Some New Results on Prime Graphs

Transcription

1 Ope Joural of Discrete Mathematics, 202, 2, Published Olie July 202 ( Some New Results o Prime Graphs Samir Vaidya, Udaya M Prajapati 2 Saurashtra Uiversity, Rajkot, Idia 2 St Xavier s College, Ahmedabad, Idia samirkvaidya@yahoocoi, udaya64@yahoocom Received May 6, 202; revised Jue, 202; accepted Jue 25, 202 ABSTRACT We ivestigate prime labelig for some graphs resulted by idetifyig ay two vertices of some graphs We also itroduce the cocept of strogly prime graph ad prove that the graphs C, P, ad, are strogly prime graphs Moreover we prove that W is a strogly prime graph for every eve iteger 4 eywords: Prime Labelig; Prime Graph; Strogly Prime Graph Itroductio We begi with fiite, udirected ad o-trivial graph G VG ( ), EG ( ) with the vertex set VG ( ) ad the edge set EG ( ) Throughout this work C deotes the cycle with vertices ad P deotes the path o vertices I the wheel W C the vertex correspodig to is called the apex vertex ad the vertices correspodig to C are called the rim vertices, where The star, is a graph with oe vertex of degree called apex ad vertices of degree oe called pedat vertices Throughout this paper VG ( ) ad EG) ( deote the cardiality of vertex set ad edge set respectively For various graph theoretic otatio ad termiology we follow Gross ad Yelle [] while for umber theory we follow Burto [2] We will give brief summary of defiitios ad other iformatio which are useful for the preset ivestigatios Defiitio : If the vertices are assiged values subject to certai coditio(s) the it is kow as graph labelig Vast amout of literature is available i prited as well as i electroic form o differet kid of graph labelig problems For a dyamic survey of graph labelig problems alog with extesive bibliography we refer to Gallia [] Defiitio 2: A prime labelig of a graph G is a ijective fuctio f : V( G),2,,, VG ( ) such that for every pair of adjacet vertices u ad v, gcd f( u), f( v) The graph which admits a prime labelig is called a prime graph The otio of a prime labelig was origiated by Etriger ad was discussed i a paper by Tout et al [4] May researchers have studied prime graphs For eg Fu ad Huag [5] have proved that P ad, are prime graphs Lee et al [6] have proved that W is a prime graph if ad oly if is eve Deretsky et al [7] have proved that C is a prime graph Defiitio : Let u ad v be two distict vertices of a graph G A ew graph Guv, is costructed by idetifyig (fusig) two vertices u ad v by a sigle ew vertex x such that every edge which was icidet with either u or v i G is ow icidet with x i Guv, Vaidya ad aai [8] have established that the graph obtaied by idetifyig ay two vertices u ad v (with duv (, ) ) of C ( 4 ) is a prime graph The switchig ivariace of various prime graphs is discussed by Vaidya ad Prajapati [9] I the preset paper we ivestigate further results o prime graphs Bertrad s Postulate 4: For every positive iteger there is a prime p such that p 2 2 Prime Labelig of Some Graphs Theorem 2: The graph obtaied by idetifyig ay two vertices of, is a prime graph Proof: The result is obvious for, 2 Therefore we start with Let v 0 be the apex vertex ad v, v2,, v be the cosecutive pedat vertices of, Due to the ature of, two vertices ca be idetified i followig two possible ways: Case : The apex vertex v0 is idetified with ay of the pedat vertices (say v ) Let the ew vertex be u 0 ad the resultat graph be G The degvi, for i 2,,, ad degu0 as there is a loop icidet at u 0 Defie f : V( G), 2,, as f vi i for i 2,,, ad f( u0 ) Obviously f is a ijectio ad gcd f( u), f( v) for every pair of adjacet vertices Copyright 202 SciRes

2 00 S VAIDYA, U M PRAJAPATI u ad v of G Hece G is a prime graph Case 2: Ay two of the pedat vertices (say v ad v ) are idetified Let the ew vertex be u ad the resultat graph be G So i G, degv i, for i, 2,, 2, degu 2 ad degv0 Defie f : V( G), 2,, as f vi i for i 0,, 2,, 2 ad f u Obviously f is a ijectio ad gcd f( u), f( v) for every pair of adjacet vertices u ad v of G Hece G is a prime graph Illustratio 22: The prime labelig of the graph obtaied by idetifyig the apex vertex with a pedat vertex of,7 is show i Figure Illustratio 2: The prime labelig of the graph obtaied by idetifyig two of the pedat vertices of,7 is show i Figure 2 Theorem 24: If p is a prime ad G is a prime graph of order p the the graph obtaied by idetifyig two vertices with label ad p is also a prime graph Proof: Let f be a prime labelig of G ad i be the label of the vertex v i for i, 2,, p Moreover u be the ew vertex of the graph G which is obtaied by idetifyig v ad vp of G Defie f : u, v2, v,, vp, 2,, p as f vi if x vi, i 2,,, p f( x) if x u i if x vi, i 2,,, p The f( x) if x u Obviously f is a ijectio For a arbitrary edge e uv of G we claim that gcd f( u), f( v) To prove our claim the followig cases are to be cosidered Case : If u u the gcd f( u), f( v ) gcd f( u ( ) ), f v gcd, f( v) Case 2: If u u ad v u the gcd f( u), f( v ) gcd f( u), f( u) gcd f ( u), Case : If u u ad v u the u vi, v vj for some i, j 2,,, p with i j the gcd f( u), f( v ) gcd f v, f v i j gcd f v, f v i j as v i ad v j are adjacet vertices i the prime graph G with the prime labellig f Thus i all the possibilities f admits a prime labelig for G Hece G is a prime graph Illustratio 25: I the followig Figures ad 4 prime labelig of a graph G of order 5 ad the prime labelig for the graph G obtaied by idetifyig the vertices of G with label ad 5 are show Theorem 26: The graph obtaied by idetifyig ay two vertices of P is a prime graph Proof: Let v, v 2,, v be the vertices of Let P u be the ew vertex of the graph G obtaied by idetifyig two distict vertices v a ad v b of P The G is othig but a cycle (possibly loop) with at the most two Figure The prime labelig of the graph obtaied by idetifyig the apex vertex with a pedat vertex i,7 Figure 2 The prime labelig of the graph obtaied by idetifyig two of the pedat vertices i,7 Figure The prime labelig of a graph of order five Figure 4 The prime labelig of the graph obtaied by idetifyig the vertices of Figure with label ad 5 Copyright 202 SciRes

3 S VAIDYA, U M PRAJAPATI 0 paths attached at u Such graph is a prime graph as proved i Vaidya ad Prajapati [0] Illustratio 27: I the followig Figures 5-9 prime labeligs for P5 ad the graphs obtaied by idetifyig two vertices i various possible ways are show Strogly Prime Graphs Defiitio : A graph G is said to be a strogly prime Figure 5 A prime labelig of P 5 Figure 6 A prime labelig of the graph obtaied by idetifyig v ad v 2 of P 5 of Figure 5 Figure 7 A prime labelig of the graph obtaied by idetifyig v ad v of P 5 of Figure 5 Figure 8 A prime labelig of the graph obtaied by idetifyig v ad v 4 of P 5 of Figure 5 Figure 9 A prime labelig of the graph obtaied by ideti- fyig v ad v 5 of P 5 of Figure 5 graph if for ay vertex v of G there exits a prime labelig f satisfyig f() v Observatio 2: is a strogly prime graph as ay vertex of ca be assiged label ad the remaiig vertices ca be assiged label 2 ad as show i Figure 0 Observatio : If e is a edge of 4 the G 4 e is a prime graph (see Figure ) but it is ot a strogly prime graph If possible either of the vertices of G with degree 2 ca be assiged the label Suppose v is assiged the label the available vertex labels for t he remaiig three vertices of G v are 2, ad 4 Cosequetly the vertices correspodig to the labels 2 ad 4 are adjacet i G v See Figure 2 Observatio 4: Every spaig subgraph of a strogly prime graph is a strogly prime graph Because every spaig subgraph of a prime graph is a prime graph as proved by Seoud ad Youssef [] Theorem 5: Every path is a strogly prime graph Proof: Let v, v2,, v be the cosecutive vertices of P If v a is ay arbitrary vertex of P the we have the followi g possibilities: Case : If v a is either of the pedat vertices (say va v ) the the fuctio f : V P, 2,, de- f vi i, for all i, 2,,, is a prime la- fied by belig for P with f va f v Case 2: If v a is ot a pedat vertex the a j for some j 2,,, the the fuctio f : V P, 2,, defied by i j if i, 2,, j; f( vi ) i j if i j, j,,, is a prime labelig with f va f v j Thus from the cases d escribed above P is a strogly prime graph Illustratio 6: It is possible to assig label to arbitrary vertex of P 5 i order to obtai differet prime labelig as show i Figures -8 Theorem 7: Every cycle is a strogly prime graph Proof: Let v, v2,, v be the cosecutive vertices of C Let v be ry vertex of a arbitra C The v vj for some j, 2,, The fuctio f : V C, 2,, defied by i j if i,2,, j; f vi i j if i j, j,,, is a prime labelig for C with f() v f v j Thus f admits prime labelig as well a s it is possible to assig label to ay arbitrary vertex of C That is C, is strogly prime graph Theorem 8:, is a strogly prime graph Proof: For, 2 the respective graphs P 2 ad P are strogly prime graphs as proved i the Theorem 5 Copyright 202 SciRes

4 02 S VAIDYA, U M PRAJAPATI Figure 7 A prime labelig of P 5 havig v 5 as label Figure 0 A strogly prime labelig of Figure 8 A prime labelig of P 5 havig as label For 2 let v 0 be the apex vertex ad v, v2,, v be the pedat vertices of, If x is ay arbitrary vertex of, the we have the followig possibilities: Case : If x is the ap ex vertex the x v0 The the fuctio f : V,, 2,, defied by f vi i for i 0,, 2,, is a prime labelig o, with f( x) f v 0 Case 2: If x is oe of the pedat vertices the Figure A prime labelig of e x v j for some j, 2,, Defie 4 f : V, 2,,,, f v 0 p, wher e p is the largest prime less tha or equal to ad the remaiig vertices are distictly labeled from 2,,, p Accordig to Bertrad s postulate Figure 2 4 e is ot a strogly prime graph Figure A prime labelig of P 5 havig v as label Figure 4 A prime labelig of P 5 havig v 2 as label Figure 5 A prime labelig of P 5 havig v as label Figure 6 A prime labelig of P 5 havig v 4 as label p 2 Therefore p is co-prime to every i- teger from, 2,,, p Thus every edge e ab is icidet to the apex vertex v 0 whose label is p, thus gcd f( a), f( b) gcd p, f( b) or gcd f( a), f( b) gcd f( a), p Hece this fuctio f admits a prime labelig o, with f( x) f v j Thus from all the cases described above, is a strogly prime graph Illustratio 9: It is possible to assig label t o arbitrary vertex of,7 i order to obtai prime labelig as show i Figures 9 ad 20 Theorem 0: W is a strogly prime graph for every eve positive iteger 4 Proof: Let v 0 be the apex vertex ad v, v 2,, v be the cosecutive rim ve rtices of W Let x be a arbitrary vertex of W We have the followig possibilities: Case : x is the apex vertex of W that is x v0 The the fuctio f : V W, 2,, defi ed as f v i i (), 0,, 2,, i Obviously f is a ijectio For a arbitrary edge e ab of G we claim that gcd f( a), f( b) To prove our claim the followig subcases are to be cosidered i, 2,, Subcase (): if e v i v i+ for some v 5 Copyright 202 SciRes

5 S VAIDYA, U M PRAJAPATI 0 Figure 9 A prime labelig of,7 with the apex vertex as label The oly differece betwee the defiitio of labelig fuctios of () ad (2) is the labels ad p are iterchaged The clearly f is a ijectio For a arbitrary edge e ab of G we claim that gcd f( a), f( b) T o prove our claim the followig subcases are to be cosidered Subcase (2): If e v0vi for some i, 2,, the gcd f v0, f vi gcd p, f vi as p is co-prime to every ite ger from, 2,, p Subcase (2): If e vv i i for some i, 2,, p p, p,, the gcd f vi, f vi gcd( i, i 2) as i ad i 2 are cosecutive positive itegers Subcase (): If e vv i i for i p 2 the gcd f vi, f vi gcd f vp 2, f v p gcd( p,) Subcase (4): If e vv i i for i p t he gcd f v, f v f vp, v i i gcd f p gc d(, p ) Thus i all the possibilities described above f admits prime labelig as well as it is possible to assig label to ay arbitrary vertex of W That is, W is a strogly prime graph for every eve positive iteger 4 Illustratio : It is possible to assig label to ar- vertex of W bitrary 8 i order to obtai prime labelig as show i Figures 2 ad 22 Corollary 2: The friedship graph ( C ) is a strogly prime graph Figure 20 A prime labelig of,7 with a pedat vertex as label the, gcd f v f v gcd( i, i ) gcd( i, i 2) as i ad i 2 are cosecutive positive itegers Subcase (2): if e vv the gcd f v, f v gcd(,2) as is a odd iteger ad it is ot divisible by 2 Subcase (): if e v0v i for some i, 2,, the gcd f v0, f v i gcd(, i ) Case 2: x is oe of the rim vertices W e may assume that x v p where p is the largest prime less tha or eq ual to Accordig to the Bertrad s Postulate such a prime p exists with p 2 Defie i i a fuctio f : V W, 2,,, as p, if i 0; f vi i, if i, 2,, p ;, if i p (2) Figure 2 A prime labelig of W 8 with the apex vertex as label Figure 22 A prime labelig of W 8 with a rim vertex as label Copyright 202 SciRes

6 04 S VAIDYA, U M PRAJAPATI ( ) Proof: The friedship graph C is a oe poit uio of copies of C It ca also be thought as a graph obtaied by deletig every alterate rim edge of W 2 Beig a spaig subgraph of strogly prime graph ( ) W 2, C is a strogly prime graph Corollary : The star,2 is a strogly prime grap h Proof:,2 is obtaied from strogly prime graph W 2 by deletig all the rim edges of the W 2 Beig a spaig subgraph of strogly prime graph W 2,,2 is a strogly prime graph 4 Cocludig Remarks 5 Ackowledgemets The prime umbers ad their behaviour are of great im- portace as prime umbers are scattered ad there are arbitrarily large gaps i the sequece of prime umbers If these characteristics are studied i the frame work of graph theory the it is more challegig ad excitig as well Here we ivestigate several results o prime graphs This discussio becomes more iterestig i the situatio whe two vertices of a graph are idetified We also itroduce a cocept of strogly prime graph As every prime graph is ot a strogly prime graph it is very excitig to ivestigate graph families which are strogly prime graphs We ivestigate several classes of prime graph which are strogly prime graph The authors are highly thakf ul to the aoymous referee for valuable commets ad costructive suggestios REFERENCES tios, CRC Press, Boca Rato, 999 [] J Gross ad J Yelle, Graph Theor y ad Its Applica- [2] D M Burto, Elemetary Number Theory, 2d Editio, Brow Publishers, New York, 990 [] J A Gallia, A Dyamic Survey of Graph Labelig, The Electroic Joural of Combiatorics, Vol 8, 20 [4] A Tout, A N Dabboucy ad Howalla, Prime Labelig of Graphs, Natioal Academy Sciece Letters, Vol, 982, pp [5] H L Fu ad C Huag, O Prime Labelligs, Discrete Mathematics, Vol 27, No -, 994, pp 8-86 doi:006/002-65x(92) [6] S M Lee, I Wui ad J Yeh, O the Amalgamatio of Prime Graphs, Bulleti of the Malaysia Mathematical Scieces Society (Secod Series), Vol, 988, pp [7] T Deretsky, S M Lee ad J Mitchem, O Vertex Prime Labeligs of Graphs, I: J Alvi, G Chartrad, O Oellerma, A Schwek, Eds, Graph Theory, Combiatorics ad Applicatios: Proceedigs of the 6th Iteratioal Coferece Theory ad Applicatios of Graphs, Wiley, New York, 99, pp [8] S Vaidya ad aai, Prime Labelig for Some Cycle Related Graphs, Joural of Mathematics Research, Vol 2, No 2, 200, pp [9] S Vaidya ad U M Prajapati, Some Switchig Ivariat Prime Graphs, Ope Joural of Discrete Mathematics, Vol 2, No, 202, pp 7-20 doi:0426/ojdm [0] S Vaidya ad U M Prajapati, Some Results o Prime ad k-prime Labelig, Joural of Mathematics Research, Vol, No, 20, pp /6696 [] M A Seoud ad M Z Youssef, O Prime Labelig of Graphs, Cogressus Numeratium, Vol 4, 999, pp Copyright 202 SciRes