AN ALGORITHM FOR ODD GRACEFULNESS OF THE TENSOR PRODUCT OF TWO LINE GRAPHS

Transcription

1 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 AN ALGORITHM FOR ODD GRACEFULNESS OF THE TENSOR PRODUCT OF TWO LINE GRAPHS M. Ibrah Moussa Faculty of Coputers & Inforaton, Benha Unversty, Benha, Egypt oussa_6060@yahoo.co ABSTRACT An odd graceful labelng of a graph G ( V, E ) = s a functon f : V ( G) {0,,,... E( G) } such that f ( u) ) s odd value less than or equal to E( G) for any u, v V ( G ). In spte of the large nuber of papers publshed on the subect of graph labelng, there are few algorths to be used by researchers to gracefully label graphs. Ths work provdes generalzed odd graceful solutons to all the vertces and edges for the tensor product of the two paths P n and P denoted P n P. Frstly, we descrbe an algorth to label the vertces and the edges of the vertex set V ( P P ) and the edge set E ( P P ) respectvely. Fnally, we prove that the graph Pn P s odd graceful for all ntegers n and. KEY WORDS Vertex labelng, edge labelng, odd graceful, Algorths. n n. INTRODUCTION Let G s a fnte sple graph, whose vertex set denoted V ( G), and the edge set denoted E ( G) The order of G s the cardnalty n = V ( G ) and the sze of G s the cardnalty q = E ( G). We wrte uv E ( G) f there s an edge connectng the vertces u and v n G. A path graph P n sply denotes the graph that conssts of a sngle lne. In other words, t s a sequence of vertces such that fro each of ts vertces there s an edge to the next vertex n the sequence. The frst vertex s called the start vertex and the last vertex s called the end vertex. Both of the are called end or ternal vertces of the path. The other vertces n the path are nternal vertces. The tensor product of two graphs G andg denoted G G ts vertex set denoted V ( G G) = V ( G) ` V ( G) consder any two ponts u = ( u, u), and v = ( v, v ) n V ( G G ) the edge set denoted E ( G G ) where E ( G G ) = {( u, v )( u, v ): u u E ( G ) & v v E ( G ) } DOI : 0.5/graphoc.0.30

2 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 The tensor product s also called the drect product, or conuncton. The tensor product was ntroduced by Alfred North Whtehead and Bertrand Russell n ther Prncpa Matheatca [3]. The graph G = ( V, E) conssts of a set of vertces and a set of edges. If a nonnegatve nteger f ( u) s assgned to each vertex u, then the vertces are sad to be labeled. G = ( V, E ) s tself a labeled graph f each edge e s gven the value f ( e) = f ( u) ) where u and v are the endponts of e. Clearly, n the absence of addtonal constrants, every graph can be labeled n nfntely any ways. Thus utlzaton of labeled graph odels requres poston of addtonal constrants whch characterze the proble beng nvestgated. An odd graceful labelng of the graph G wth n = V ( G ) vertces and q = E ( G) edges s a one-toone functon f of the vertex set V ( G) nto the set{0,,,..., q } wth ths property: f we defne, for any edge uv the functon f ( uv ) = f ( u) ) the resultng edge label are{,3,q }.A graph s called odd graceful f t has an odd graceful labelng. The odd graceful labelng proble s to fnd out whether a gven graph s odd graceful, and f t s odd graceful, how to label the vertces. The coon approach n provng the odd gracefulness of specal classes of graphs s to ether provde forulas for odd gracefully labelng the gven graph, or construct desred labelng fro cobnng the faous classes of odd graceful graphs. The study of odd graceful graphs and odd graceful labelng defnton was ntroduced by Gnanaoth [9], she proved the followng graphs are odd graceful: the graph C K s oddgraceful f and only f even, and the graph obtaned fro Pn P by deletng an edge that ons to end ponts of the P n paths, ths last graph knew as the ladder graph. She proved that every graph wth an odd cycle s not odd graceful. Ths labelng has been studed n several artcles. In 00 Moussa [9] have presented the algorths that showed the graph Pn C s odd graceful f s even. For further nforaton about the graph labelng, we advse the reader to refer to the brllant dynac survey on the subect ade by J. Gallan n hs dynac survey [8]. In ths paper, we frst explctly defned an odd graceful labelng of Pn P, Pn P3, Pn P4 and Pn P5 then, usng ths odd graceful labelng, descrbed a recursve procedure to obtan an odd graceful labelng of the graph P n P. Fnally; we presented an algorth for coputng the odd graceful labelng of the tensor graph P n P, the correctness of the algorth was proved at the end of ths paper. The reander of ths paper s organzed as follows. In secton, we gave a nce set of applcatons related wth the odd graceful labelng. In secton 3 defned the tensor graph of two path graphs, and then drew the tensor graph Pn P on the plane. Secton 4 descrbed the algorth of odd graceful labelng of the tensor graph P n P and then proved the correctness of the algorth. The coputatons cost s coputed at the end of ths secton. Secton 5 s the concluson of ths research.

3 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0. THE APPLICATION RANGES The odd graceful labelng s one of the ost wdely used labelng ethods of graphs. Whle the labelng of graphs s perceved to be a prarly theoretcal subect n the feld of graph theory and dscrete atheatcs, t serves as odels n a wde range of applcatons. The codng theory: The desgn of certan portant classes of good non perodc codes for pulse radar and ssle gudance s equvalent to labelng the coplete graph n such a way that all the edge labels are dstnct. The node labels then deterne the te postons at whch pulses are transtted. The x-ray crystallography: X-ray dffracton s one of the ost powerful technques for characterzng the structural propertes of crystallne solds, n whch a bea of X-rays strkes a crystal and dffracts nto any specfc drectons. In soe cases ore than one structure has the sae dffracton nforaton. Ths proble s atheatcally equvalent to deternng all labelng of the approprate graphs whch produce a prespecfed set of edge labels. The councatons network addressng: A councaton network s coposed of nodes, each of whch has coputng power and can transt and receve essages over councaton lnks, wreless or cabled. The basc network topologes are nclude fully connected, esh, star, rng, tree, bus. A sngle network ay consst of several nterconnected subnets of dfferent topologes. Networks are further classfed as Local Area Networks (LAN), e.g. nsde one buldng, or Wde Area Networks (WAN), e.g. between buldngs. It ght be useful to assgn each user ternal a node label, subect to the constrant that all connectng edges (councaton lnks) receve dstnct labels. In ths way, the nubers of any two councatng ternals autoatcally specfy (by sple subtracton) the lnk label of the connectng path; and conversely, the path label unquely specfes the par of user ternals whch t nterconnects [0]. Cellular networks are ost successful coercal applcaton of wreless networks. In a cellular network, a servce coverage area s dvded nto saller square or hexagonal areas referred to as cells. Each cell s served by a base staton. The base staton s able to councate wth oble statons such as cellular telephones, usng ts rado transcever. A oble swtchng center (MSC) connects the base staton wth the publc swtched telephone network (PSTN). There are two possble cell confguratons to cover the servce area, the frst s a hexagonal confguraton whch s coonly used where each cell has sx neghborng cells and the second s a esh confguraton whch s ore sple where four neghbors can be assued for each cell (horzontal and vertcal ones) [,, 3, 4]. Mesh nets are consdered good odels for largescale networks of wreless sensors that are dstrbuted over a geographc regon. Many cellular systes operate under constrants. One of these constrants s nterodulaton constrant, whch nterferes wth the use of frequency gaps that are ultples of each other [5]. The channel assgnent under nterodulaton constrant s related to graceful labelng of graphs [0]. The challenge concernng channel assgnent s to gve axal channel reuse wthout volatng the 3

4 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 constrants so that blockng s nal [6]. Fgure 0 llustrates a tensor graph P n P wth each vertex n the set V( Pn P ) represents a base staton n a cell and the edges represent geographcal adacency of cells. The black dots n ths fgure are the vertces of the tensor graph, whch correspond to the base statons, and the connected black lnes are the edges of the tensor graph, whch correspond to adacent cells n the network. Fgure 0. Two-densonal network topology wth the esh confguraton Rado Channel Assgnents: The vertex-labelng of graphs wth nonnegatve ntegers provdes a natural settng to study probles of rado channel assgnent.the frequency spectru allocated to wreless councatons s very lted, so the cellular concept was ntroduced to reuse the frequency. Each cell s assgned a certan nuber of channels. To avod rado nterference, the channels assgned to one cell ust be dfferent fro the channels assgned to ts neghborng cells. However, the sae channels can be reused by two cells that are far apart such that the rado nterference between the s acceptable. By reducng the sze of cells, the cellular network s able to ncrease ts capacty, and therefore to serve ore subscrbers. The notaton offset s needed when t uses the graph labelng to fnd channel assgnents to forbd repeated labelng. In 000 Chartrand, Erwn, Harary, and Zang descrbed the graph labelngs wth specfc constrants related to the daeter of the graph are called rado labelngs [7]. On the other sde, soe of the algorths for channel assgnent wth general constrants are based on graph labelng. For further nforaton about the algorths for channel assgnent wth general constrants, we advse the reader to go to [0] For further nforaton about other applcatons of labeled graphs, such applcatons nclude radar, crcut desgn, astronoy, data base anageent, on autoatc drllng achne, deternng confguratons of sple resstor networks and odels for constrant prograng over fnte doans (see e.g., [, 4, 5, 7 and 8]). Labeled graph also apply to other areas of atheatcs (see e.g., [, 6]). 4

5 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 3.THE TENSOR OF TWO PATHS The tensor graph Pn P has a total nuber of vertces equals n and ( n )( ) edge. Let l, l,..., l be dstnct parallel lnes lyng n a plane. Let s, s,..., s be nfnte set of ponts on l, l,..., l respectvely. We consder the vertex set of the graph Pn P s parttoned nto vertex sets V, V,..., V drawn so that the vertces le on dstnct consecutve ponts of s, s,..., s respectvely. An arbtrary nuber =,,..., the set s has the vertces v for all =,,..., n, ths representaton shown n Fgure. v v v v 3 v 3 v vn v n vn v n 3 3 vn v n v v v n v n v v v n Fgure the graph P n P 4.VARIATION OF ODD GRACEFUL LABELING OF n P 4.. Odd Gracefulness of n P Theore Pn P s odd graceful for every nteger n >. Proof As shown n the above secton, let the vertex set of the graph Pn P s parttoned nto two dsont sets V and V and drawn so that the vertces le on dstnct consecutve ponts of s and srespectvely. The total nuber of vertces equals n and total nuber of edges equals ( n ). For any vertex v s and v s the odd graceful labelng functons ) : s {0,,,..., 4n 3} and f defned respectvely by ( v ): s {0,,,...,4 n 3} P P v n ) = odd n even f v ( ) odd = (4n 3) even 5

6 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 The weghts of the vertces n the set s belong to the even nubers 0,, 4,,4(n-)- and the weghts of the vertces n the set s belong to the odd nubers, 3, 5,,4(n-)-. The edge s labelng nduced by the absolute value of the dfference of the vertex s labelng. The reader can easly fnd out, that the vertex labels are dstnct and all the edge labels are dstnct odd labels. Thus, f s odd graceful labelng of Pn P. Fgure shows the ethod labelng of the graph P P ths coplete the proof Odd gracefulness of Pn P3 Fgure the graph P8 P Theore Pn P3 s odd graceful for every nteger n >. Proof The vertex set of the graph Pn P3 s parttoned nto three dsont sets V, V and V3 drawn so that the vertces le on dstnct consecutve ponts of s, s and s 3 respectvely. The total nuber of vertces equals 3n and total nuber of edges equals 4( n ). For any vertex 3 v s, v s and v s f defned respectvely by 3 ( v ) 3 the odd graceful labelng functons f ( ), odd odd 3 (4n 5) + odd ) = ) = ) = (6n 4) even (8n 7) even n even v f ( v ) The weghts of the vertces n the set s and s3 are the even nubers 0,, 4,, 8(n-)- and the weghts of the vertces n the set s are the odd nubers, 3, 5,,8(n-)-. The edge s labelng nduced by the absolute value of the dfference of the vertex s labelng. The reader can easly fnd out, that the vertex labels are dstnct and all the edge labels are dstnct odd labels. Thus, f s odd graceful labelng of Pn P3. Fgure3 shows the ethod labelng of the graph P 8 P 3 ths coplete the proof. and 6

7 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March Odd gracefulness of Pn P4 Fgure 3 the graph p8 p3 Theore 3 Pn P4 s odd graceful for every nteger n >. Proof: The vertex set of the graph Pn P4 s parttoned nto four dsont sets V, V, V3 and V 4 drawn so that the vertces le on dstnct consecutve ponts of s, s, s 3 and s4 respectvely. The total nuber of vertces equals 4n and total nuber of edges equals 6( n ). For any vertex 4 3 v s, v s, v s and v the odd graceful labelng functons f ( ), ), ) 3 3 s 4 4 and f ( ) defned respectvely as follows: v o d d o d d ) = ) = ( 0 n 8 ) e v e n ( n ) e v e n 3 4 n + 5 o d d 4 ( n ) + o d d ) = ) = 6 n 4 e v e n 6 n 5 e v e n The weghts of the vertces n the set s and s3 are the even nubers 0,, 4,, (n-)- and the weghts of the vertces n the set s and s4 are the odd nubers, 3, 5,,(n-)-. The edge s labelng nduced by the absolute value of the dfference of the vertex s labelng. The reader can easly fnd out, that the vertex labels are dstnct and all the edge labels are dstnct odd labels. Thus, f s odd graceful labelng of Pn P4. Fgure4 shows the ethod labelng of the graph P 8 P 4 ths coplete the proof. v Fgure 4 the graph p8 p4 7

8 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March Odd gracefulness of Pn P5 Theore 4 Pn P5 s odd graceful for every nteger n >. Proof The vertex set of the graphpn P5 s parttoned nto four dsont sets V, V, V3, V4 andv 5 drawn so that the vertces le on dstnct consecutve ponts of s, s, s3, s 4 and s5 respectvely. The total nuber of vertces equals 5n and total nuber of edges equals 8( n ). For any vertex v, v, v, v and v 5 s 5 the odd graceful labelng 3 4 s s s 3 s 4 functons f ( ), f v v 5 f f f and f ( ) defned respectvely as follows: 3 4 ( v ), ( v ), ( v ) 3 ( ) ) = ) 4 ( ) f v odd odd 4n + 5 odd = = (4n ) even (6n 5) even 0n 8 even n + odd = 0n 9 even v 5 8n + 9 odd ) = 4( r 3)( n ) (n 4) even The weghts of the vertces n the set s, s 3 and s5 are the even nubers 0,, 4, 6,,6(n-)- and the weghts of the vertces n the set s and s4 are the odd nubers, 3, 5,,6(n-)-. The edge s labelng nduced by the absolute value of the dfference of the vertex s labelng. The reader can easly fnd out, that the vertex labels are dstnct and all the edge labels are dstnct odd labels. Thus, f s odd graceful labelng of Pn P5. Fgure5 shows the ethod labelng of the graph P P5 ths coplete the proof. 8 Fgure 5 the graph p8 p5 8

9 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 Theore 5 Pn P6 s odd graceful for every nteger n >. Proof The vertex set of the graph Pn P6 s parttoned nto four dsont sets V, V, V3, V4, V5 and V6 drawn so that the vertces le on dstnct consecutve ponts of s, s, s3, s, s 5 and s6 respectvely. The total nuber of vertces equals 6n and total nuber of edges equals0( n ) For any vertex v s, v s, v s3, v s4, v s5 and v 6 s6 the odd graceful labelng functon f v, ( ) 6 f v f v f v f v and ) defned respectvely as follows: ( ), ( ), ( ), ( ) o d d o d d ) = ) = ( 8 n 6 ) e v e n ( 0 n 9 ) e v e n 3 4 n + 5 o d d 4 ( n ) + o d d ) = ) = 4 n e v e n 4 n 3 e v e n 5 6 n + 7 o d d 6 4 n + 4 o d d ) = ) = 8 n 6 e v e n 8 n 7 e v e n The weghts of the vertces n the set s, s 3 and s5 are the even nubers 0,, 4, 6,, 0(n-)- and the weghts of the vertces n the set s, s 4 and s 6 are the odd nubers, 3, 5,,0(n-)-. The edge s labelng nduced by the absolute value of the dfference of the vertex s labelng. The reader can easly fnd out, that the vertex labels are dstnct and all the edge labels are dstnct odd labels. Thus, f s odd graceful labelng of Pn P6. Fgure6 shows the ethod labelng of the graph P 7 P 6 ths coplete the proof. 4 Fgure 6 the graph P7 P6 9

10 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March The proposed sequental Algorth We are ntroducng an algorth for the odd graceful labelng of the tensor graph P n P. The algorth runs n three passes; they and ther steps can run n a sequental or a parallel way. In the frst pass, the algorth coputes the odd label functon for the vertces le on lne l, see algorth for the ntal procedure.. Draw the graph P n P so that the vertces V, V,..., V le on dstnct consecutve ponts of l, l,..., l respectvely. For each vertex v on the frst level l nuber the vertex as ) = ( o d d ) ) = 4 ( ) ( n ) ( n ) ( e v e n ) Algorth : Procedure Intalzaton In the second pass, the algorth labels the vertces on the lnes l, l 3, l 4,... l, t labels the vertces on the lne l wth the labels, 4 ( n ) ( 4 n 3 ), 3, 4 ( n ) ( 4 n ),..., 4 ( n ) + ( ( o d d ) ), 4 ( ) ( n ) ( n 4 ) ( e v e n ),..., It labels the vertces on the lne l 3 wth the labels 4 ( n ), 4 ( n ) 6 ( n ), 4 n, 4 ( n ) 6 n,..., 4 ( n ) + ( ( o d d ) ), 4 ( ) ( n ) ( n 4 ) ( e v e n ),... The algorth traverses to the next lnes and t repeats prevous processes as shown n algorth untl vsts all the vertces on the lne l. If s odd and ore than 3; the vertces le on the lne l assgn values accordng to step 3 n algorth, otherwse f 3 these vertces assgn values accordng to functons n steps, n algorth. 0

11 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0. For each level l, s odd and ore than three, nuber ts vertces by ) = ( + ) ( n ) + ( ) ( o d d ) ) = 4 ( ) ( n ) ( ( ) n ( + ) ) ( e v e n ). For each level l, s even and ore than or equal two, nuber ts vertces ) = ( ) ( n ) + ( o d d ) ) = 4 ( ) ( n ) ( ( ) n ( ) ) ( o d d ) 3. If ( s odd and > 3 ) v by ( + 3 ) n ( + 4 ) + o d d ( f v ) = 4 ( n ) ( ( 3 ) n ( ) ) e v e n Algorth,,,,... In the last pass, the label s dstrbuton of the edge between the lnes l l l 3 l 4 l s nduced by the absolute value of the dfference of the vertex s labelng; the followng algorth gves the odd edge labelngs.. The edge s labels between the lne l and the lne l, nduced by + v ) = 4 ( ) ( n ) ( ) ( o d d ) v ) = 4 ( ) ( n ) ( 3 ) ( o d d ) v ) = 4 ( ) ( n ) ( n ) ( ) ( e v e n ). The label s dstrbuton of the edge between the lnes l, l 3,... l s defned consecutvely by + + v ) = ( )( n ) ( ) (, o d d ) + v ) = ( )( n ) ( 3 ) (, o d d ) + + v ) = 4 ( )( n ) ( ) (, e v e n ) + v ) = 4 ( )( n ) ( 3 ) (, e v e n )

12 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 3. If ( s odd and > 3 + v ) = 4 n 3 ( o d d ) v ) = 4 ( n ) ( 3 ) ( o d d ) + v ) = n ( e v e n ) v ) = n ( ) ( e v e n ) 4. Muffle all the edge s labels that contan a vertex not n the vertex set V( P P ) vz. : v + ) when = n, and + 5. The resultng labelng s odd graceful. v ) when =, for all =,,..., + Algorth 3 Theore 6 The tensor graph P n P s odd graceful for all n and. Proof: The vertex set of the graph P n P parttoned nto the vertex sets V = { v, v,..., v }, V = { v, v,..., v },..., V = { v, v,..., vn } le on dstnct consecutve ponts n n l, l,..., l respectvely, wth edges only between the vertces n of V + ( le on lne l + ) for all n V (le on lne l )and, so the graphs P n P are -partte graphs. The vertces le on the lne l are labeled consecutvely wth even values fro the followng nuberng f v ) = 4 ( )( n ) ( n ) =, 4, ( ) =, =, 3, 5,. The vertces le on the odd lnes l, = 3,5,... are labeled consecutvely wth even values fro the followng nuberng ) = ( + )( n ) + ( ), =, 3,5, ) = 4( )( n ) (( ) n ( + ) ), =, 4,6,... The vertces le on the even lnes l, =,4,6,... are labeled consecutvely wth odd values fro the followng nuberng ) = ( )( n ) +, =,3,5,. ) = 4( )( n ) (( ) n ( )), =, 4, 6,..

13 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 Maxu and Mnu Values between the resultng vertex labels ) are; ) = ax ax ), ax ), ax ) = 4( n )( ) = q n n n ) = n n ), n ), n ) = 0 n n n The reader can fnd out easly that the functon f s one to one the vertex set of P n P, and the vertex labels are assgned unquely as shown n algorth. Thus, the labels functon ) {0,,,..., q } for all n,. Because the edges only between the vertces n V, V 3, V 5,..., (are labeled wth even values) and n V, V 4, V 6,..., (are labeled wth odd values), then the absolute values of the dfference of the vertex s labelngs are odd value as shown n algorth 3. It reans to show that the labels of the edges of Pn P are all the ntegers of the nterval [, q-] = [, 4( n )( ) ]. To prove that all edges labels are dfferent, wthout loss of generalty, suppose that n s even nuber, we have to consder the followng cases () Step of Algorth 3; { ± } ax ax v ) = 4( n )( ) = q f = n { ± } n n v ) = 4( n )( ) + f = n n The edges labels are odd, dstnct and are nubered between the above axu and nu values accordng to the decreasng ( = n s odd value) 4( n )( ),..., ( n )( 3) + 3,( n )( 3) +,..., ( n )( 3) 3,..., 4( n )( ) + () Step of Algorth 3; + ax ax v± ) = v ) = 4( n )( ) n + n n v ± ) = n v n ) = n 3

14 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 The edges labels are odd, dstnct and are nubered between the above axu and nu values accordng to the decreasng ( = n s odd value) 4( n )( ), 4( n )( ) 3,..., ( n )( 7),..., ( n )( 7) 5,..., 3,. () Step 3 of Algorth 3; ths step s runnng only f n s odd nuber, so t nduced a subset of the edge s labelng, whch nduced n Step. { ± } n ax ax v ) = v ) = 4n 5 = q { } n n ± n n n v ) = v ) = The edges labels are odd, dstnct and are nubered between the above axu and nu values accordng to the decreasng sequence 4n 5, 4n 7,..., n 3, n 5,..., n,..., 3,. We consder ths subset v± ), n of the edge s labelng nduced n Step 3 f and only f n s odd nuber, and we consder the subset v ± ), n of the edge s labelng nduced n Step f and only f n s even nuber. It s easy to see that n any of the above cases, the edge labels of the graph Pn P for all and n are all dstnct odd ntegers of the nterval [,q-]= [, 4( n )( ) ] the above algorth gves an odd gracefulness of Pn P The algorth s traversed exactly once for each vertex n the graph P n P, snce the nuber of vertces n the graph equals n then at ost O ( n) te s spent n total labelng of the vertces and edges, thus the total runnng te of the algorth s O ( n). The parallel algorth for the odd graceful labelng of the graph P n P, based on the above proposed sequental algorth s buldng easly. Snce all the above three algorths,,and 3 are ndependent and there s no reason to sort ther executng out, so they are to on up parallel n the sae te pont. 5. CONCLUSION It s desred to have generalzed results or results for a whole odd graceful class f possble. But tryng to fnd a general soluton, t frequently yelds specalzed results only. Ths work presented the generalzed solutons to obtan the odd graceful labelng of the graphs obtaned by tensor product Pn P of two path graphs. After we ntroduced an odd graceful labelng of Pn P, Pn P3, Pn P4 and Pn P5, we descrbed a sequental algorth to label the vertces and the edges of the graph Pn P. The sequental algorth runs n lnear wth total runnng te equals O ( n). The parallel verson of the proposed algorth, as we showed, exsted and t s descrbed shortly. 4

15 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 REFERENCES [] A. Brauer, A proble of addtve nuber theory and ts applcaton n electrcal engneerng, J. Elsha Mtchell Scentfc SOC., VOl. 6, pp , August 945. [] J. Leech, On the representaton of,,...,n by dfferences. J.LondonMath. Soc., vol. 3, pp , Apr [3] Whtehead, Alfred North, and Bertrand Russell (90, 9, 93) Prncpa Matheatca,vol3,Cabrdge: Cabrdge Unversty Press. nd edton, 95 (Vol. ), 97 (Vols, 3). [4] D. Fredan, Proble 63-., A Sharp autocorrelaton publshed functon, SIAM Rev., vol. 5, p. 75, July 963. Solutons and coents, SIAM Rev., vol. 7, pp , Apr. 965; SIAM Rev., vol. 9, pp , July 967. [5] L. H. Harper, DSIF ntegrated crcut layout and soperetrc probles JPL Space Progras suary 37-66, vol., pp. 37-4, Sept [6] J. C. P. Mller, Dfference bases, three probles n addtve nuber theory, n Coputers n Nuber Theory, A. D. L. Atkn and B. J. Brch, Eds. London: Acadec Press, 97, pp [7] J.A. Bondy, U.S.R. Murty, Graph Theory wth Applcatons, Elsever North-Holland, 976. [8] Bloo, G. S. and Golob, S. W. Applcatons of Nubered Undrected Graphs, Proceedngs of the Insttute of Electrcal and Electroncs Engneers 65 (977) [9] R.B. Gnanaoth (99), Topcs n Graph Theory, Ph.D. Thess, Madura Kaara Unversty, Inda. [0] Dorne R. and Hao J.-K., Constrant handlng n evolutonary search: A case study of the frequency assgnent, Lecture Notes n Coputer Scence, 4, 80 80, 996. [] I. F. Akyldz, J. S. M. Ho, and Y.-B. Ln, Moveent-based locaton update and selectve pagng for PCS networks, IEEE/ACM Transactons on Networkng, 4, 4, , 996. [] A. Abutaleb and V. O. K. L, Locaton update optzaton n personal councaton systes, Wreless Networks, 3, 05 6, 997. [3] Jung H. and Tonguz O. K., Rando spacng channel assgnent to reduce the nonlnear nterodulaton dstorton n cellular oble councatons, IEEE Transactons on Vehcular Technology,48, 5, , 999. [4] G. Chartrand, D. Erwn, F. Harary, and P. Zang, Rado labelngs of graphs, Bulletn of the Insttute of Cobnatorcs and ts Applcatons, 000. [5] I. F. Akyldz, Y.-B. Ln, W.-R. La, and R.-J. Chen, A new rando walk odel for PCS networks, IEEE Journal on Selected Areas n Councatons, 8, 7, 54 60, 000. [6] J. L, H. Kaeda and K. L, Optal dynac oblty anageent for PCS networks, IEEE/ACM Transactons on Networkng, 8, 3, 39 37, 000. [7] L. Narayanan. Channel assgnent and graph ultcolorng, n I. Stoenovc (Ed.), Handbook of Wreless Networks and Moble Coputng, New York: Wley, 00. 5

16 Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (GRAPH-HOC) Vol.3, No., March 0 [8] J. Gallan A dynac survey of graph labelng, the electronc ournal of cobnatorcs 6 (009), #DS6 [9] M. I. Moussa AN ALGORITHM FOR ODD GRACEFUL LABELING OF THE UNION OF PATHS AND CYCLES Internatonal ournal on applcatons of graph theory n wreless ad hoc networks and sensor networks (Graph-Hoc), Vol., No., M arch 00. 6