Iteratioal Joural of Pure ad Applied Mathematics Volume 97 No. 4 014, 515-55 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijpam.v97i4.11 PAijpam.eu RADIO NUMBER FOR CROSS PRODUCT P (P ) Chah Yog Jug 1, Waqas Nazeer, Saima Nazeer 3, Arif Rafiq 4 ad Shi Mi Kag 5 1 Departmet of Busiess Admiistratio Gyeogsag Natioal Uiversity Jiju, 660-701, KOREA,4 Departmet of Mathematics Lahore Leads Uiversity Lahore, 54810, PAKISTAN 3 Departmet of Mathematics Lahore College for Wome Uiversity Lahore, 54600, PAKISTAN 5 Departmet of Mathematics ad RINS Gyeogsag Natioal Uiversity Jiju, 660-701, KOREA Abstract: A Radio labelig of the graph G is a fuctio g from the vertex set V(G) of G to N {0} such that f(u) f(v) diam(g)+1 d G (u,v), where diam(g) ad d G (u,v) are diameter ad distace betwee u ad v i graph G, respectively. The radio umber r(g) of G is the smallest umber k such that G has radio labelig with max{f(v) : v V(G)} = k. We ivestigate radio umber for the cross product of P ad P. AMS Subject Classificatio: 05C1, 05C15, 05C78 Key Words: chael assigmet, radio labelig, radio umber, cross product Received: August 8, 014 Correspodece author c 014 Academic Publicatios, Ltd. url: www.acadpubl.eu
516 C.Y. Jug et al 1. Itroductio I 1980, Hale [5] preseted the idea for radio frequecy assigmet problems. Later i 001, Chartrad et al. [] applied this idea for assigmet of chaels to FM radio statio. These assigmet have bee made o the fact that frequecies eed to be assiged to the chaels such that there is miimum iterferece. The geographically closed radio statios should be assiged differet frequecies to avoid iterferece. The iterferece graph is developed to solve the chaels assigmet problems by covertig a assigmet of chaels i to graph labelig. I iterferece graph, there is a edge betwee two vertices if the correspodig trasmitter have major iterferece. It is assumed that the distace betwee vertices is two if two trasmitters have mior iterferece i iterferece graph ad if the vertices are at distace three or beyod the there is o iterferece betwee trasmitters. I other words, the adjacet vertices represet the very close trasmitters ad those vertices which are at distace two apart represets close trasmitters. A pair of trasmitters which has small iterferece must receive differet chaels ad two trasmitters which has large iterferece must receive chaels that are at least two apart are suggested i [1] by Robert. Motivated through this problem Griggs ad Yeh [4] relates the chaels with o-egative iteger by itroducig L(, 1)-labelig, which is defied as follows: Defiitio 1.1. A distace two labelig (or L(, 1)-labelig) of a graph G = (V(G),E(G)) isafuctiogfromvertex set V(G) tothesetofoegative itegers such that the followig coditios are satisfied: (1) g(u) g(v) if d(u,v) = 1, () g(u) g(v) 1 if d(u,v) =. The differece betwee the largest ad the smallest label assiged by g is called the spa of g ad the miimum spa over all L(,1)-labelig of G is called the λ-umber of G, deoted by λ(g). The L(, 1)-labelig has explored i the past two decades by may researchers like Chag ad Kuo [1], Georges ad Mauro [3], Sabaki [13], Vaidya ad Batva [14], Vaidya et al. [16], Wag [17] ad Yeh [18], [19]. But as time passed, practically it has bee observed that the iterferece amog trasmitters might go beyod two levels. Radio labelig exteds the umber of iterferece level cosidered i L(, 1)-labelig from two to largest possible iterferece amog trasmitter, i.e. the diameter of G which is defied
RADIO NUMBER FOR CROSS PRODUCT P (P ) 517 as follows: Defiitio 1.. The diameter of a graph is deoted by diam(g) ad defied as the maximum distace betwee ay two vertices, that is, diam(g) = max{d(u,v);u,v G}. Here d(u,v) is distace betwee u ad v which is defied as follows: Defiitio 1.3. Let G be a coected graph, the distace d(u,v) betwee ay pair of vertices u,v is the legth of the shortest path betwee them. Motivated through the problem of chael assigmet of FM radio statios Chartrad et al. [] itroduced the cocept of radio labelig which also kow as Multi-level distace labelig of graph as follows: Defiitio 1.4. A radio labelig which is also kow as multilevel distace labelig of G is a fuctio g : V(G) N {0} such that the iequality g(u) g(v) diam(g)+1 d(u,v) holds for ay pair of distict vertices u,v. The spa of g is the differece of the largest ad the smallest chaels used, max u,v V(G) {g(v) g(u)}. The radio umber of G is deoted by r(g) ad is defied as the miimum spa of radio labelig of G. Note that whe diam(g) is two tha radio labelig ad distace two labelig are idetical. The radio labelig is studied i the past decade by may researches like Liu [6], Liu ad Xie [7], [8], Liu ad Zhu [9] ad Vaidya ad Vihol [15]. Moreover, the radio umber for path ad cycles was determied i [9], for the square of paths was ivestigated by Liu ad Xie [8], for the square of a cycle [7]. Radio Number for geeralized prism graph was studied i [10] ad a geeralized gear graph was discussed i [11], where lower boud of radio umber is determied. Radio labelig for some cycle related graphs are studied by Vadiya ad Vihol [15]. I this paper, we completely determie the radio umber of cross product for P ad P. Through our this discussio, the order of P (P ) is. I Theorem. we determie the radio umber for P (P ). Theorem.3 ad Theorem.5 give us lower ad upper boud for radio umber of P (P ); 3, ad fially i Theorem.6 we get r(p (P )) = +1.. Mai Results The cross product of graphs G ad H deoted by G(H), is the graph with the vertex set V(G) V(H) = {(u,v) : u V(G), v V(H)}, where (u,x)
518 C.Y. Jug et al is adjacet to (v,y) wheever (i) u 1 = v 1 ad u v or (ii) u 1 v 1 ad u v. Case 1: For P (P ), whe is odd, let v 0 ad u 0 be the ceters. Let v 1,v,,v 1 be the vertices o the left side ad v 1,v,,v 1 be the vertices o the right side with respect to ceter v 0 ad u 1,u,,u 1 vertices o the left side ad u 1,u,,u 1 with respect to ceter u 0. So for P (P ), V(P (P )) = V V U U, where V = {v 0,v 1,v,,v 1 U = {u 0,u 1,u,,u 1 be the be the vertices o the right side }, V = {v 0,v 1,v,,v 1}, }, U = {u 0,u 1,u,,u 1 Case : For P (P ), whe is eve, let v 0 ad v 0, u 0 ad u 0 bethe ceters. Let v 1,v,,v 1 be the vertices o the left side with respect to ceter v 0 ad v 1,v,,v 1 be the vertices o the right side with respect to ceter v 0 ad u 1,u,,u 1 be the vertices o the left side with respect to ceter u 0 ad u 1,u,,u 1 be the vertices o the right side with respect to ceter u 0. So for P (P ), V(P (P )) = V V U U, where V = {v 0,v 1,v,,v 1 }, V = {v 0,v 1,v,...,v 1}, U = {u 0,u 1,u,...,u 1 }, U = {u 0,u 1,u,,u 1}. We say two vertices u ad v are o opposite side i P (P ), if u V V ad v U U. Defiitio.1. Thelevel fuctio l from V(P (P )) to set of oegative itegers from a ceter vertex c is defied as l(u) := {d(u,c);c is a ceter vertex} for ay u V(P (P )). Note that, i P (P ), the maximum level fuctio is 1 if is odd ad 1 if is eve. Observatios. We made followig observatio for P (P ), }. (a) (b) d(u,v) V(P (P )) = { l(u)+l(v), if is odd; l(u)+l(v)+1, if is eve.
RADIO NUMBER FOR CROSS PRODUCT P (P ) 519 Theorem.. Let P (P ) be the cross product of P ad P. The r(p (P )) = 4. Proof. Radio labelig of P (P ) as show i figure. 1 3 0 4 Figure 1: r(p (P )) = 4 Theorem.3. Let P (P ) be the cross product of P ad P for 3. The r(p (P )) +1. Moreover, the equality holds if ad oly if there exist a radio labelig g with orderig u 1,u,..., of vertices of P (P ) such that g(u 1 ) = 0 < g(u ) < g(u 3 ) < < g(u ), all the followig holds, for all 1 i 1, (a) u i ad u i+1 are o opposite sides, (b) {u 1,u } = {c 1,c }, where c 1,c are ceter vertices. The Proof. Let g be a optimal radio labelig for P (P ), where g(u 1 ) = 0 < g(u ) < g(u 3 ) < < g(u ). g(u i+1 ) g(u i ) (d+1) d(u i,u i+1 ) for all 1 i 1. Summig these 1 iequalities, we get 1 r(p (P )) = g(u ) ( 1)(d+1) d(u i,u i+1 ) (.1) i=1
50 C.Y. Jug et al Case (a): For P (P ), whe is odd, we have 1 i=1 d(u i,u i+1 ) [l(u i )+l(u i+1 )] 1 i=1 = u V (G) u V (G) l(u) l(u 1 ) l(u ) l(u). (.) Substitutig (.) i (.1), we get r(p (P )) = g(u ) ( 1)(d+1) u V(G) l(u), sice d = 1 ad u V(G) l(u) = 1, so ( ) 1 r(p (P )) ( 1)() = +1. Case (b): For P (P ), whe is eve, we have 1 i=1 d(u i,u i+1 ) [l(u i )+l(u i+1 )+1] 1 i=1 = u V(G) u V(G) l(u) l(u 1 ) l(u )+( 1) l(u)+( 1). (.3) Substitutig (.3) i (.1), we get r(p (P )) = g(u ) ( 1)(d+1) u V(G) l(u) ( 1), sice d = 1 ad u V(G) l(u) =, so ( ) r(p (P )) ( 1)() ( 1) = +1. Thus, from above two cases we have desired result.
RADIO NUMBER FOR CROSS PRODUCT P (P ) 51 Theorem.4. Let g be a assigmet of distict o-egative itegers to V(P (P )) ad {u 1,u,u 3,,u } be the orderig of V(P (P )) such that g(u i ) < g(u i+1 ) defied by g(u 1 ) = 0 ad g(u i+1 ) = g(u i )+d+1 d(u i,u i+1 ). The g is a radio labelig ad for ay 1 i, the followig holds. (a) d(u i,u i+1 ) +1 if is odd, (b) d(u i,u i+1 ) +1 ad d(u i,u i+1 ) d(u i+1,u i+ ) if is eve. Proof. Let g(u 1 ) = 0 ad g(u i+1 ) = g(u i ) + d + 1 d(u i,u i+1 ), for ay 1 i 1, ad let for each i = 1,,..., 1, g i = g(u i+1 ) g(u i ). We wat to prove that g is a radio labelig, if (a) ad (b) holds, that is, for ay j i, g(u j ) g(u i ) d+1 d(u j,u i ) Case (a): Whe is odd, we have d = 1 ad let (a) holds ad we take i > j, the g(u i ) g(u j ) = g j +g j+1 + +g i 1 = (i j)(d+1) d(u j,u j+1 ) d(u j+1,u j+ ) d(u i 1,u i ) ( ) +1 (i j)() (i j) by usig (1) ( ( )) +1 = (i j) ( ) 1 = (i j) d+1 d(u i,u j ). Case (b): Whe is eve, let (b) holds ad we take i > j g(u i ) g(u j ) = g j +g j+1 + +g i 1 = (i j)(d+1) d(u j,u j+1 ) d(u j+1,u j+ ) d(u i 1,u i )
5 C.Y. Jug et al If i j = eve, the If i j = odd, the g(u i ) g(u j ) (i j)(d+1) i j ( ) +1 ( ) = (i j)() (i j) i j ( ) = (i j) i j d+1 d(u i,u j ). g(u i ) g(u j ) (i j)(d+1) d+1 d(u i,u j ). ( ) i j +1 +1 i j ( ) i j +1 ( ) Thus, i both the cases g is a radio labelig ad hece the result. Theorem.5. Let P (P ) be the cross product of P ad P for 3. The r(p (P )) +1. Proof. Here we cosider followig two cases. Case 1: Whe is odd, defie g : V(P (P )) {0,1,,..., +1} by g(u i+1 ) = g(u i )+d+1 l(u i ) l(u i+1 ) as per orderig of vertices give below. +1 +1 +1 1 v Rk v L1 v Rk v L1 v R(k 1) 1 v 0 +1 +1 v L v R(k 1) +1 +1 v Lk v R1 +1 v Lk +1 v L 1 1 v 0 1 v R1 Case : Whe is eve, defie g : V(P (P )) {0,1,,..., +1} by
RADIO NUMBER FOR CROSS PRODUCT P (P ) 53 g(u i+1 ) = g(u i )+d l(u i ) l(u i+1 ) as per orderig of vertices give below. v L0 + v R(k 1) + v R(k 3) v R(k 1) + v L1 + v L1 + v L(k 1) v R(k ) + + v L 1 v L(k 1) v R0 v L0 v R(k ) v R0 + v L v R(k 3) Sice g satisfy coditios of Theorem.4, so g is radio labelig with spa +1, hece r(p (P )) +1. Theorem.6. Let P (P ) be the cross product of P ad P. The r(p (P )) = +1. Proof. The proof follows from Theorem.3 ad Theorem.5. Example.7. I Figure, orderig of the vertices ad optimal radio labelig of P 9 (P ) is show. v 0 v 4 v 1 u 4 u 1 v 3 v u 3 u v v 3 u u 3 v 1 v 4 u 1 u 4 u 0 = r(p 9 (P )). u4 u u 3 u1 u 0 68 51 34 17 73 64 47 30 13 u 1 u u 3 u 4 60 43 6 9 0 56 39 5 v4 v3 v v v 1 0 v 1 v v 3 v 4 Figure : r(p 9 (P ) = 73
54 C.Y. Jug et al Example.8. I Figure 3, orderig of the vertices ad optimal radio labelig of P 10 (P ) is show v 0 v 4 v 1 v 3 v v v 3 v 1 v 4 v 0 u 0 u 4 u 1 u 3 v u u 3 u 1 u 4 u 0 = r(p 10 (P )). u 4 u 3 86 77 u u1 u0 u 0 68 59 50 91 u 1 u 8 73 u 3 u 4 64 55 36 7 18 9 v 4 v v3 0 v v 1 0 41 3 v 0 v 1 3 14 v v 3 5 v 4 Figure 3: r(p 10 (P )) = 91 Refereces [1] G.J. Chag, D. Kuo, The L(, 1)-labelig problem o graphs, SIAM J. Discrete Math., 9 (1996), 309-316. [] G. Chartrad, D. Erwi, P. Zhag, F. Harary, Radio labeligs of graphs, Bull. Ist. Combi. Appl., 33 (001), 77-85. [3] J.P. Georges, D.W. Mauro, Labelig trees with coditio at distace two, Discrete Math., 69 (003), 17-148, doi: 10.1016/S001-365X(0)00750-1 [4] J.R. Griggs, R.K. Yeh, Labelig graphs with coditio at distace, SIAM J. Discrete Math., 5 (199), 586-595. [5] W.K. Hale, Frequecy assigemet: theory ad applicatio, Proc. IEEE, 68 (1980), 1497-1514. [6] D.D.F. Liu, Radio umber for trees, Discrete Math., 308 (008), 1153-1164, doi: 10.1016/j.disc.007.03.066
RADIO NUMBER FOR CROSS PRODUCT P (P ) 55 [7] D.D.F. Liu, M. Xie, Radio umber for square cycles, Cogr. Numer., 169 (004), 105-15. [8] D.D.F. Liu, M. Xie, Radio umber for square paths, Ars Combi., 90 (009), 307-319. [9] D.D.F. Liu, X. Zhu, Multi-level distace labelig for paths ad cycles, SIAM J. Discrete Math., 19 (005), 610-61. [10] P. Martiez, J. Ortiz, M. Tomova, C. Wyels, Radio umbers for geeralized prism graphs, Discuss Math. Graph Theory, 31 (011), 45-6. [11] M.T. Rahim, M. Farooq, M. Ali, S. Ja, Multi-level distace labeligs for geeralized gear graph, It. J. Math. Soft Comput., (01), 57-63. [1] F.S. Roberts, T-colorig of graphs: recet results ad ope problems, Discrete Math., 93 (1991), 9-45, doi: 10.1016/001-365X(91)9058-4 [13] D. Sakai, Labelig chordal: distace two coditio, SIAM J. Discrete Math., 7 (1994), 133-140. [14] S.K. Vaidya, D.D. Batva, Labelig cacti with a coditio at distace two, Le Matematiche, 66 (011), 9-36. [15] S.K. Vaidya, P.L. Vihol, Radio labelig for some cycle related graphs, It. J. Math. Soft Comput., (01), 11-4. [16] S.K. Vaidya, P.L. Vihol, N.A. Dai, D.D. Batva, L(, 1)-labelig i the cotexamplet of some graph operatios, J. Math. Res., (010), 109-119. [17] W.F. Wag, The L(, 1)-labelig of trees, Discrete Appl. Math., 154 (006), 598-603, doi: 10.1016/j.dam.005.09.007 [18] R.K. Yeh, Labelig Graphs with a Coditio at Distace Two, Ph.D. Thesis, Departmet of Mathematics, Uiversoty of South Carolia, Columbia, South Carolia, 1990. [19] R.K. Yeh, A survey o labelig graphs with coditio at distace two, Discrete Math., 306 (006), 117-131, doi: 10.1016/j.disc.005.11.09
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