Polygon Graphs of Girth 6

Full Length Research Paper

Polygon graphs of Girth 6

Shoaib U. Din 1 and Khalil Ahmad 2*

1Department of Mathematics, University of the Punjab Lahore, Pakistan. 2Department of Mathematics, University of Management and Technology, Lahore, Pakistan.

Accepted 30 March, 2011

G m, p This paper introduces a family of simple bipartite graphs denoted by P ( ) , named and constructed as polygon graphs. These polygon graphs are bicubic simple graphs possessing Hamiltonian cycles. Some important results are proved for these graphs. The girth of these graphs is G 3,1 counted as 6( m ≥ 3 ). P ( ) Polygon graph is isomorphic to a famous Pappus graph. Since polygon graphs are bipartite, therefore they can be used as Tanner graphs for low density parity check codes. This family of graphs may also be used in the design of efficient computer networks.

Key words: Polygon graph, girth, Hamiltonian cycle, bicubic graph.

INTRODUCTION

A graph G is a triple, consisting of a vertex set V= V(G) , by its (cyclic) sequence of vertices, the above cycle C an edge set E=E(G) and a map that associates to each might be written as x0 …x k-1 x0. The length of a cycle is edge two vertices (not necessarily distinct) called its end the number of its edges (or vertices). The cycle of length points. A loop is an edge whose end points are equal. k Multiple edges are edges having same end points. A k is called k-cycle and denoted by C . The length of the simple graph is one having no loops or multiple edges. shortest cycle (contained) in a graph G is the girth g (G) To any graph, we can associate the adjacency matrix A of G, the maximum length of a cycle is G and is its which is an n ×n matrix ( n=IvI ) with rows and columns circumference. If G does not contain a cycle, we set the indexed by the elements of vertex set and the (x,y) -th former to ∞ the latter to zero (Reinhard, 2000). Cubic entry is the number of edges connecting x and y. A path graphs also known as trivalent graphs are intensively is a non-empty graph studied in graph theory. The bipartite cubic graphs widely known as bicubic graphs took special interest. Tait P = (V, E) (1888), conjectured that every planer cubic graph contains Hamiltonian cycle, it was challenged by Tutte where, (1946), by a counter example, a 46-vertex graph now named for him (Tutte, 1946). In 1971, Tutte conjectured V= { x0, x 1,…………x k} , E = { x0x1,……x k-1 xk} that all bicubic graphs are Hamiltonian; however, Horton provided a 96-vertex counter example (Horton, 1982). 01Where the xi’s are all distinct. Study of cubic graphs is one of the favorite topics for graph theorists. One main reason is that they have wide The vertices x0 and xk are linked by P and are called its application. For example, it follows from the result of ends; the vertices x1,… x k-1 are the inner vertices of P. Malle et al. (1994) that, almost every finite simple group The number of edges of a path is its length. If P = x0, … has a cubic Cayley graph. Moreover, the generalization xk-1 is a path with k> 3, then the graph C: = P + x k x0 is to the graphs of degree k>3 does not appear to be more known as cycle. As with paths, we often denote a cycle difficult than the case k=3. Finally, the cubic case is the only one where we have specific examples that improve significantly on the best general results currently available incorrect format (Biggs, 1998). Polygon graphs *Corresponding author. E-mail: [email protected] are constructed by connecting odd number of copies of 198 J. Eng. Technol. Res.

polygons of same size in a delicate manner. These denoted by G( m, p ) and is named as a polygon graph. graphs are defined only for polygons with an even P number of sides and denoted by GP () m, p , where 2m is Example 1: If m = 2, p = 2, GP (2 , 2) (Figure 1) the number of sides and 2p +1 is the number of copies of polygon P. These graphs are bicubic with some GP (3,1 ) is isomorphic to a famous bicubic symmetric distance- interesting properties. Principle of mathematical induction regular Pappus graph with 18 vertices. It has representations as is used to prove the existence of Hamiltonian cycles in presented in Figure 2. these graphs. It is shown that, these graphs have girth 6. It is a simple fact that cubic Hamiltonian graphs have at least two Hamiltonian cycles. Finding such a cycle is NP- RESULTS hard in general, and no polynomial time algorithm is known for the problem of finding a second Hamiltonian Theorem 1 cycle, when one such cycle is given as part of the input (Cristina, 1999). However, in the case of polygon graph, it Let GP ( m, p ) be a polygon graph then: is easy to find Hamiltonian cycles.

i) |V | = 2m (2p+1 ), | E | = 3m (2p+1 ); METHODS OF CONSTRUCTING POLYGON GRAPHS GP GP

ii) GP ( m, p ) is a cubic graph; Let P be a polygon with 2m ( m ≥ 2 ) sides. Place 2p+1 copies iii) GP ( m, p ) is a bipartite graph. of P denoted by ′ ′ ′ in parallel, PPP,,,12⋅⋅⋅⋅ PPPp ,,, 12 ⋅⋅⋅⋅ P p such that P is in the middle, p copies on the right side of P say P, P , ⋅⋅⋅⋅ P and p copies on the left side of P Proof 1 2 p ′ ′ ′ i) Since each polygon has 2m number of vertices and say P1, P 2 , ⋅⋅⋅⋅ P p as follows. there are ( 2p+1 ) copies of polygons in GP ( m, p ) . PP′,, ′⋅⋅⋅⋅ PPPP ′ ,,,, ⋅⋅⋅⋅ P . Vertices of P, P′ and P are 12p 12 p i i Hence, total number of vertices equals: (0) ( 0) ( 0 ) (i) ( i) ( i ) denoted by v1, v 2 ,⋅⋅⋅⋅ , v 2 m , u1, u 2 ,⋅⋅⋅⋅ , u 2 m , |V | = 2m (2p+1 ) (i) ( i) ( i ) GP w1, w 2 ,⋅⋅⋅⋅ , w 2 m , i=1, ⋅⋅⋅ , p respectively. Simple connected V 1 GP graph GP is constructed by drawing edges such that, an even | E | = ∑ d() v k GP vertex is connected with odd vertex and an odd vertex with an even 2 k =1 one in the following manner.

Where d( v k ) is the degree of kth vertex. Algorithm

2m (2 p + 1) Edges between vertices of different polygons 1 1 = ∑ d( v i ) = 3{} 2m (2 p + 1) = 3m (2 p + 1) 2 i=1 2 a) For i< p connect m vertices of P with m vertices of P i i −1 and m vertices of P i with m vertices of Pi +1 . Similarly, connect ii) By definition of GP ( m, p ) , it is clear that each vertex of ′ ' a polygon is connected with two vertices of the same m vertices of P i with m vertices of Pi −1 and m vertices of polygon and with one vertex of a polygon, either on its ′ P i with m vertices of Pi +1 . right or on its left. Hence, degree of each vertex becomes three. So, G m, p is a regular simple graph of degree b) For i= p , m vertices of P i are already joined with m P ( ) vertices of Pi −1 , where the remaining m vertices of P i are joined three, that is GP ( m, p ) is a cubic graph. ′ iii) There exists a vertex labeling such that, each even with the remaining m vertices of P i . vertex is connected with three odd vertices and each odd vertex is connected with three even vertices, therefore

Edges between vertices within a polygon vertices of GP can be colored using only two colors. As

every two colorable graph is bipartite. Hence, GP is a Only sides of polygon represent edges in GP . This simple graph is bipartite graph with equal number of vertices in each part.

Din and Ahmad 199

1 1 1 1 9 1 6 7 1 2

4 3 5 8 1 1 1 1

2 1 Figure 1. Polygon graph Gp (2 , 2).

Figure 2. Polygon graph Gp (3 , 1).

Theorem 2 χ G m, t = 2 , ( G m, p is bipartite) ( P ( )) P ( )

Let GP ( m, p ) be a polygon graph, then ∆(GP ( m, t )) = 3 (GP ( m, p ) is cubic). χ Gmp( ,) ≤ ∆ Gmp( , ) ( P) ( P ) Theorem 3

Proof G m, p is a Hamiltonian graph that is, it contains a P ( ) Since G m, p is a bipartite graph, therefore there is Hamiltonian cycle!. P ( ) no odd cycle in it. Also, since by definition GP ( m, p ) is Proof not complete, so according to Brooks theorem (1914).

Let m = 2, we use mathematical induction on p, to prove χ Gmp( ,) ≤ ∆ Gmp( , ) ( P) ( P ) the result. For p= 1, GP (2,1) contains the Hamiltonian 200 J. Eng. Technol. Res.

9 10 6 7 1 2

4 3 5 8 12 11

Figure 3. Polygon graph Gp(2,1).

path. 1 – 2 – 3 – 4 – 11 – 12 – 9 – 10 – 7 – 8 – 5 – 6 as arbitrary m , G( m, p ) has Hamiltonian cycle. shown in Figure 3. P 2, Let GP ( p) contain Hamiltonian cycle for p≤ k that is; Theorem 4

(0) ( 0) ( 0) ( 0 ) (1) ( 1) ( 1) ( 1 ) v1, v 2 , v 3 , v 4 , u3, u 2 , u 1 , u 4 ……… Let GP ( m, p ) be a simple graph, for m ≥ 3 , girth of (k) ( k) ( k) ( k ) (k) ( k) ( k) ( k ) u3, u 2 , u 1 , u 4 , w1, w 2 , w 3 , w 4 ,…………. GP ( m, p ) is 6 that is, gG( P ( mp,)) = 6. (k) ( k) ( k) ( k ) (0) w, w , w , w v 1 2 3 4 1 Proof

Now we prove that GP (2, p ) contains Hamiltonian cycle Since GP ( m, p ) is a bipartite graph. The girth must be for p= k + 1. Replace edges between P and P ′ by k k an even number. There are two types of cycles in ′ ′ drawing edges between Pk , Pk+1 and Pk , Pk+1 . Now we GP ( m, p ) . have two vertices of P and two vertices of P ′ each k+1 k+1 of degree two; now connect these vertices to make the Cycles containing vertices of one polygon following path. Since there are 2m (with m ≥ 3 ) vertices in each polygon, (0) ( 0) ( 0) ( 0 ) (1) ( 1) ( 1) ( 1 ) the length of the cycle must be greater than or equal to 6. v1, v 2 , v 3 , v 4 , u3, u 2 , u 1 , u 4 ……… (k) ( k) ( k) ( k ) (k+1) ( k + 1) ( k) + 1( k + 1 ) gG( P ( mp,)) = 6 u3, u 2 , u 1 , u 4 , u3, u 2 , u 1 , u 4

(k+1) ( k + 1) ( k + 1) ( k + 1 ) (k) ( k) ( k) ( k ) w1, w 2 , w 3 , w 4 w1, w 2 , w 3 , w 4 ,…………. (k) ( k) ( k) ( k ) (0) Cycles involving the vertices of more than one w1, w 2 , w 3 , w 4 v1 polygon is a Hamiltonian path. Similarly, it could be proved that for In this case, the shortest cycle must involve at least two Din and Ahmad 201

vertices from two adjacent polygons that is, two edges Theorem 7 from each polygon. Moreover, one edge is required to switch from one Let G( m, p ) be a simple graph. Then, there are at polygon to the other and one edge to come back making P a total of 6.   most m(2 p + 1 ) 2m( 2 p + 11) −  cycles in GP .

gG( P ( mp,)) = 6 Proof

Theorem 5 Since G( m, p ) is a cubic graph. So when a path enters P a vertex v i there are two options for the path to proceed Let GP ( m, p ) be a polygon graph. Then there are at to the next vertex, since the total number of vertices is least 2p + 1 number of cycles of length 2m . ( ) 2m( 2 p + 1 ) . Therefore, total number of different paths

from v i to v i equals Proof 2m (2 p + 1)  All cycles containing the vertices of a polygon are of   =mp (2 + 1) [2 mp (2 + 1) -1] length 2m . The cycles containing vertices of more than 2  one polygon of length 2m may also exist.

However the number of cycles which involve vertices of Since at initial vertex that is v i , there is one more option, only one polygon equals (2p + 1 ) ,(since GP ( m, p ) , at the last there is one less option. Hence, total number of cycles is mp (2+ 1) [2 mp (2 + 1)-1] . contains (2p + 1 ) polygons). Hence, there will be at least

2p + 1 cycles of length 2m . Theorem 8

Theorem 6 Every edge of GP ( m, p ) is contained in an even number

of Hamiltonian cycles. Let GP ( m, p ) be a polygon simple graph. Then, there is at least one cycle of length Proof

2m( 2 p + 1 ) . Since G( m, p ) is a cubic graph that is, degree of each P

vertex is three let vvi j∈ EG( P (( mp , )) Proof 1 1 Let GP( mp, ) = G Pij − vv . Now in GP except vi and From theorems 5 and 3, G m, p is a Hamiltonian P ( ) v j all vertices are of degree three, whereas vi and v j are graph for all possible values of m and p , so there is a 1 the only vertices of degree two. G Contains two longest Hamiltonian cycle that is a closed path contain each P vertex (except first) exactly once. Since there are paths from vi and v j , hence GP ( m, p ) posses two 2m 2 p + 1 vertices in G m, p . So, there must be at ( ) P ( ) Hamiltonian cycles containing vi v j least one cycle of length 2m( 2 p + 1 ) .

Conclusions

Corollary 1 Polygon graph is a new discrete structure. We worked

out basic properties. Polygon graphs are bipartite, there- Let GP ( m, p ) be a simple graph. The circumference of fore could be used as Tanner graphs, to generate low , 2 2 1 density parity check codes. Moreover, polygon graphs GP ( m p ) is m( p + ) . would be attractive due to their simple construction

202 J. Eng. Technol. Res.

algorithm with a wide range of a choice of parameters. In Cristina CB, Miklos S, Zsolt (1999). “On the Approximation of finding this paper, we presented polygon graphs of girth 6. It Hamiltonian Cycle in Cubic Hamiltonian Graphs” J. Algorithms, 31(1): 249-268. could be extended to the graphs of girth more than 6. Malle GG, Saxel J, Weigel T (1994). "Generators of classical groups" Geom. Dedicata, 49: 85-116. Horton HJD (1982). "On Two-Factors of Bipartite Regular Graphs." REFERENCES Disc. Math. , 41: 35-41. Tutte WT (1946). "On Hamiltonian Circuits." J. London Math. Soc., 21: Biggs NN (1998). "Construction of cubic graphs with large girth". 98-101. Electronic J. Combinatorics, p. 5.