Hamiltonian Cycle in Complete Multipartite Graphs

Transcription

1 Annls of Pure nd Applied Mthemtics Vol 13, No 2, 2017, ISSN: X (P), (online) Pulished on 18 April 2017 wwwreserchmthsciorg DOI: Annls of Hmiltonin Cycle in Complete Multiprtite Grphs Joseph Vrghese Kureethr Deprtment of Mthemtics, Christ University Bengluru , Krntk, Indi E-mil: Received 26 Mrch 2017; ccepted 14 April 2017 Astrct This pper dels with the necessry nd sufficient conditions for complete multiprtite grph to hve Hmiltonin cycle Keywords: Hmiltonin cycle; multiprtite grph; triprtite grph AMS Mthemtics Suject Clssifiction (2010): 05C10 1 Introduction Finding cycles of ritrry length in connected grphs nd complete multiprtite grphs hs een n interesting type of prolem from the elementry to the high order reserch in the study of grphs A spnning cycle in grph is nmed s the Hmiltonin Cycle fter the fmous Irish Mthemticin Sir Willim Rown Hmilton The prolem of finding Hmiltonin cycle in n ritrry lrge grph is NP-complete Most fmous sufficient conditions for the existence of Hmiltonin cycle in grph re due to Dirc (1952) nd Ore (1960) Billington rought out n extensive survey of cycle decompositions of complete multiprtite grphs [1] A list of the numer of Hmiltonin Cycles in vrious types of grphs is prepred y Weisstein [7] An extensive survey of the Hmiltonin Prolem is due to Gould [2] Hmiltonin prolem is lso distnce prolem A link etween distnce in grph nd colouring is explored in [4] Mny results ssocited with complete multiprtite grphs re found in [5] nd [6] Ro Li rings out surprising connection etween spectrl rdius nd some Hmiltonin properties of grphs [3] In this pper we see necessry nd sufficient condition for the existence of Hmiltonin cycle in complete multiprtite grphs 2 Lrgest cycle in complete triprtite grph The length of the lrgest cycle in complete triprtite of prticulr nture is given elow Lemm 21 If p, q nd r re the crdinlities of the prtite sets of complete triprtite grph G= K p,q,r nd if sum of ny pir mong p, q nd r is less thn the third numer, then the length of the lrgest cycle in G is twice the sum of the those pir Proof: Since p, q nd r re the crdinlities of the prtite sets, without loss of generlity, let (p+q)<r To get the lrgest cycle we my try to use the mximum numer of vertices 223

2 Joseph Vrghese Kureethr in G This mximum is reched only when we consider the sets with p vertices nd q vertices together s single set nd the set with r vertices s the second set Hence, prcticlly, it works s iprtite grph with one set contining p+q vertices nd the second set with r vertices Since (p+q)<r, the lrgest cycle possile is of size 2(p+q) 3 Existence of Hmiltonin cycle The following lemm descries the coincidence of the lrgest cycle s the spnning cycle (Hmiltonin Cycle) Lemm 31 Let p, q nd r e the crdinlities of the prtite sets of complete triprtite grph G= K p,q,r If p q r nd if r< (p+q), then G hs Hmiltonin cycle ie, G hs cycle of length p+q+r Proof: Let P, Q nd R e the prtite sets of G with crdinlities p, q nd r, respectively Without loss of generlity, let us ssume tht p<q<r It is given tht r < (p+q) Hence, let q=p+ nd r=q+ ie, q = p + & r = p + + (1) As r < (p+q), we hve (p++) < (p+p+), using (1) ie, <p ie, p > 0 (2) We cn express p s, p = ( p ) + (3) Using (3), q nd r cn e rewritten s, q = ( p ) + + (4) nd r = ( p ) (5) Using the ove prtitioning of the integers p, q nd r, sets P, Q nd R cn e prtitioned s follows: P = P( p ) P (6) where P( p ) = p nd P = where Q( p ) = p, Q = nd Q = where R( p ) = p, R = 1 Q = Q( p ) Q Q (7) R = R R R R (8), R = nd 2 ( p ) 1 2 R = Let V ( P( p ) ) = { u1, u 2,, u( } nd V ( P ) = { u( + 1, u( + 2,, u p} Let V ( Q( p ) ) = { v1, v2,, v( }, V ( Q ) = { v( + 1, v( + 2,, v p} nd Q ) = { v, v,, v } V ( p+ 1 p+ 2 q 224

3 Hmilton Cycle in Complete Triprtite Grphs Let V ( R( p ) ) = { w1, w2,, w( }, V ( R ) = { w( ) 1, ( ) 2,, } 1 p + w p + w p, R ) = { w, w,, w } nd R ) = { w, w,, w } V ( p+ 1 p+ 2 q V ( 2 q+ 1 q+ 2 r The construction of the cycle is s follows: We construct simple pth with u 1 P ( p ) P s the initil vertex Choose v1 Q( p ) Q nd w1 R( p ) R s the next two vertices of the simple pth, in tht order After w 1, we choose u2 P( p ) Continuing in this mnner, choosing vertices in the incresing order of the suffices of the vertices from P( p ), Q( p ) nd R( p ) in cyclic mnner, we terminte the simple pth t w( R( The length of the simple pth thus formed is 3( p ) 1 (9) In extending the ove pth, we now consider the sets Q nd R only We djoin the vertex w( p ) to v p+1 Q Q Shuttling etween the vertices of Q nd R, we terminte the simple pth t w R Hence, the length of the simple pth is thus q extended to n dditionl length of 2 s, Q = R = (10) At this level, the vertices of P, Q, R 1 nd R 2 re not used in the simple pth As ech of these sets hs the sme crdinlity, nd since R nd R re susets of R, we extend the simple pth in the following mnner Adjoin w tou q P P ( + 1 Alternting etween the vertices ech of P nd djoin w to v p Q Q ( R we rech the vertex w 2 p R 1 Shuttling etween the vertices ech of Q nd R we rech the vertex w 2 r R 2 We Hence the simple pth is now extended with n extr length 4 (11) As of now, using (9), (10) nd (11), the totl length of the simple pth is 3 ( p ) (12) Since the pth is terminted t w, we cn complete the cycle y djoining r R 2 it to the initil vertex of the pth u 1 P Totl length of the cycle thus formed is 3(p-) , using (12)ie, the cycle is of length 3p+2+ However, 3 p = p + p + p + + +, using (1) ie, 3p = p + ( p + + ) + ( p + ) ie, 3 p = p + q + r 225

4 Joseph Vrghese Kureethr Hence, the cycle thus formed is Hmiltonin Cycle 4 Min results The discussions in the previous sections led us to the most importnt results of this pper We cn estimte the length of the longest cycle in complete multiprtite grph We strt with complete triprtite grph nd then we extend the result to complete multiprtite grphs Theorem 41 Let G=K p,q,r e complete triprtite grph such tht p q r Then the length p + q + r, if r p + q of the lrgest cycle in G is 2( p + q), if r > p + q Proof: Given tht the grph G is complete triprtite Oviously, length of ny cycle will not exceed p+q+r, s this gives the totl numer of vertices in the network As p q r, let us explore the different possiilities in this inequlity Cse 1: p=q=r In this cse, oviously, the length of the lrgest cycle is p+q+r Cse 2: p = q <r There cn e three different possiilities here, viz, r<(p+q), r=(p+q) nd (p+q)<r When r<(p+q), we will get cycle of length p+q+r, using Lemm 31 If r=(p+q), s p=q, we will get without much difficulty, cycle of length p+q+r When (p + q) <r, we will get cycle of length 2(p+q), using Lemm 21 Cse 3: p<q=r Here s q>p, nd q=r, q-p nd r-pvertices will contriute 2(q-p) length of the cycle nd the remining prt will contriute 3p to the length of the Hmiltonin cycle Hence the totl length of the Hmiltonin cycle is 2(q-p)+3p ie, =2q-2p+3p ie, =p+2q ie, = p+q+r Cse 4: p<q<r Su Cse 1: (p+q)<r This is delt with in Cse 2 Su Cse 2: (p+q)=r This is lso delt with in Cse 2 Su Cse 3: r<(p+q) This cse is delt with in Lemm 31 Hence the proof Theorem 42 A complete multiprtite grph G of t lest three vertices is Hmiltonin if nd only if the crdinlity of no prtite set is lrger thn sum of the crdinlities of ll the other prtite sets Proof: Suppose tht crdinlities one of the prtite sets is lrger thn the sum of the crdinlities of ll the other prtite sets As there re only less numer of vertices in ll 226

5 Hmilton Cycle in Complete Triprtite Grphs other prtite sets put together, we cnnot form Hmiltonin cycle y piring elements of the this set nd tht of the other sets Conversely, ssume tht crdinlity of no prtite set is lrger thn sum of the crdinlities of ll other sets We cn regroup the prtite sets s three sets nd y pplying Theorem 41, form Hmiltonin cycle However, n lternte proof is given elow Let the complete m-prtite grph hve the m prtite sets with crdinlities n1 n2 n m 1 n m It is given tht n m n1 + + nm 1 Let nm 2 = p, nm 1 = qnd nm = r Without loss of generlity, let us ssume tht p < q < r As r < p + q, let r + = p + q Hence, we hve r = ( p ) + q Let the prtite sets with crdinlities p, q nd r e P, Q nd R respectively We now see how circle of length p+q+r exists mong the vertices of the sets P, Q nd R Let the vertex sets corresponding to R, P nd Q e w, w,, w }, u, u,, u } 227 { 1 2 r { 1 2 p nd v, v,, v } respectively As r = ( p ) + q, we cn re-lel the vertices of R, P { 1 2 q nd Q s follows: R = { w1, w2, wp, wp + 1,, w( p + ) = p, wp+ 1,, w( p+ q ) = r} = u, u, u, u,, u } nd P { 1 2 p p + 1 ( p + ) = p Q { v( p ) + 1,, v( p ) + = p, v p+ 1,, v( p ) + q= q = } We get the cycle connecting ll these vertices s descried elow: We strt simple pth from w1 to u p, lternting the vertices of R nd P in the incresing order of their suffices From u p, we extend the pth to w ( p ) + 1 From w ( p ) + 1 we extend the pth to w p+ 1 shuttling mong the sets R, P nd Q connecting the vertices of them in the incresing order of their suffices At this stge, ll the vertices of P re prt of the simple pth Hence, from w p+ 1we extend the pth to v q, lternting the vertices of R nd Q in the incresing order of their suffices Now, we hve simple pth of length p + q + r 1 We now connect vq to w 1 y n edge to complete cycle of length p + q + r It is esy to construct the Hmiltonin cycle using the vertices of other prtite sets As ech of the other prtite sets hs vertices less thn tht of the set P, s the cycle reches vertex of the set P, it cn e extended to vertices of other prtite sets efore it reches the set R Hence, in this wy Hmiltonin cycle cn e identified in complete multiprtite grph if the crdinlity of no prtite set is lrger thn sum of the crdinlities of ll the other prtite sets 5 Conclusion Although there re mny results coming up frequently regrding the existence of Hmiltonin cycle in vrious fmilies of grphs, precise nswer is still elusive for generl grphs This rticle is n ttempt to explore the existence of Hmiltonin cycle in

6 Joseph Vrghese Kureethr complete multiprtite grphs The results might e otined through vrious other methods However, simple lgeric proof is descried here Acknowledgement The uthor would like to express his pprecition to Ms Roop Vrghese of Indin Institute Science Eduction nd Reserch, Bhopl, whose internship under the uthor motivted him to work out the mtter discussed in this pper REFERENCES 1 EJBillington, Multiprtite grph decomposition: cycles nd closed trils Le Mtemtiche, LIX Fsc I II, (2004) RJGould, Recent dvnces on the Hmiltonin prolem: survey III Grphs nd Comintorics, 30 (2014) RLi, Spectrl rdius nd some Hmiltonin properties of grphs, Annls of Pure nd Applied Mthemtics, 9 (2015) PTMrykutty nd KAGermin, Open distnce pttern edge coloring of grph, Annls of Pure nd Applied Mthemtics, 6 (2014) JVrghese nd AAntonysmy, On doule continuous monotonic decomposition of grphs, Journl of Computer nd Mthemticl Sciences, 1 (2010) JVrghese nd AAntonysmy, On modified continuous monotonic decomposition of tensor product of grphs, Interntionl Journl of Contemporry Mthemticl Sciences, 5 (2010) EWWeisstein, Hmiltonin Cycle, MthWorld -- A Wolfrm We Resource 228