American Journal of Mathematics and Statistics 2013, 3(3): 124-129 DOI: 10.5923/j.ajms.20130303.04
Polygon Semi Designs
Shoaibu Din1, Khalil Ahmad2,*
1Department of M athematics, University of the Punjab Lahore 2Department of M athematics, (University of Management and Technology Lahore, Pakistan)
Abstract In this paper a family of semi design (vk,,λ) , 2≤
is shown that these graphs have girth 6. It is a simple fact that 1. Preliminaries and Introduction cubic Hamiltonian graphs have at least two Hamiltonian cycles. A graph G is a triple consisting of a vertex set V= V(G), an An incidence structure is defined on polygon simple graph edge set E=E(G) and a map that associates to each edge two Gp(m,p) through a matrix H( mt, ) , where ro ws of H( mt, ) vertices (not necessarily distinct) called its end points. A are taken as points and columns as blocks. loop is an edge whose end points are equal. Multiple edges are edges having same end points. A simple graph is one Let Vv= {1,2,3,... } . A collect ion β of distinct having no loops or multiple edges. To any graph, we can subsets of V is called Polygon Semi Design (vk,,λ) if associate the adjacency matrix A which is an n ×n matrix (n=IvI) with rows and columns indexed by the elements of 2≤
American Journal of Mathematics and Statistics 2013, 3(3): 124-129 125
codes. The size of the smallest stopping set in LDPC codes distance-regular Pappus graph with 18 vertices. It has helps in analyzing their performance under iterative following representations. decoding, just as minimum distance helps in analyzing the performance under maximum likelihood decoding[7]. The structure of the paper is as follows. In section 2, we introduce the definitions and notation used in the later sections; it also includes some results on polygon simple graphs. In section 3 polygon semi designs are defined on polygon simple graphs, a lower bound for the size of the Fi gure 1. Polygon graph Gp (3 , 1) smallest stopping set in a PSD is proposed. Theorem 2.1: Let GP ( mp, ) be a polygon graph. then. i) | V | = 2m (2p+1) 2. Polygon Graph GP | E | = 3m (2p+1) GP Let P be a polygon with 2mm ( ≥ 2) sides. Place 2p+1 ii) GP ( mp, ) is a cubic graph ′′ ′ copies of P denoted by PP,1 , P 2 ,⋅⋅⋅⋅ Ppp ,,, PP12 ⋅⋅⋅⋅ P iii) GP ( mp, ) is a bipartite graph. in parallel such that P is in the middle, p copies on the right Proof: side of P say PP12,,⋅⋅⋅⋅ Pp and p copies on the left side of P i) Since each polygon has 2m number of vertices and there
′′ ′ are (2p+1) copies of polygons in GP ( mp, ). say PP12,,⋅⋅⋅⋅ Pp as follows. | V | = 2m (2p+1) ′′ ′ GP P1, P 2 ,⋅⋅⋅⋅ Ppp ,,, PPP12 , ⋅⋅⋅⋅ P. VG ′ 1 P Vertices of P, Pp and Pp are denoted by | E | = dv( ) where dv is the degree of GP ∑ k ( k ) (00) ( ) ( 0) (ii) ( ) ( i) 2 k =1 vv, ,,⋅⋅⋅⋅ v , uu, ,,⋅⋅⋅⋅ u 12 2m 12 2m kth vertex. (ii) ( ) ( i) ⋅⋅⋅⋅ respectively. Simple 2mp (2+ 1) ww12, ,, w 2m ip=1, ⋅⋅⋅ , 1 1 = dv()= 3{ 2mp (2+ 1)} = 3mp (2+ 1) connected graph is constructed by drawing edges, such ∑ i GP 2 i=1 2 that an even vertex is connected with odd vertex and an odd ii) By definition of it is clear that each vertex vertex with even one in the following manner. GP ( mp, ) i. Edges between vertices of different polygons. of a polygon is connected with two vertices of the same a) For j
b) For jp= m vertices of Pj are already joined with connected with three even vertices, therefore vertices of GP can be colored using only two colors. As every two colorable m vertices of Pj−1 where the remaining m vertices of Pj graph is bipartite. Hence GP is bipartite graph with equal are joined with remaining m vertices of ′ . Pj number of vertices in each part. = ⋅⋅⋅ where jp1, , Theorem 2.2: GP ( mp, ) is a Hamiltonian graph i.e. it ii. Edges between vertices within a polygon contains a Hamiltonian cycle!.
Only sides of polygon represent edges in GP Proof: Let m = 2, we use mathematical induction on p, to prove the result. For p= 1 GP (2,1) contains Hamiltonian This simple graph is denoted by GP ( mp, ) is named as path. polygon graph. 1 – 2 – 3 – 4 – 11 – 12 – 9 – 10 – 7 – 8 – 5 – 6 as shown G (3,1) is isomorphic to a famous bicubic symmetric in fig 2. P
126 Shoaibu Din et al.: Polygon Semi Designs
9 10
7 6
1 2
4 3
8 5
12 11
Fi gure 2. Polygon graph Gp(2,1)
Let G (2, p) contain Hamiltonian cycle for pk≤ m ≥ 3 , Then the girth of G mp, is 6 i.e . P P ( ) i.e. gG( ( mp,)) = 6. (0000) ( ) ( ) ( ) (1111) ( ) ( ) ( ) P vvvv1,,, 234, uuuu3,,, 21 4……… Proof: Since GP ( mp, ) is a bipartite graph. The girth (kkkk) ( ) ( ) ( ) (kkkk) ( ) ( ) ( ) uuuu3,,, 21 4, wwww1,,, 234,……… must be an even number. We consider two cases depending (kkkk) ( ) ( ) ( ) (0) on the types of cycles in wwww1,,, 234v1 GP ( mp, ) Case 1: Cycles containing the vertices of one polyg on. Now we prove that GpP (2, ) contains Hamiltonian Since there are 2m ( m ≥ 3 ) vertices in each polygon, cycle for pk= +1 gG( p ( mp,6)) ≥ Replace edges between P and P ′ by drawing edges k k Case 1: Cycles invol ving the vertices of more than one between P , P and P ′ , P ′ . Now we have two polyg on. k k +1 k k +1 In this case the shortest cycle must involve at least two ′ vertices from two adjacent polygons i.e. two edges from each vertices of Pk +1 and two vertices of Pk +1 each of degree polygon. Moreover one edge is required to switch from one two; now connect these vertices to make the following path. polygon to the other and another edge to come back. Thus (0000) ( ) ( ) ( ) (1111) ( ) ( ) ( ) vvvv1,,, 234, uuuu3,,, 21 4……… gG( P ( mp,6)) = (kkkk) ( ) ( ) ( ) (kkkk+1) ( + 1) ( ) ++ 11( ) uuuu3,,, 21 4, uuuu3,,, 21 4 (kkkk++++1111) ( ) ( ) ( ) (kkkk) ( ) ( ) ( ) wwww1,,, 234wwww1,,, 234,………. 3. Polygon Semi Design (kkkk) ( ) ( ) ( ) (0) wwww1,,, 234v1 3.1. Defini tions is a Hamiltonian path. Let H( mt, ) where tp=21 + be a matrix whose Similarly it could be proved that for arbitrary m , columns represent odd labeled vertices and rows represent has Hamiltonian cycle. GP ( mp, ) even labeled vertices of a polygon simple graph Theorem 2.3: Let be a simple graph, where GP ( mp, ) GP ( mp, ) or vice versa, such that
American Journal of Mathematics and Statistics 2013, 3(3): 124-129 127
1 if ith odd vertex is incident with jth even vertex H( mt, ) = h = ij mt× mt 0 Otherwise
Example 3.1. Refer to the polygon graph GP (2 , 1) in figure 2. 1 3 5 7 9 11 2 1 1 0 0 1 0 1 4 1 1 0 0 0 1 2 H (2,3) 6 1 0 1 1 0 0 3 = 8 0 1 1 1 0 0 4 10 0 0 1 0 1 1 5 12 0 0 0 1 1 1 6
1 2 3 4 5 6
, Theorem 3.1: Let PSD (vk,,λ) be a Polygon Semi k 1 s and according to the theorem 3.2.2 each row also , Design. Then vb= .i.e. PSD is symmetric. contain rks= ,1 condition 2 in definition means that if Proof: F ro m theorem 5. 1 total number of vertices equals we pick any two columns there are at most λ rows in 22m( p+= 12) mt , where tp=21 + which is an even which there is a 1 in both these columns. For m ≥ 3 any number. Hence if we label all vertices from 1,2,...... 2mt , two rows(columns) of H( mt, ) have at most one 1 in since even label vertices are taken as points i.e. v = mt common i.e. in the same column(row). So λ =1 and odd vertices as blocks b= mt , Hence vb= Although by theorem 3.2.1 PSD is symmetric, it does not Theorem 3.2 If PSD (λ,,vk)is a polygon semi design mean that its incidence matrix should be symmetric. then each element of the base set occurs in r blocks, where In an incidence matrix A of the PSD (vk,,λ) not just bk= vr and kr= each row but also each column has exactly ks1' and not Proof: To prove the claim we count in two ways the just every two rows but also every two columns have at most cardinality of the set. t λ 1's in common therefore also A is an incidence X={( xB,|) x ∈∈ VB , β} matr ix for s o me (vk,,λ) PSD, Also notice that by for each ∈ the block can be chosen in xV B r fisher’s in equality, the transpose of an incidence matrix different ways, hence by product rule ||X= vr ; on the design could be an incidence matrix of a design only if other hand, for each of the b blocks x can be chosen in k bv= . different ways, again by product rule ||X= bk , So 3.3. Stopping Sets bk= vr Since = Definiti on: A set S= { BB, ,...... B} of blocks in a vb 12 s ⇒ = kr polygon semi design (vk,,λ) is called stopping set if
3.2. Incidence Matri x s ∀∈p UBi , If PSD is a (vk,,λ) polygon semi design then the i=1
Spi=∈∈{ B Sp| B i} is of order at least 2. Let smin binary bv× matrix Aa= ( ij ) where be the size of smallest non empty stopping set. 1 if ith block of β contains jth point of v .
aij = 0 other wise 3.3.1. Bounds For Stopping Sets Is called incidence matrix of the PSD. Theorem 3.3.1 Let Dm( , 3,1) m≥ 3 be a polygon Of course such a matrix is by no means unique, but symmetric semi design. If is a stopping set in then. depends on the order in which we write the blocks and points. S D By definition, each column contains sSmin =| | ≥ 4.
128 Shoaibu Din et al.: Polygon Semi Designs
Proof As be a stopping set such that = and S sS|| therefore each point in ∑ must be incident with at least 2 v ||∪=Bm i.e. number of points is all blocks of . ∈ i S BSi blocks in ∑ . So ∑ is a stopping set. v v Count the set X={( pB,,) B ∈∈ S p B} in two Suppose by contradiction for some p ∈∑ p lies in v different ways. Since there are points in ∪ B and m ∈ i BSi two blocks of ∑ , and v lies in more three blocks of each point lies in at least two blocks BS∈ by definition of v . So the total number of such pairs is at least . So there is a pair of points , which lies in S (,pB ) ∑ pvand 2m . v more than one blocks of PSD, this contradicts the fact that On the other hand there are s blocks in S and each each pair of points lies in at most one block. Hence each block containing three points so there we exactly 3s such point in can be incident in at most one block in pairs e xist. ∑ v ⇒≥ 32sm
Now by inclusion exclusion principle ∑ . v m≥−3 s∑ || BBijij ∩ ≠ Cor ollar y 3.3.1 In (vk, ,2) Polygon Semi Design Bi, Bj∈ S Sb≤−3 min s Proof: For each vV∈ , |∑ |3= s ∑ |BBij∩≥ |, ij ≠ v ≥− 2 ms3 Bi, Bj∈ S 2 b =||||||β =∑∑ + Any two blocks intersect in most two points. vv − s ⇒=−||3b ⇒Sb ≤−3 3ss≥− 23 ∑ min 2 v 36s≥− s ss( − 1) 4. Conclusions ss−≥13 s ( ) Polygon simp le graph is a new discrete structure. We s −≥13 worked out for some basic properties. Polygon graphs are s ≥ 4 bipartite, therefore could be used as Tanner graphs to generate low density parity check codes. A polygon semi Let λ polygon semi design, for ∈ we (vk,, ) vV design is again a recent development in combinatorial define as the collection of blocks s uch mathematics. Some results are proved on these designs. The ∑ Bi ∈ β v bounds proved for stopping sets, are to be used as a performance indicator for the LDPC codes defined on that ∈ also is the set vBi ∑ β \,∑ polygon semi designs, these codes are widely used for error v v deduction and correction. ∑ =∈∉{Bβ | vB} v Theorem 3.3.2 Let (v, 3,1) be a polygon semi design, REFERENCES with r = 3 then ∑ =∈∉{Bβ | vB} is a stopping v [1] Tutte, W. T. "On Hamiltonian Circuits." J. London Math. Soc. set. 21, 98-101, 1946 Proof: Let vV∈ , for each p ∈ , p should be in at [2] Horton, J. D. "On Two-Factors of Bipartite Regular Graphs." ∑ Disc. Math. 41, 35-41, 1982. v most 1 block of ∑ ,[by definition of polygon semi design [3] J Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, v England: Oxford University Press, 1998 any pair of points should be contained by at most one block]. [4] Reinhard Diestel. Graph Theory. Electronic Edition 2000. c Sine each vV∈ lies in e xact ly r = 3 , blocks in PSD, Springer-Verlag New York 1997, 2000
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