Adjacent vertex distinguishing total coloring of tensor product of graphs

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1 America Iteratioal Joural of Available olie at Research i Sciece Techology Egieerig & Mathematics ISSN Prit): ISSN Olie): ISSN CD-ROM): AIJRSTEM is a refereed idexed peer-reviewed multidiscipliary ad ope access oural published by Iteratioal Associatio of Scietific Iovatio ad Research IASIR) USA A Associatio Uifyig the Scieces Egieerig ad Applied Research) Adacet vertex distiguishig total colorig of tesor product of graphs 1 REzhilarasi ad KThirusagu 1 Ramaua Istitute for Advaced Study i Mathematics Uiversity of Madras Cheai Idia Departmet of Mathematics SIVET College Gowrivakkam Cheai Idia Abstract: I this paper we prove the adacet vertex distiguishig total colorig coecture for the graphs amely tesor product ad -tesor product of the graphs Path by Path ad Path by Cycle i detail Also we prove that P C ad K admits adacet vertex distiguishig total colorig ad their color classes are discussed Keywords: simple graph adacet vertex distiguishig total colorig adacet vertex distiguishig total chromatic umber tesor product -tesor product I Itroductio A graph G cosists of a set of vertices V G) ad a set of edges E G) All graphs cosidered i this paper are simple ad fiite For every vertex u v V G) the edge coectig two vertices is deoted by uv EG) ad their distace deoted by d G u v) is the legth of the shortest path betwee u ad v Let G) deote the maximum degree of a graph G If the edge set is empty the G is ull graph For basic termiology ad cocepts of graph theory we refer [] [3] A graph is coected if ay two of its vertices are the edpoits of oe of its paths Otherwise it is discoected For discoected graphs it is ofte useful to cosider the coected subgraphs that are maximal with respect to iclusio They are uiquely determied ad called coected compoets or simply compoets If every compoet of a discoected graph cosists of a sigle vertex we say the graph is totally discoected The totally discoected graph o vertices is deoted by D For graphs G ad 1 G we let G1 G deotes their uio that is V G1 G ) = V G1 ) V G ) ad E G1 G ) = E G1 ) E G ) Throughout the paper we deote the path graph cycle graph complete graph with vertices by P C ad K respectively Defiitio 11 Let G1 V1 E1) ad G V E) be two coected graphs [5] The tesor product of G1 ad G deoted by G G 1 G is the graph with vertex set V G) V G1 ) V G ) ad the edge set ad the edge set EG) = {u v) adacet to ' v' ) d G v v' ) 1 Defiitio 1 u d G u u' ) 1 ad 1 [1] Let - tesor product G G 1 G of G1 ad G has the vertex set V G) V G1 ) V G ) ad the edge set E G 1 G ) = {u v) adacet to u ' v' ) d G u u' ) ad d G v v' ) 1 Defiitio 13 [8] A total k-colorig of G is a mappig f: VG) EG) {1 k k + such that ay two adacet or icidet elemets i VG) EG) have differet colors A proper total k-colorig of G is adacet vertex distiguishig or a total-k-avd-colorig if C f u) C f v) wheever uv EG) where C f v) is the color set or color class) of the vertex v with respect tof) we deote C f v) as Cv) Cv) = fv) f vw) vwe G) ad C v) = {1 k \Cv) The miimum umber of colors required to give a adacet vertex distiguishig total colorig abbreviated as AVDTC) to G is deoted by χ avt G) The well-kow AV DTC coecture madeby Zhag et al [8] says that every simple graph G has χ avt G) G) + 3 More recetly Wag [6] ad Che idepedetly cofirmed this coecture for graph G with G) 3 Xiag'e Che[7] proved χ avt G) 6 for graphs with maximum degree AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 5

2 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 G) = 3 If G is a bipartite graph the χ avt G) G)+ Also JHulga [4] has give a short proof for a upper boud o the adacetvertex distiguishig total chromatic umber of graphs of maximum degree 3 II AVDTC of P C ad ZHANG Zhogfu [8] preseted the cocept of adacet vertex distiguishig total colorig of some special graphs I this sectio we propose a procedure for AV DTC of P C ad K ad also we discuss their color sets Let v 1 v v be the vertices of the graph G The vertices ad edges are colored from the color set {1 k usig the fuctio f Here fv i )deotes the color ofthe vertex v i adfv i v ) deotes the color of the edgev i v 3 = 3 Theorem 1 The graph P admits AV DTC ad χ avtp ) ={ 4 4 Proof Let v 1 v v be the vertices of the path P Defief: VP ) U EP ) {1 kas follows It ca be easily proved for = 3 i = 1 For = 3 fv i ) = i for i = 13 ad fv i v i+1 ) ={ 1 i = The color classes are Cv 1 ) = Cv 3 ) = {31 ad Cv ) = {13 1 if i 1 mod ) Now we prove for 4 For 1 i fv i ) ={ if i 0 mod ) 3 if i 1 mod ) For 1 i -1 fv i v i+1 ) ={ 4 if i 0 mod ) Moreover for i 0 mod ) Cv i )= {34 ad Cv i+1 )= {431 {3 if 0 mod ) Both Cv 1 )ad Cv )are two-elemet set Cv 1 )= {31 ad Cv )={ {41 if 1 mod ) Clearly the color set of ay two adacet vertices are differet 3 = 3 χ avtp ) ={ 4 4 Theorem The graph C admits AVDTC ad C ) = 4 for 4 avt Proof: Let v 1v v be the vertices of the cycle C with vertices ad edges Defie f: VC) EC) {1 k as follows Case-1 If 1mod) For 1 i -1 1 fori 1mod) f v ) i fori 0mod) 3 1mod) For 1 i - ) fori f v v 1 ad fv i i )=4 fv v 1)= fv -1v )=1 4 fori 0mod) The color classes are C v ) C v ) {31 C v ) {14 For i Case- If 0mod) 1 1 {431 fori 1mod) C v i ) {34 fori 0mod) For 1 i 1 fori 1mod) f vi ) fori 0mod) For 1 i mod) ) fori f vivi 1 4 fori 0mod) ad fv v 1)=4 {431 if i 1mod) The color classes are C v i ) {34 if i 0mod) ) = 4 Therefore avt C for 4 Theorem 3 The graph K admits AVDTC ad 1 if 0mod) K ) = avt if 1mod) 1) Proof: Let v 1v v be the vertices of K with vertices ad edges K AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 53

3 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 VK )={ v 1v v 3 ad Case-1 If 0mod ) E K ) { v v 1 i ad i fv i)= for 1 i i fviv)=i+ for i+=+1 i For i+ +1 fori 0mod) f viv ) i ) mod 1) fori 1mod) The color classes are for i 1 C v i ) { i C v i ) i 1 Case- If 1mod ) For i+ + The color classes are fv i)= for 1 i fviv)=i+ for i+=+ i fori 0mod) f viv ) i ) mod ) fori 1mod) _ C v i ) i for 1 i _ ; ) _ C v i 1 i 1 ; for i C v 1 ) 1 Clearly the color set of ay two adacet vertices are differet 1 if 0mod) avt K ) = if 1mod) III AV DTC of G1 G Next we preset the AV DTC of tesor product of two graphs G 1 ad G Let {v 1 v v m be the vertices of G 1 ad {w 1 w w be the vertices of G By the defiitio of tesor product we obtai a graph with vertex set {u i for i = 1 m ad = 1 Now we start by ivestigatig a product of two path P m ad P m Theorem 31 The tesor product P m ad P admits AV DTC ad 3 for m = ad = 3 4 for m = ad 4 avt Pm P ) ={ 5 for m = 3 ad 3 6 for m 4 ad 4 Proof Let the vertex set of P m ad P are {v 1 v v m ad {w 1 w w respectively The vertices ad edges are colored by defiig f : V P P ) E P P ) {1 k The vertex ad edge set of P m P is give by VP m P ) = {u i i = 1 m = 1 m m1 m1 EP m P ) = u ui1 ) ui 1 u 1)) 1 1) i1 i1 m Clearly P m P has m vertices ad m-1)-1) edges First we prove for m = the graph G = P P has two compoets each of them is path of legth 1 For m = ad = fu 1 ) = 1; fu ) = ; fu 1 u 1 ) = fu 1 1 u ) = 3 for = 1 The color classes of the graph P P is Cu 1 1 ) = Cu 1 ) = {31 ad Cu 1 ) = Cu ) = {3 For = 3 fu ) = fu 1 ) = for = 13 fu 1 u 1 ) = ; fu 1 1 u ) = 3 fu u 1 3 ) = fu 1 u 3 ) = 1 The color classes of the graph P P 3 for i = 1 Cu i 1 ) = Cu i 3 ) = {13 ad Cu i ) = {13 Next for 4 fu i ) = 1; for i = 1 ad 1 For i = 1 ad = 13-1) fu i u i+1 +1 ) = i + ; fu i+1 u i +1 ) = i + 3 The color classes are Cu 1 1 ) = {31 Cu 1 ) = {41 Cu 1 ) = {4 Cu ) = {3 Cu 1 ) = {341 Cu ) = {34 for = 3 1) 3 = 3 The χ avt P P )={ 4 4 AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 54

4 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 Next we discuss for m = 3 ad 3 Cosider the graph G = P 3 P with 3 vertices ad 4-1) edges fu i ) = i; for i = 13 ad = 13 For i = 1 ad 1-1) 5 if i + 5 1) 0mod 5) fu i u i+1 +1 ) = i + ad fu i+1 u i +1 )= { i + 5 1)mod 5) otherwise The color classes of P 3 P are Cu 1 1 ) = {31 Cu 1 ) = {51 Cu 1 ) = {45 Cu ) = {13; Cu 3 1 ) = {13 Cu 3 ) = {43 For -1) Cu 1 ) = {351 Cu ) = {1345; Cu 3 ) = {143 χ avt P 3 P ) = 5 for 3 Now we cosider the geeral graph P m P for m 4 ad 4 1 if i 1 mod ) For 1 i m ad 1 ; fu i ) ={ if i 0 mod ) For 1 i m-1 ad if i 1 mod ) fu i u i+1 +1 )={ 4 if i 0 mod ) ad fu 5 if i 1 mod ) i+1 u i +1 )={ 6 if i 0 mod ) The color classes are Cu 1 1 ) = {31 Cu 1 ) = {51 Cu 1 ) = {351 for = 3-1 For i m-1 ad -1 {3456 for i 0 mod ) {54 for i 0 mod ) Cu i )={ ad Cu {34561 for i 1 mod ) i 1 )={ {361 for i 1 mod ) {36 if i 0 mod ) Cu i )={ {451 if i 1 mod ) {35 if m is eve ad Cu m )={ {461 if m is odd {61 if m is odd Cu m 1 )={ {5 if m is eve {3 m ad eve ad Cu m )={ {41 m ad odd Therefore the colour classes of ay two adacet vertices are differet 3 for m = ad = 3 4 for m = ad 4 χ avt P m P ) ={ 5 for m = 3 ad 3 6 for m 4 ad 4 Hece the theorem Theorem 3 The tesor product P m C admits AV DTC ad 4 for m = ad 3 5 for m = 3 ad 3 χ avt P m C ) ={ 6 for m 4 ad 4 Proof Let the vertex sets of P m ad C are {v 1 v v m ad {w 1 w w respectively The vertex ad edge set of the graph P m C is give by VP m C ) ={u i i = 1 m = 1 m1 m1 u ui1 1) ui 1 u 1)) 1 1) EP m C ) = i1 i1 m 1 m1 u u i1 1) ui 1 u 1)) i1 i1 Clearly P m C ) has m vertices ad m - 1)) edges First we prove for m = ad 3 The graph G = P C ) has vertices ad edges G = P C ={ C if 1 mod ) The vertices ad C if 0 mod ) edges are colored by defiig f: V P m C ) E P m C ){1 k fu i ) = i ; for i = 1 ad = 13 For i = 1 ad 1-1) fu i u i+1 +1 ) = i + ; fu i+1 u i +1 ) = i + 3 For i = 1 = fu i+1 +1 u i ) = i + ; fu i +1 u i+1 ) = i + 3 The color classes are Cu 1 ) = {341 Cu ) = {34 for = 1 The χ avt P C ) = 4 3 Next we discuss the case m = 3 ad 3 The graph P 3 C has 3 vertices ad 4 edges AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 55

5 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 fu i ) = i ; for i = 13 ad = 13 For i = 1 ad 1-1) 5 if i + 5 1) 0mod 5) fu i u i+1 +1 ) = i + ; ad fu i+1 u i +1 ) { i + 5 1)mod 5) otherwise 5 if i = 1 For i = 1 ad = fu i u i+1 +1 ) = i + ad f u i+1 u i +1 )={ 1 if i = For 1 the color classes are Cu 1 ) = {351 Cu ) = {1345; Cu 3 ) = {143 χ avt P 3 C ) = 5; 3 Next we prove for the geeral graph P m C for m 4 ad 4 V P m C ) = m ad E P m C ) = m - 1) For 1 i m ad 1 1 for i 1 mod ) fu i )={ for i 0 mod ) For 1 i m - 1) ad 1-1) 3 if i 1 mod ) 5 if i 1 mod ) fu i u i+1 +1 )={ ad fu 4 if i 0 mod ) i+1 u i +1 )={ 6 if i 0 mod ) For 1 i m - 1) ad = 5 if i 1 mod ) fu i+1 u i +1 )={ 6 if i 0 mod ) 3 if i 1 mod ) ad fu i+1 +1 u i )={ 4 if i 0 mod ) The color classes are Cu 1 ) = {351 for = 1 For i m - 1) ad 1 {3456 if i 0 mod ) Cu i )={ ad Cu {34561 if i 1 mod ) m )={ {461 if m is odd {35 if m is eve Here the color classes of ay two adacet vertices are differet Thus 4 for m = ad 3 5 for m = 3 ad 3 χ avt P m C ) ={ Hece the theorem 6 for m 4 ad 4 IV AVDTC of G1 G I this sectio we discuss a detailed procedure for - tesor product of some graphs ad their AV DTC The -tesor product of two graphs G 1V 1;E 1) ad G V ;E ) is deoted by G1 G Let G P ad 1 m G P be the two path graphs with m ad vertices respectively The graph P is a ull graph isolated Pm Pm) { v1 v vm vertices) if G 1 ) or G ) Cosider the vertex set V ad V P ) { w1 w w respectively Sice P P is a ull graph we cosider m 3 Theorem 41 The graph P m P admits AVDTC ad 3 for m = 34 ad = for m = 34 ad 7 χ avt P m P ) = 5 for m = 56 ad 5 6 for m 7 ad 7 { Proof: Let the vertex sets of P m ad P are {v 1v v m ad {w 1w w respectively The vertex ad edge set of P m P is give by V Pm P ) { u i 13 m ad 13 m m E Pm P ) u ui uiu 13 ) i1 i1 The graph P m P has m vertices ad -)m-) edges First we prove for m=34 ad =34 AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 56

6 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 If 1 mod ) the graph P 3 P has two path graphs each of legth 1 ad with D vertices These D vertices are are u ) =13 If 0 mod ) the graph P 3 P has four path graphs each of legth with D vertices The graph P 4 P has four path each of legth 1 ad 1 for 1 mod ) ad it has eight path graphs each of legth for 0 mod ) The vertices ad edges are colored by defiig f: V P m P ) E P m P ) {1 k We discuss the case for m=34 =34 ad m 1 fori 1 For 1 f u ) otherwise For 1 i m- ad 1 - f u u ) f u u ) 3 Next For the case =56 We have for 1 i m ad 1 f u 1 for 1 ) for 34 3 for 56 3 for 1 For 1 i m-) ad 1 -) f ui u ) f u u ) 1 otherwise 1 fori 1 Now for 7 we have for 1 i m ad 1 f u ) ad fori 34 For 1 i m-) ad 1 -) f u ) i u 3 ad f u ) i u 4 Nextwe discuss the case for m=56 ad 5 For 1 i m ad 1 f u 1 fori 1 ) fori 34 ad 3 otherwise For 1 i m-) ad 1 -) we have 3 fori 1 f u u ) 5 fori 1 ad f ui ui 4 otherwise ) 1 otherwise Now we discuss the color classes of P 3 P for 3 The color classes of P 3 P 3 for =13 Cu 1)={13 Cu 3)={3 Cu 1)={1 Cu 3)={ ad Cu )={1 for =13 {31 fori 1 The color classes of the graph P 3 P 4 for =134 we have C u ) {1 fori {3 fori 3 The color classes of the graph P 3 P 5 for i=13 we have {31 for 15 {1 for 1 C u ) {13 for 3 ad i C u ) { for 34 {3 for 4 {3 for 5 The color classes of the graph P 3 P 6 for i=13 we have {1 for 1 {13 for 156 C u ) ad C u ) {13 for 34 { for 34 {3 for 56 The color classes of the graph P 3 P 7 are Cu )={1 {13 for 1 {4 for 1 C u ) {134 for 34 ad 1 C u ) {34 for 34 3 {14 for 1 {3 for 1 Next for m=4 ad =4 For 1 the color classes of the graph P 4 P 4 are AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 57

7 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 {31 fori 1 C u ) {3 fori 34 {31 for 15 The color classes of the graph P 4 P 5 for i=134 C u ) {13 for 3 i {3 for 4 {31 for 156 The color classes of the graph P 4 P 6 for i=134 C u ) {13 for 34 {31 for 1 Now we discuss the color classes of P 4 P for 7 For i=1 C u ) {134 for 34 {14 for 1 {4 for 1 ad for i=34 C u ) {34 for 34 {3 for 1 The color classes of the graph P m P for m=5 ad 5 we have {13 for 1 {45 for 1 C u ) {135 for 34 C u 1 3 ) {1345 for 34 {34 for 1 {13 for 1 {13 for 1 {31 for 1 C u ) {134 for 34 C u 5 ) {134 for 34 {34 for 1 {14 for 1 {4 for 1 C u4 ) {34 for 34 {3 for 1 The color classes of P m P for m = 6 ad 5 we have {13 for 1 {45 for 1 For i=1 C ui ) {135 for 34 ; For i=34 C u ) {1345 for 34 {15 for 1 {13 for 1 {13 for 1 For i=56 C ui ) {134 for 34 {34 for 1 Now for the geeral graph P m P m fori 1mod4) For 1 i m ad 1 f u ) fori 03mod4) For 1 i m-) ad 1 -) 3 1mod4) ) fori 5 1mod4) f u u ad i i 4 fori 03mod4) ) fori f ui u 6 fori 03mod4) Now we discuss the color classes for i=1 of the graph P m P for m 7 7 C u ) {13 for 1 {135 for 34 {15 for 1 AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 58

8 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 {136 for 1 {13456 for3 if i 1 mod4) For 3 i m- {145 for 1 C u ) {45 for 1 {3456 for3 if i 03mod4) {36 for 1 {16 for 1 If m mod 4) the C um 1 ) C um ) {146 for 3 {14 for 1 {5 for 1 If m 0mod 4) the C um 1 ) C um ) {35 for 3 {3 for 1 If m 1mod 4) the {16 for 1 {5 for 1 ad C um ) {146 for 3 C um 1 ) {35 for 3 {14 for 1 {3 for 1 If m 3mod 4) the {16 for 1 {5 for 1 ad C um 1 ) {146 for 3 C um ) {35 for 3 {14 for 1 {3 for 1 Clearly the color classes of ay two adacet vertices are differet Cotiuig the process we have 3 for m = 34 ad = for m = 34 ad 7 χ avt P m P ) = 5 for m = 56 ad 5 Hece the theorem 6 for m 7 ad 7 { 3 form 34 ad 4 Theorem 4 The graph P m C admits AVDTC ad 4 form 34 ad 4 avt Pm C ) = 5 for m 56 ad 5 6 for m 7ad 7 Proof: The graph P m C has the vertex V P C ) { u i 13 m ad 13 m m m u u u u for 1 ) ad the edge set E i 1 i 1 Pm C ) m m u u u u for 1 i1 i1 P m C has m vertices ad m-) edges The vertices ad edges are colored by defiig f: V P m C ) E P m C ) {1 k First for m=34 ad 4 For 1 i m ad 1 1 for i 1 f ui ) otherwise For =4 for 1 i m-) ad 1 -) f u u ) f u u ) 3 Next for m = 34 ad > 4 The graph P m C has D vertices with four cycles each of legth whe 0mod 4) or a cycle of legth whe 13mod 4) or two cycles each of legth whe mod 4) The graph P m C has 4 vertices ad 4 edges ad it has all bipartite compoets with two cycles each of legth for 13 mod 4) eight cycles each of legth for 0 mod 4) ad four cycles each of legth for mod4) For 1 i m-) ad 1 -) f u u ) 3; f u u ) 4ad AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 59

9 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 For = -1) f u u ) 3; f u u ) 4 {134 fori 1 The color set for 1 of the graph P3 C for > 4 C ui ) {34 for i 3 {1 fori The color set for 1 of the graph P4 C {134 fori 1 for > 4 C u ) {34 fori 34 1 fori 1 Now we discuss the case for m=56 ad 5 For 1 i m ad 1 f u ) fori 34 3 otherwise For 1 i m-) ad 1 -) 3 for i 1 f ui ui ) ad 5 fori 1 f u u 4 otherwise ) i 1 otherwise 5 fori 1 For =-1) we have f ui ui ) ad 3 fori 1 f u u ) 1 otherwise 4 otherwise {135 fori 1 The color classes of P 5 C for 5 we have 1 {143 fori 5 C u ) {1345 fori 3 {34 fori 4 {135 fori 1 The color classes of P 6 C for 5 we have for 1 C u ) {1345 fori 34 {143 fori 56 Now we prove the geeral graph for m 7 ad 7 For 1 i m ad 1 1 fori 1mod4) f u ) fori 03mod4) For 1 i m-) ad 1 -) 5 fori 1mod4) ad 5 1mod4) f u u ) ) fori f u u 6 fori 03mod4) 6 fori 03mod4) For =-1) 5 fori 1mod4) f u u ) ad 3 fori 1mod4) i i f u u ) i 6 fori 03mod4) 4 fori 03mod4) The color classes are C u i ) {135 for i 1 ad 1 For 3 i m- {13456 if i 1 mod4) C u ) {3456 if i 03mod4) If m 1 mod 4) ad 1 {35 if i m 1 C u ) {146 if i m If m mod 4) ad 1 C u ) C u ) {146 m1 m um 1 ) C um ) If m 0 mod 4) ad 1 C {35 If m 3 mod 4) ad 1 {35 if i m C u ) {146 if i m 1 The color classes are differet for ay two adacet vertices 3 for m 34 ad 4 4 for m 34 ad 4 Hece the theorem avt Pm C ) = 5 for m 56 ad 5 6 for m 7 ad 7 Coclusio: We foud the adacet vertex distiguishig total chromatic umber of tesor product ad -tesor product of path by path ad path by cycle Also we are workig i the directio of fidig theadacet vertex distiguishig total chromatic umber of -tesor product of path P 3 with some special graphs like wheel star su let pa graph ad complete bipartite graph V Refereces [1] UP Acharya ad HS Mehta "-Tesor product of graphs"iteratioal Joural of Mathematics ad Scietific Computig Vol4 No1 014) AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 60

10 REzhilarasi et al America Iteratioal Joural of Research i Sciece Techology Egieerig & Mathematics41) September- November 018 pp 5-61 [] NL Biggs Algebraic Graph Theory Cambridge Uiversity Press d editio Cambridge 1993 [3] Harary F Graph Theory Addiso-Wesley Readig Mass 1969) [4] Joatha Hulga "Cocise proofs for adacet vertex-distiduishig total colorigs" Discrete Mathematics ) [5] KKParthasarathy Basic Graph Theory Tata McGraw Hill 1994 [6] H Wag "O Adacet-Vertex- Distiguishig total chromatic umbers of graphs with 4 = 3" Jcomb Optim14 007) [7] Xiag' e Che "O Adacet-Vertex- Distiguishig total colorig umbers of graphs with 4 = 3" Discrete Mathematics ) [8] ZHANG Zhogfu CHEN Xiag 'e L Jigwe YAO Big LU Xizhog ad WANG Jiafag "O Adacet-Vertex- Distiguishig total colorig of graphs" Sciece i Chia SerA Mathematics 005 Vol 48 No3) AIJRSTEM 18-41; 018 AIJRSTEM All Rights Reserved Page 61