UPPER AND LOWER BOUNDS OF THE BASIS NUMBER OF KRONECKER PRODUCT OF A WHEEL WITH A PATH AND A CYCLE

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1 J.Edu. Sci, Vol. (9) No.(2) 2007 UPPER AND LOWER BOUNDS OF THE BASIS NUMBER OF KRONECKER PRODUCT OF A WHEEL WITH A PATH AND A CYCLE Ghassa T. Marougi Ayha A. Khalil Dept. Math./College of Computers & Mathematics Scieces/ Mosul Uiversity Received 27/7/2006 Accepted 3/0/2006 الملخص يعرف العدد الا ساس b(g) لبيان G عل ى ان ه اص غر ع دد ص حيح موج ب k بحي ث ان لG قاعدة ذات ثني ة k لف ضاء دارات ه. ف ي ه ذا البح ث س وف ن درس القي د الاعل ى والاص غرللعدد الا ساس لجداء Kroecker للعجلة مع الدرب والدارة حيث توصلنا إلى النتاي ج الا تية: 3 b( P ) 4, m 4 ad 3, 3 b ( C ) 5, m 4, 3. ABSTRACT The basis umber, b(g),of a graph G is defied to be the smallest positive iteger k such that G has a k-fold basis for its cycle space. We ivestigate upper ad lower bouds of the basis umber of Kroecker product of a wheel with a path ad a cycle. It is proved that 3 b( ) 4, m 4 ad 3, ad 3 b ( C ) 5, m 4, 3. 64

2 UPPER AND LOWER BOUNDS OF THE BASIS. INTRODUCTION. Throughout this paper, we cosider oly fiite, udirected ad simple graphs. Our termiology ad otatios will be stadard except as idicated. For udefied terms, see [3]. Let G be a coected graph, ad let e, e 2,, e q be a orderig of the edges. The ay subset S of edges correspods to a (0,)-vector (a, a 2,.., a q ) i the usual way, with a i = if e i S ad a i =0 otherwise, for i=,2,..q. These vectors form a q-dimesioal vector space, deoted by (Z 2 ) q over the field Z 2. The vectors i (Z 2 ) q which correspod to the cycles i G geerate a subspace called the cycle space of G, ad deoted by ξ(g). It is well kow that dim ξ(g)= γ (G)=q-p+k, where p is the umber of vertices, k is the umber of coected compoets ad γ(g) is the cyclomatic umber of G. A basis for ξ(g) is called h-fold if each edge of G occurs i at most h of the cycles i the basis. The basis umber of G, deoted by b(g), is the smallest positive iteger h such that ξ(g) has a h-fold basis, ad such a basis is called a required basis of G ad deoted by B r (G). If B is a basis for ξ(g) ad e is a edge of G, the the fold of e i B, deoted by f B (e) is defied to be the umber of cycles i B cotaiig e. Defiitio: Let G=(V,E) be a simple graph with order ad vertex set V={p,p 2,..., p }. the adacecy matrix of G, deoted by A(G) is the matrix defied by : A(G)=[a i ] where a i = 0,if the edge p i, p i E,, otherwise a i is called the adacecy umber of the pair (v i, v ) of vertices. Defiitio: Let the vertex sets of the graphs G ad H be {p i i=,2,,m} ad {q =,2,,} resp.,the the Kroecker product [8], G H,is the graph with vertex set {(p i, q ): for i=,2,,m ad =,2,,} such that the adacecy umber of the pair (p i, q ), (p k, q l ) is the product of the adacecy umbers of (p i,p k )i G ad (q,q l ) i H. G H is also called direct product(teser product) of G ad H,ad may be deoted by G.H []. G H is also defied as V( G H )=V(G )xv( H ) E( G H )={(p i, q ) (p k, q l ) p i p k E(G) ad q q l E(H)}. The Kroecker product is commutative (up to isomorphism) ad associative [7]. 65

3 Ghassa T. Marougi & Ayha A. Khalil The first importat result of the basis umber occured i 937 whe MacLae [5] proved that a graph G is plaar if ad oly if b(g) 2. I 98, Schmeichel [6] proved that for 5,b(K )=3, ad for m, 5, b(k m, )=4 except for K 6,0, K 5, ad K 6, i which =5,6,7 ad 8. Moreover, i 982, Baks ad Schmeichel [2] proved that for 7, b(q )=4, where Q is the -cube. The purpose of this paper is to determie upper ad lower bouds of the basis umber of Kroecker product of a wheel with a path ad a cycle. 2.. O the Basis Number Of Wm. I this sectio, we obtai upper ad lower bouds for the basis umber of kroecker product of a wheel with a path.let the vertex sets of C m ad P be Z m ad Z respectively, where Z deotes the additive group of residues modulo. Let the cycle C m be 0,,2,,m-,0. The followig lemma is eeded i the proof of the followig theorem which is due to Weichsel [8]. Lemma:If G ad H are coected graphs the the Kroecker product G H is coected if ad oly if either G or H cotais a odd cycle. Let be the oi of a cycle 0 2 L( m 2) 0 with the vertex α ad let P = 0 2L ( ).( See Fig.) α (m-2) Fig P Theorem 2. For m 4 ad 3, 3 b ( ) 4. Proof: Oe ca easily observe from Fig.2, that P3 cotais a subgraph H homeomorphic to K 3,3. Thus by Kuratowskis theorem [3], P 3 is o plaar, sice P3 is a subgraph of Wm for 3 ; the by MacLaes theorem [5], b ( P ) 3. 66

4 UPPER AND LOWER BOUNDS OF THE BASIS α (m-3)0 (m-2)0 α 0 (m-3) α 0 (m-3) (m-2) 00 (m-2) α (m-3)2 (m-2)2 α2 (m-2)2 (m-2)0 P 3 H Fig.2: P3 To complete the proof we fid a 4-fold basis ) for ξ ( Wm We have two cases: Case (): m is eve. Let B = Cm ) N P M,. where C m ) is the basis for ξ ( Cm ) discussed i Theorem 2.2.[4] i which m- is odd, that is, Cm ) = { i,( i )( + ), i( + 2),( i + )( + ), i : i Z m ad = 0,,..., 3} { S}, where S = 00,,20,3,..., ( m 2)0,0,0,2,30,..., ( m 2),00, N = { i,( i + )( + ), α, i( + ),( i + ), + ), i : i = 0,,2,..., m 3 ad = 0,,2,..., 2}, P = {( m 2),0( + ), α,( m 2)( + ),0, + ),( m 2) : = 0,,..., 3} ad M = { α, i( + ), ( i + ), ( i + 2)( + ), α ad + ), i, ( i + )( + ), ( i + 2), + ): i = 0,2,4,..., m 4 ad = 0,,..., 2}. It is clear that B = ( m )( 2) + + ( m 2)( ) + ( 2) + ( m 2)( ) = 3m 4m = γ ( P We shall prove that B is idepedet. First, the cycles of N P ad M are idepedet for each = 0,,..., 2 sice ay liear combiatio of cycles i N P or M for some i = 0,,..., m 2 cotais edges of the form, i,( i + )( + ) or i( + ),( i + ). That is, ay liear combiatio of cycles i N P ad M is ot equal to zero modulo (2). Moreover, for all = 0,,..., 2, every cycle of N P cotais a edge of the form α, i( + ) or + ), i for some 67

5 Ghassa T. Marougi & Ayha A. Khalil i =,3,5,..., m 2 which is ot preset i ay cycle of M. Also the cycles i N P M satisfy N P M ) ( N P M ) = Φ for all k where ( k k k N is defied as follows: It is clear that the vertex set of Wm ca be partitioed ito V 0, V,..., V, where V = {( i, ) : i = α,0,,2,..., m 2}. Notice that V ( Wm ) = { α,0,,2,..., m 2}. Now, N is the cycle of N that oi a vertex of V to a vertex of V +, for each = 0,,..., 2. By a similar method, we defie P ad M. Moreover, for every ocosective itegers ad k i {0,,,-2}, every cycle i N P M is edge-disoit with every cycle i N k Pk M k. Furthermore, if C i is ay cycle i N P M, = 0,,..., 3 the C i cotais the edge i,( i + )( + ) which is ot cotaied i ay cycle i N + P+ M +. This shows that N P M is idepedet. Moreover, the cycles of N P M for all = 0,,..., 2 are idepedet from the cycle of C m ) because if C is i ay cycle geerated from cycles i N P M, the C cotais i a edge of the form α, i( + ) or + ), i for some i = 0,,..., m 2 which is ot preset i ay cycle of C m ). Thus ) is idepedet set of cycles ad so it is a basis for ξ ( Wm We ow cosider the fold of ). Partitio, the edge set of Wm P ito i,( i + )( + ) or i( + ),( i + ) ad α, i( + ) or + ), i i which i Z m ad = 0,,..., 2. Thus if e is ay edge i Wm of the form i,( i + )( + ) or i( + ),( i + ), the 2,, C m P ) N P M ad so 4. P ) While, if e is ay edge i Wm of the form α, i( + ) or + ), i,the = 0, C m P ad so ) N P 2, 2 M 68

6 UPPER AND LOWER BOUNDS OF THE BASIS f ( e) 4. P ) Therefore, ) is a 4-fold basis. Case (2): m is odd. Let Wm ) = B ( Cm ) M N. Where B ( C m P ) is a basis for ξ ( Cm ) discussed i[,theorem, case(2)] amely, B ( C P ) = { i, ( i + )( ),( i + 2), ( i + )( + ), i m : i Z m, =,2,..., 2 ad ( i) is { i, ( i + )( ),( i + 2), ( i + )( + ), i : i Z m, =,2,..., 2 ad ( i) is eve} odd}, M = { α, i( + ), ( i + ), ( i + 2)( + ), α ad + ), i, ( i + )( + ), ( i + 2), + ) : i = 0,2,4,... m 3 mod( m ) ad = 0,,2,..., 2} ad N is same as Case (). It is clear that Wm ) = ( m )( 2) + ( m )( ) + ( m 2)( ) = m 2m m m + + m m = 3m 4m = γ ( Wm As i the proof of Case (), we ca show that ) is idepedet set of cycles ad so it is a basis for ξ ( Wm ) of fold 4,that is B = P Note that if m is eve ad m 4,the P2 is plaar graph,(see Fig.3). 69

7 Ghassa T. Marougi & Ayha A. Khalil (m-2) α0 3 α (m-2)0 (m-3) Fig.3 Hece b( Wm P2 ) = 2 for eve m 4. While, if m is odd ad m 5, the P2 cotais a subgraph K homeomorphic to K 3, 3. Therefore by MacLaes theorem [5],the graph is oplaar (see Fig.4). P 2 70

8 UPPER AND LOWER BOUNDS OF THE BASIS α (m-4)0 (m-3)0 (m-2)0 α 0 2 (m-4) (m-3) (m-2) K α0 0 (m-2) (m-4)0 α 0 (m-3) Fig.4: P2 Hece b( Wm P2 ) = 3 for eve m 5. We coclude the followig table m b P ) ( m 4, m is eve 2 2 m 4, m is eve 3 3 or 4 m 5, mis odd 2 3 m 5, mis odd 3 3 or 4 7

9 Ghassa T. Marougi & Ayha A. Khalil 2.2. O the Basis Number Of Wm. I this sectio, we obtai upper ad lower bouds for the basis umber of kroecker product of a wheel with a cycle. Theorem 3. For m 4, 3, we have 3 b ( ) 5. Proof: Sice Wm is a subgraph of Wm for all m 4,ad 3, the by Theorem 2, we have Wm is oplaar ad so by MacLaes theorem [5], we have b ( C ) 3.For m=4 ad =3,( see Fig.5). α α 0 2 α Fig.5: W4 P3 To complete the theorem we establish a 5-fold basis ) for ξ ( Wm We have two possibilities for m. () m is eve. The cosider the followig set of cycles i Wm : Wm ) = Cm ) N M. Where C m ) is a basis for ξ ( Cm ) discussed i Theorem 2.3.[4], where m- is odd, that is, Cm ) = Cm ) B { S, S 2}, where Cm ) = { i,( i )( + ), i( + 2),( i + )( + ) : i Z m ad = 0,,..., 3} { S}, S = 00,,20,3,...,( m 2)0,0,0,2,30,...,( m 2),00 B = { i, ( i )( + ), i( + 2), ( i + )( + ), i : i = 0,,..., m 3 mod( m ) ad = 2, (mod )}, S = ( m 2)( 2),( m 3)( ),( m 2)0, 0( ),( m 2)( 2) S = 00,( m 2)( ),( m 3)0,( m 4)( ),...,0,0( ),( m 2)0,( m 3)( ),...,20,( N = { i,( i + )( + ), α, i( + ),( i + ), + ), i : i Z m ad Z }. ad 2 ),00. 72

10 UPPER AND LOWER BOUNDS OF THE BASIS M = { α, i( + ),( i + ),( i + 2)( + ), α ad + ), i,( i + )( + ),( i + 2), + ) : i = 0,2,4,..., m 4 ad It is clear that W m Z }. ) = m + + ( m ) + ( m 2) = 3m 4 + = γ ( W C ) m We will prove that B C ) is idepedet. It is clear that 2 ( N = ( N P ) { N } ad = ( M ) { M } = 0 2 M, where N P M for = 0 = 0,,..., 2 are as metioed i the proof of Theorem 2. As i the proof of Theorem 2, N M is idepedet. Moreover for all i Z m, N M cotais the edge i ( ),( i + ) 0, which is ot cotaied i 2 ( = 0 N P M Thus N M is idepedet set of cycles. Furthermore N M is idepedet from the cycles of C m C ) sice for all Z, if Ci is ay cycle geerated from cycles of N M, the C i cotais the edge of the from α, i( + ) or + ), i for some i Z m which is ot preset i ay cycle of C m Thus Wm ) = Cm ) N M, is idepedet ad so it is a basis. We ow cosider the fold of Partitio the edge-set of Wm C ito i,( i + )( + ) or i( + ),( i + ) ad α, i( + ) or + ), i for i Z m ad Z. Therefore if e is ay edge i Wm of the form i,( i + )( + ) or i( + ),( i + ), the 3,, C m C ) N M ad so 5. C ) While if e is ay edge of the form α, i( + ) or + ), i the = 0, 2, 2 C m C ) N M ad so 4. C ) Thus, the basis B C ) is of fold 5. ( 73

11 Ghassa T. Marougi & Ayha A. Khalil (2) m is odd, the cosider the followig set of cycles i Wm : Wm ) = B ( C P ) { F, F m i i : i Z m } N M, where B ( C m P ) is a basis for ξ ( Cm ) metioed i [, Theorem, case (2)] ad F i, F i are idepedet cycles [, Theorem 2, case ()]. That is, B ( C m P ) = { i, ( i + )( ), ( i + 2), ( i + )( + ), i : i Z ad ( i) eve } { i, ( i + )( ), ( i + 2), ( i + )( + ), i : F m i Z =,2,..., 2 ad ( i) odd } {00,,20,3,..., ( m 3)0, ( m 2),00}, i = {0i,( i ),2i,...,( m 2)( i ),0 i}, F = {0i,( i + ),2i,...,( m 2)( i + ),0 i}, N i ad M, =,2,..., 2 m { i,( i + )( + ), α, i( + ),( i + ), + ), i : i = 0,,..., m ad } = 3 = { α, i( + ),( i + ), ( i + 2)( + ), α ad + ), i, ( i + )( + ),( i + 2), + ) : i = 0,2,4,... m 3 mod( m ) ad Z }. is clear that W C ) = ( m )( 2) + + 2( m ) + ( m 2) + ( m ) m = m 2m m 2 + m 2 + m = 3m 4 + = γ ( Wm ). As i possibility (), we ca prove that ) is a 5-fold basis for ξ ( Wm Remark. I cotrast to upper bouds of the basis umbers of Wm ad Wm C give i Theorem 2 ad Theorem 3, oe ca coecture that the upper boud for the basis umber of kroecker product of two wheels ad W is b ( W ) 0. Coecture: (i) What is b( K m K )? where ad W are subgraphs of K m ad K. resp. (ii) What did you coecture about b G G )? ( 2, Z It 74

12 UPPER AND LOWER BOUNDS OF THE BASIS REFERENCES. [] A.A.Ali, The basis umber of the direct product of paths ad cycles, Ars combiatoria, Vol.27 (989), pp [2] J.A.Baks ad E.F.Schmeichel, The basis umber of the -cube, J.Comb.Theory, Ser.B, Vol.33, No.2, (982), pp [3] F.Harary, Graph Theory, 3 rd. pritig, Addiso-Wesely, Readig, Massachusetts, (972). [4] A.A.Khaleel, O the basis umber of Kroecker product of some special graphs,m.sc.thesis,mosul Uiversity,(2004). [5] S.MacLae, A combiatorial coditio for plaar graphs, Fud. Math., Vol.28, (937), pp [6] E.F.Schmeichel, The basis umber of a graph,j.comb.theory, Ser.B, Vol.30,No.2 (98), pp [7] H.H.Teh, ad H.P., Yap, Some costructio problems of homogeous graphs, Bull. Math. Soc., Nayag Uiv., (964), pp [8] P.M.Weichsel, The kroecker product of graphs, Proc,Amer. Math.Soc., Vol.3, (962), pp