 , f ( v 2 n 7 ) 4 n 24 , f ( v 2 n 6 ) 4 n 22 , f ( v 2 n 5 ) 4 n 16 , f ( v 2 n 4 ) 4 n 12 , f ( v 2 n 3 ) 4 n 14 , f ( v 2 n 2 ) 4 n 8 , f ( v 2 n 1 ) 4 n 2 , f ( v 2 n ) 4 n + 2 .
The labels of the vertices of C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) , respectively, are as in case (1).
Case (6): n 11 mod 12 .
First, we label the edges of the paths P m ( k ) , 1 k n beginning with the edges of the path P m ( 1 ) the same as in case (1).
Second, we label the edges of the circles C n ( k ) , 1 k m beginning with the edges of the innermost circle C n ( 1 ) , then the edges of outer circle C n ( m ) , and then the edges of the circles C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) .
Finally, we label the edges of the circles C m ( m 1 ) , C m ( m 3 ) , , C m ( 2 ) .
Label the edges of the circle C n ( 1 ) as follows: f ( v 1 v 2 ) = 2 n ( m 1 ) + 4 , f ( v 2 v 3 ) = 2 n ( m 1 ) + 2 , f ( v 3 v 4 ) = 2 n ( m 1 ) + 6 , f ( v 4 v 5 ) = 2 n ( m 1 ) + 8 , f ( v 5 v 6 ) = 2 n ( m 1 ) + 12 , f ( v 6 v 7 ) = 2 n ( m 1 ) + 10 , f ( v 7 v 8 ) = 2 n ( m 1 ) + 14 , f ( v 8 v 9 ) = 2 n ( m 1 ) + 16 , f ( v 9 v 10 ) = 2 n ( m 1 ) + 20 , f ( v 10 v 11 ) = 2 n ( m 1 ) + 18 , f ( v 11 v 12 ) = 2 n ( m 1 ) + 22 , f ( v 12 v 13 ) = 2 n ( m 1 ) + 24 , f ( v 13 v 14 ) = 2 n ( m 1 ) + 28 , , f ( v n 8 v n 7 ) = 2 m n 16 , f ( v n 7 v n 6 ) = 2 m n 14 , f ( v n 6 v n 5 ) = 2 m n 10 , f ( v n 5 v n 4 ) = 2 m n 12 , f ( v n 4 v n 3 ) = 2 m n 8 , f ( v n 3 v n 2 ) = 2 m n 6 , f ( v n 2 v n 1 ) = 2 m n 2 , f ( v n 1 v n ) = 2 m n 4 , f ( v n v 1 ) = 2 m n .
Label the edges of the circle C n ( 2 ) as follows: f ( v n + i v n + i + 1 ) = 4 n ( m 1 ) + 2 i , 1 i n 2 , f ( v 2 n 1 v 2 n ) = 4 m n , f ( v 2 n v n + 1 ) = 4 m n 2 .
Label the edges of C n ( m ) , C n ( m 2 ) , C n ( m 4 ) , , C n ( 3 ) and C n ( m 1 ) , C n ( m 3 ) , C n ( m 5 ) , , C n ( 4 ) as in case (1).
Thus, the labels of corresponding vertices mod ( 4 m n 2 n ) will be:
The labels of the vertices of the circle C n ( 1 ) are as follows: f ( v 1 ) 6 , f ( v 2 ) 4 m n 4 n + 10 , f ( v 3 ) 4 m n 4 n + 14 , f ( v 4 ) 4 m n 4 n + 22 , f ( v 5 ) 4 m n 4 n + 30 , f ( v 6 ) 4 m n 4 n + 34 , f ( v 7 ) 4 m n 4 n + 38 , f ( v 8 ) 4 m n 4 n + 46 , f ( v 9 ) 4 m n 4 n + 54 , f ( v 10 ) 4 m n 4 n + 58 , f ( v 11 ) 4 m n 4 n + 62 , f ( v 12 ) 4 m n 4 n + 70 , , f ( v n 7 ) 4 n 44 , f ( v n 6 ) 4 n 36 , f ( v n 5 ) 4 n 32 , f ( v n 4 ) 4 n 28 , f ( v n 3 ) 4 n 20 , f ( v n 2 ) 4 n 12 , f ( v n 1 ) 4 n 8 , f ( v n ) 4 n 4 .
The labels of the vertices of the circle C n ( 2 ) are as follows: f ( v n + 1 ) 4 , f ( v n + i ) 4 i + 2 , 2 i n 2 , f ( v 2 n 1 ) 4 n , f ( v 2 n ) 4 n + 2 .
The labels of the vertices of C n ( 3 ) , C n ( 4 ) , , C n ( m 2 ) , C n ( m 1 ) , C n ( m ) , respectively, are as the same as in case (1).
Illustration: An e.e.g.l. of the cylinder grid graphs C 9 , 3 , C 7 , 9 , C 7 , 11 , C 7 , 13 , C 7 , 15 , C 7 , 17 and C 7 , 19 is shown in Figure 14.

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 C m × C n and rectangular P m × P n 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.

Acknowledgments

The authors are grateful to the anonymous reviewers for their helpful comments and suggestions for improving the original version of the paper.

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. Cylinder grid graph C m , n .
Figure 1. Cylinder grid graph C m , n .
Symmetry 11 00584 g001
Figure 2. The cylinder grid graph C m , n , m is even and n 2 .
Figure 2. The cylinder grid graph C m , n , m is even and n 2 .
Symmetry 11 00584 g002
Figure 3. An edge even graceful labeling (e.e.g., l.) of the cylinder grid graphs C 8 , 11 and C 8 , 12 .
Figure 3. An edge even graceful labeling (e.e.g., l.) of the cylinder grid graphs C 8 , 11 and C 8 , 12 .
Symmetry 11 00584 g003aSymmetry 11 00584 g003b
Figure 4. The cylinder grid graph C 3 , n , n 1 mod 6 .
Figure 4. The cylinder grid graph C 3 , n , n 1 mod 6 .
Symmetry 11 00584 g004
Figure 5. The cylinder grid graph C 3 , n , n 3 mod 6 .
Figure 5. The cylinder grid graph C 3 , n , n 3 mod 6 .
Symmetry 11 00584 g005
Figure 6. The cylinder grid graph C 3 , n , n 5 mod 6 .
Figure 6. The cylinder grid graph C 3 , n , n 5 mod 6 .
Symmetry 11 00584 g006
Figure 7. An e.e.g., l. of the cylinder grid graphs C 3 , 25 , C 3 , 27 and C 3 , 29 .
Figure 7. An e.e.g., l. of the cylinder grid graphs C 3 , 25 , C 3 , 27 and C 3 , 29 .
Symmetry 11 00584 g007aSymmetry 11 00584 g007b
Figure 8. An e.e.g., l. of the cylinder grid graph C 3 , 5 .
Figure 8. An e.e.g., l. of the cylinder grid graph C 3 , 5 .
Symmetry 11 00584 g008
Figure 9. The cylinder grid graph C m , n , m is odd greater than 3 and n 2 .
Figure 9. The cylinder grid graph C m , n , m is odd greater than 3 and n 2 .
Symmetry 11 00584 g009
Figure 10. The cylinder grid graph C m , 2 .
Figure 10. The cylinder grid graph C m , 2 .
Symmetry 11 00584 g010
Figure 11. An e.e.g., l. of the cylinder grid graph C m , 4 .
Figure 11. An e.e.g., l. of the cylinder grid graph C m , 4 .
Symmetry 11 00584 g011
Figure 12. An e.e.g., l. of the cylinder grid graphs C 9 , 2 , C 9 , 4 , C 7 , 10 , C 7 , 12 , C 7 , 14 , C 7 , 16 , C 7 , 18 , and C 7 , 20 .
Figure 12. An e.e.g., l. of the cylinder grid graphs C 9 , 2 , C 9 , 4 , C 7 , 10 , C 7 , 12 , C 7 , 14 , C 7 , 16 , C 7 , 18 , and C 7 , 20 .
Symmetry 11 00584 g012aSymmetry 11 00584 g012bSymmetry 11 00584 g012cSymmetry 11 00584 g012d
Figure 13. The cylinder grid graph C m , 3 , m is odd, m 3 .
Figure 13. The cylinder grid graph C m , 3 , m is odd, m 3 .
Symmetry 11 00584 g013
Figure 14. An e.e.g.l. of the cylinder grid graphs C 9 , 3 , C 7 , 9 , C 7 , 11 , C 7 , 13 , C 7 , 15 , C 7 , 17 and C 7 , 19 .
Figure 14. An e.e.g.l. of the cylinder grid graphs C 9 , 3 , C 7 , 9 , C 7 , 11 , C 7 , 13 , C 7 , 15 , C 7 , 17 and C 7 , 19 .
Symmetry 11 00584 g014aSymmetry 11 00584 g014bSymmetry 11 00584 g014cSymmetry 11 00584 g014d

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