The L(2, 1)-Labeling on -Product of Graphs

Transcription

1 Annals of Pure and Appled Mathematcs Vol 0, No, 05, 9-39 ISSN: X (P, (onlne Publshed on 7 Aprl 05 wwwresearchmathscorg Annals of The L(, -Labelng on -Product of Graphs P Pradhan and Kamesh Kumar* Department of Mathematcs and Statstcs, Gurukula Kangr Vshwavdyalaya, Hardwar (UK, Inda *Correspondng Author E-mal: ppradhan4@gmalcom E-mal: kameshkumar0@gmalcom Receved 3 February 05; accepted 8 February 05 Abstract The L(, -labelng (or dstance two labelng of a graph G s an nteger labelng of G n whch two vertces at dstance one from each other must have labels dfferng by at least and those at dstance two must dffer by at least The L(, - labelng numberλ G of G s the smallest number k such that G has an L(, -labelng max{ f ( v: v V ( G} = k wth max f v : v V G k In ths paper, upper bound for the L(, -labelng number for the -product of two graphs has been obtaned n terms of the maxmum degrees of the graphs nvolved Degrees of vertces, vertex of maxmum degree and number of vertces of maxmum degree have been dscussed n the -product of two graphs Keywords: Channel assgnment, L(,-labelng, L(,-labelng number, graph -product AMS Mathematcs Subject Classfcaton (00: 05C78 Introducton The concept of L(, -labelng n graph come nto exstence wth the soluton of frequency assgnment problem In fact, n ths problem a frequency n the form of a nonnegatve nteger s to assgn to each rado or TV transmtters located at varous places such that communcaton does not nterfere Hale [6] was frst person who formulated ths problem as a graph vertex colorng problem Latter, Grggs and Yeh [5] ntroduced L(, -labelng on a smple graph Let G be a graph wth vertex set V ( G A functon f : V ( G Z + {0}, where Z + s a set of postve ntegers, s called L(, -labelng or dstance two labelng f f ( u f ( v when d( u, v = and f ( u f ( v when d( u, v =, where d s dstance between u and v n G A k- L(, -labelng s an L(, -labelng such that no labels s greater than k The L(, -labelng number ofg, denoted by λ ( G or λ, s the smallest number k such that G has a k- L(, -labelng The L(, - labelng has been extensvely studed n recent past by many researchers [see,,, 4, 7, 8, -] The common trend n most of the research paper s ether to determne the value of L(, -labelng number or to suggest bounds for partcular classes of graphs 9

2 P Pradhan and Kamesh Kumar Grggs and Yeh [5] provded an upper bound + for a general graph wth the maxmum degree Later, Chang and Kuo [] mproved the upper bound to +, whle Kral and Skrekarsk [0] reduced the upper bound to + Furthermore, recently Gonccalves [4] proved the bound + whch s the present best record If G s a dameter graph λ( G The upper bound s attanable by Moore graphs (dameter graphs wth order + (Such graphs exst only f =,3,7 and possbly 57; [5] Thus Grggs and Yeh [5] conjectured that the best bound s for any graph G wth the maxmum degree (Ths s not true for = For example, ( K = but λ ( K = Graph products play an mportant role n connectng many useful networks Klavzar and Spacepan [9] have shown that -conjecture holds for graphs that are drect or strong products of nontrval graphs After that Shao, et al [3] have mproved bounds on the L(, -labelng number of drect and strong product of nontrval graphs wth refned approaches Shao and Shang [5] also consder the graph formed by the Cartesan sum of graphs and prove that the λ -number of L(, -labelng of ths graph satsfes the -conjecture (wth mnor exceptons In ths paper, we have consdered the graph formed by the -product of graphs [3] and obtaned a general upper bound for L(, -labelng number n term of maxmum degree of the graphs In the case of -product of graphs, L(, -labelng number of graph holds Grggs and Yeh s conjecture [5] wth mnor exceptons A labelng algorthm A subset X of V ( G s called an -stable set (or -ndependent set f the dstance between any two vertces n X s greater than, e { d >, u, v X} A -stable set s a usual ndependent set A maxmal -stable subset X of a set Y s a -stable subset of Y such that X s not a proper subset of any -stable subset ofy Chang and Kuo [] proposed the followng algorthm to obtan an L(, -labelng and the maxmum value of that labelng on a gven graph Algorthm: Input: A graph G = ( V, E Output: The value k s the maxmum label Idea: In each step, fnd a -maxmal -stable set from the unlabeled vertces that are dstance at least two away from those vertces labelled n the prevous step Then label all the vertces n that -stable wth n current stage The label starts from 0 and then ncreases by n each step The maxmum label k s the fnal value of Intalzaton: Set X = φ; V = V ( G; = 0 Iteraton: 30

3 The L(, -Labelng on -Product of Graphs Determne Y and X Y = { u V : u s unlabeled and d v X } X s a maxmal -stable subset of Y If Y = φ then set X = φ Label the vertces of X (f there s any wth 3 V V X 4 If V φ +, go to step 5 Record the current as k (whch s the maxmum label Stop Thus k s an upper bound on λ ( G Let u be a vertex wth largest label k obtaned by above Algorthm We have the followng sets on the bass of Algorthm just defned above I = { : 0 k and d = for some v X }, e I s the set of labels of the neghbourhood of the vertex u I = { : 0 k and d for some v X }, e I s the set of labels of the vertces at dstance at most from the vertex u I3 = { : 0 k and d 3 for all v X } = {0,,, k } I e I 3 conssts of the labels not used by the vertces at dstance at most from the vertex u Then Chang and Kuo showed that λ( G k I + I3 I + I In order to fnd k, t suffces to estmate B = I + I n terms of ( G We wll nvestgate the value B wth respect to a partcular graph ( -product of two graphs The notatons whch have been ntroduced n ths secton wll also be used n the followng sectons 3 The -product of graphs The -product G H of two graphs G and H s the graph wth vertex set V ( G V ( H, n whch the vertex ( u, v s adjacent to the vertex ( u ', v ' f and only f ether u s adjacent to u ' n G and v s not adjacent to v ' n H or u s not adjacent to u ' n G and v s adjacent to v ' n H e ether uu ' E( G and vv ' E( H or uu ' E( G and vv ' E( H For example, we consder the Fgure Now, we state and prove the followng corollary to fnd out the degree of any vertex of -product of two graphs 3

4 P Pradhan and Kamesh Kumar P 4 : u u u3 u4 v ( u, v ( u, v ( u3, v ( u4, v P 3 : v ( u, v ( u, v ( u3, v ( u, v 4 v 3 ( u, v3 ( u, v3 ( u3, v3 ( u4, v3 Corollary 3 Let u = V ( G H, where u V ( G, v V ( H, = n, V ( H = n deg ( u G d V ( G = and deg ( v H = d then deg ( u ( n d d ( n d d G H = + Proof: By the defnton of - product, the vertex ( u, v s adjacent to the vertex ( u', v ' f and only f ether uu ' E( G and vv ' E( H or uu ' E( G and vv ' E( H Now number of adjacent vertces to u n G e uu ' E( G = d Number of non-adjacent vertces to v n H e vv ' E( H = n d Number of adjacent vertces to v n H e vv ' E( H = d Number of non-adjacent vertces to u n G e uu ' E( G = n d Number of vertces where uu ' E( G and vv ' E( H = d( n d And Number of vertces where uu ' E( G and vv ' E( H = d( n d Total number of vertces adjacent to u = n G H = deg ( u ( n d d ( n d d G H = 3 The maxmum degree (largest degree of G H The maxmum (largest degree of G P P 4 3 Fgure: -product of P and P 4 3 H plays an mportant role n fndng out the upper bound for the L(, -labelng To fnd out the maxmum degree of G H proceed as follows:, we 3

5 The L(, -Labelng on -Product of Graphs Let, be the maxmum degree of G, H and ', ' be the mnmum degree of G, H respectvely Let be the maxmum degree of G H Case I: If n n ( n ' + ( n ' when = ( n ' + ( n ' when > Case II: If n < n ( n ' + ( n ' when = ( n ' + ( n ' when < From the above two cases, t can be wrtten as = max[( n ' + ( n ', ( n ' + ( n ' ] 33 The number of vertces of maxmum degree n G H Let nmn and n max be the number of vertces of mnmum and maxmum degree n a graph G and nmn and n max be the number of vertces of mnmum and maxmum degree n a graph H Then Case I: n n If The number of vertces of maxmum degree n graph G H = nmn nmax If > The number of vertces of maxmum degree n graph G H = nmax nmn Case II: n < n If The number of vertces of maxmum degree n graph G H = nmax nmn If < The number of vertces of maxmum degree n graph G H = nmn nmax 4 Upper bound for the L(, -labelng number n G H In ths secton, general upper bound for the L(, -labelng number ( λ -number of -product n term of maxmum degree of the graphs has been establshed In ths regard, we state and prove the followng theorem 33

6 P Pradhan and Kamesh Kumar Theorem 4 Let,, be the maxmum degree of G H, G, H and n, n, n be the number of vertces of G H, G, H respectvely If, ( G H ( n n 6 λ + Proof: Let u = be any vertex n the graph G H Denote d = deg (, u G H d = deg G ( u, d = deg H ( v, = max deg ( G, ' = mn deg ( G, = max deg ( H, ' = mn deg ( H, ν ( G = n, and ν ( H = n Hence d = ( n d d + ( n d d (from corollary and = ( G H = max[( n ' + ( n ', ( n ' + ( n ' ] Let us consder the Fgure For any vertex u ' n G at dstance from u, there must be a path u ' u '' u of length two between u ' and u ng ; but the degree of v n H s d, e v has d adjacent vertces n H, by the defnton of -product G H, there must be d + nternally-dsjont paths(two paths are sad to be u u " u ' v ( u, v ( u", v ( u ', v v ( u, v v v 3 ( u ', v v d ( u, v d Fgure : ( u ', v d nternally-dsjont f they do not ntersect each other of length two between ( u ', v and ( u, v Hence for any vertex n G at dstance from u = whch are concded n 34

7 The L(, -Labelng on -Product of Graphs G H ; on the contrary whenever there s no such a vertex n G at dstance from u n G, there exst no such correspondng d + vertces at dstance from u = whch are concded n G H In the former case, snce such d + vertces at dstance from u = are concded n G H and hence they should be counted once only and therefore we have to subtract d + = d from the value d( whch s the best possble number of vertces at dstance from a vertex u = n G H Let the number of vertces n G at dstance from u be t t [0, d( ] Now, f we take t = d( whch s the best possble number of vertces at dstance from a vertex u n G to get the number of vertces at dstance from u = n G H, we wll have to subtract at least dd ( from the value d( For H, we can proceed n smlar way to get the number of vertces at dstance from u = n G H and n ths case subtract dd ( from the value d( Hence, the number of vertces at dstance from u = n G H wll d d ( + d d ( = d d ( + from the value d( decrease altogether By the above analyss, the number d( dd ( + s now the best possble number of vertces at dstance from u = n G H But some cases are remanng to be consdered for fndng out the best possble number of vertces at dstance from u = n G H Let ε be the number of edges of the subgraph F nduced by the neghbours of u The edges of the subgraph F nduced by the neghbours of u can be dvded nto the followng two cases u u ' v ( u, v ( u ', v v ' 35 Fgure3: ( ut, v '

8 u P Pradhan and Kamesh Kumar u ' v v ' ( u, v ' t Fgure 4: ( ut, v ' Case I: Consder the Fgure 3 for ths case For each neghbour vertex ( u ', v (where u ' s adjacent to u n G of u = and any vertex ( u, v ' (where v ' s adjacent to v n H and u t s any vertex of G whch s not adjacent to u and u ' n G ( u, ' t v must be the common neghbour vertex of ( u ', v and ( u, v there must be an edge between ( ut, v ' and ( u ', v But there are at least ( n d d neghbour vertces lke ( ut, v ' of x = and there are d neghbour vertces lke ( u ', v of u = Hence the number of edges of the subgraph F nduced by the neghbour vertces of u s at least ( n d dd e ε ( n d dd By a symmetrc analyss, the neghbour vertces of u should agan add at least ( n d dd Case II: Consder the Fgure 4 for ths case For each neghbour ( ut, v ' (where v ' s adjacent to v n H and ut s any vertex of G whch s nether equal to u nor adjacent to u and ( ut, v ' (where u t s adjacent to ut n G of u = n G H Obvously vertex ( ut, v ' s adjacent to ( ut, v ' n G H, hence there s an edge between them There are at least ( n d d neghbour vertces lke ( ut, v ' of u = n G H So at least ( n d d edges wll exst between ( ut, v ' and ( ut, v ' Hence the number of edges of the subgraph F nduced by the neghbours of u s at least ( n d d By symmetrc analyss, the neghbours of u should agan add at least ( n d d By the analyss of the above two cases, ε ( n d d d + ( n d d d + ( n d d + ( n d d t 36

9 The L(, -Labelng on -Product of Graphs Whenever there s an edge n F, the number of vertces wth dstance from u wll decrease by, hence the number of vertces wth dstance from u = n G H wll stll need at least a decrease ( n d dd + ( n d dd + ( n d d + ( n d d from the value d( dd ( + (the number d( dd ( + s now the best possble for the number of vertces wth dstance from u = n G H Hence for the vertex u, the number of vertces wth dstance from u s no greater than The number of vertces wth dstance from u s no greater than d( d d ( + ( n d d d ( n d d d ( n d d ( n d d Hence I d, I d + d( d d ( + ( n d d d ( n d d d ( n d d ( n d d Then B = I + I = d + d + d( d d ( + ( n d d d ( n d d d ( n d d ( n d d = d( + d d ( n + n + + d d 4 ( n d d ( n d d = (( n d d + ( n d d ( + d d ( n + n + + d d 4 ( n d d ( n d d Defne f ( s, t = (( n s t + ( n t s( + st( n + n + + s t 4 ( n s t ( n t s Then f ( s, t has the absolute maxmum at (, on [0, ] [0, ] f (, = (( n + ( n ( + ( n + n ( n ( n 37

10 P Pradhan and Kamesh Kumar = (( n + ( n ( + ( n + n 4 ( n ( n = (( n + ( n + ( n + ( n ( n + n 4 ( n ( n Snce s the maxmum degree of graph G H and ( n + ( n s the degree of any vertces n graph G H Therefore ( n + ( n = max[( n + ( n, ( n + ( n ] ' ' ' ' f (, n + n ( n + n 4 n + n + Then, = ( n + n 6 λ +, ( G H k B ( n n 6 where, = max[ ( n + ( n, ( n + ( n ] ' ' ' ' Corollary 4 Let be the maxmum degree of G H except for when one of ( G and ( H s Then λ( G H Proof: If one of or s then G H s stll a general graph, hence we can suppose that and (hence n 3 and n 3 Then ( n + n 6 ( = Ths mples that λ( G H k B ( n + n 6 Therefore the result follows Acknowledgement Ths research work s supported by Unversty Grant Commsson (UGC New Delh, Inda under the UGC Senor Research Fellowshp (SRF scheme to the second author REFERENCES GJChang and DKuo, The L(, -labelng on graphs, SIAM J Dscrete Math, 9 ( GJChang and et al, On L(d, -labelng of graphs, Dscrete Math, 0 ( EMEl-Kholy, ESLashn and SNDaoud, New operatons on graphs and graph foldngs, Internatonal Mathematcal Forum, 7( DGonccalves, On the L(p, -labelng of graphs, n Proc 005 Eur Conf Combnatorcs, Graph Theory Appl S Felsner, Ed, (005, JRGrggs and RKYeh, Labelng graphs wth a condton at dstance two, SIAM J Dscrete Math, 5 (

11 The L(, -Labelng on -Product of Graphs 6 WKHale, Frequency assgnment: Theory and applcaton, Proc IEEE, 68(6 ( PKJha, Optmal L(, -labelng of strong product of cycles, IEEE Trans Crcuts systems-i, Fundam Theory Appl, 48(4 ( PKJha, Optmal L(, -labelng on Cartesan products of cycles wth an applcaton to ndependent domnaton, IEEE Trans Crcuts systems-i, Fundam Theory Appl, 47( ( S Klavzar and S Spacepan, The -conjecture for L(, -labelngs s true for drect and strong products of graphs, IEEE Trans Crcuts systems-ii, Exp Brefs, 53(3 ( DKral and RSkrekovsk, A theorem about channel assgnment problem, SIAM J Dscrete Math, 6 ( DDFLu and RKYeh, On Dstance Two Labelng of Graphs, Ars Combnatora, 47 ( DSaka, Labelng Chordal Graphs wth a condton at dstance two, SIAM J Dscrete Math, 7 ( ZShao and et al, Improved bounds on the L(, -number of drect and strong products of graphs, IEEE Trans Crcuts systems-ii, Exp Brefs, 55(7 ( ZShao and RKYeh, The L(, -labelng and operatons of graphs, IEEE Trans Crcuts and systems-i, 5(4 ( ZShao and DZhang, The L(, -labelng on Cartesan sum of graphs, Appled Mathematcs Letters, ( ZShao and et al, The L(,-labelng of K,n -free graphs and ts applcatons, Appled Mathematcs Letters, ( ZShao and et al, The L(, -labelng on graphs and the frequency assgnment problem, Appled Mathematcs Letters, ( SKVadya and DDBantva, Dstance two labelng of some total graphs, Gen Math Notes, 3( ( SKVadya and DDBantva, Some new perspectves on dstance two labelng, Internatonal Journal of Mathematcs and Soft Computng, 3(3 (03, SPaul, MPal and APal, L(,-labellng of crcular-arc graph, Annals of Pure and Appled Mathematcs, 5( ( SPaul, MPal and APal, L(,-labelng of permutaton and bpartte permutaton graphs, Mathematcs n Computer Scence, DOI 0007/s SPaul, MPal and APal, L(,-labelng of nterval graphs, J Appl Math Comput DOI 0007/s