PAijpam.eu IRREGULAR SET COLORINGS OF GRAPHS

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1 Iteratioal Joural of Pure ad Applied Mathematics Volume 109 No , ISSN: (prited versio); ISSN: (o-lie versio) url: doi: /ijpam.v109i7.18 PAijpam.eu IRREGULAR SET COLORINGS OF GRAPHS P. Roushii Leely Pushpam 1, N. Srilakshmi 1, Departmet of Mathematics D. B. Jai College Cheai , Tamil Nadu, Idia Abstract: A proper colorig of a graph G is a fuctio c: V(G) {1,,...,k} for some positive iteger k. The color code of a vertex v (with respect to c) is the ordered (k+1) - tuple code(v) = (a 0,a 1,...,a k ), where a 0 is the color assiged to v ad for 1 i k, a i is the umber of vertices adjacet to v that are colored i. The colorig c is irregular if distict vertices have distict color codes ad the irregular chromatic umber χ ir(g) of G is the miimum positive iteger k for which G has a irregular k-colorig. This parameter was itroduced by Mary Radcliffe et al. We defie a irregular set colorig as follows: For a proper colorig c : V(G) {1,,...,k}, we defie the color code of a vertex v (with respect to c) by the set code(v) = {a 0,a 1,...,a k }, where a 0 is the color assiged to v ad for 1 i k, a i is the umber of vertices that are colored i. We defie the colorig c to be a irregular set colorig if distict vertices have distict color codes. The irregular set chromatic umber χ irs(g) is the miimum positive iteger k for which G has a irregular set k-colorig. I this paper we determie the exact value of χ irs(g) for paths, cycles ad wheels. AMS Subject Classificatio: 05C15 Key Words: irregular set colorig, irregular set chromatic umber 1. Itroductio The problem of studyig methods of distiguishig the vertices of a coected graph from oe aother usig graph colorig is a wide area of research. I this directio oe method of vertex-distiguishig colorig amely, irregular colorigs was itroduced ad studied by Chartrad, Mary Radcliffe, Okamoto Received: October 1, 016 Published: April 5, 016 c 016 Academic Publicatios, Ltd. url:

2 144 P. Roushii Leely Pushpam, N. Srilakshmi ad Pig Zhag i [4]. Much work related to irregular colorigs has bee doe i [1,,4,5,6]. For graph theory termiologies we i geeral follow [3]. A proper colorig of a graph G is a fuctio c: V(G) {1,,...k} of G for some positive iteger k such that o two adjacet vertices receive the same color. For a graph G ad a proper k-colorig c of G,the color code of a vertex v (with respect to c) is the ordered (k+1) - tuple code(v)= (a 0,a 1,...,a k ) where a 0 is the color assiged to v ad for 1 i k, a i is the umber of vertices adjacet to v that are colored i. The colorig c is irregular if distict vertices have distict color codes ad the irregular chromatic umber χ ir (G) of G is the miimum positive iteger k for which G has a irregular k-colorig. We defie a irregular set colorig as follows: For a proper colorig c: V(G) {1,,...,k}, we defie the color code of a vertex v (with respect to c) by the set code(v) = {a 0,a 1,...,a k }, where a 0 is the color assiged to v ad for 1 i k, a i is the umber of vertices adjacet to v that are colored i. We defie the colorig c to be a irregular set colorig if distict vertices have distict color codes. The irregular set chromatic umber χ irs (G) is the miimum positive iteger k for which G has a irregular set k-colorig. Sice every irregular set colorig is a irregular colorig of G, it follows that χ(g) χ ir (G) χ irs (G). To illustrate this cocept, cosider the path P 4 with vertices v 1,v,v 3,v 4 take i that order. We kow that χ(p 4 ) = χ ir (P 4 ) =. We claim that χ irs (P 4 ) = 3. For, if χ irs (P 4 )=, the the color sequece of the vertices of P 4 is 1,,1, ad the color codes of the vertices of P 4 are code(v 1 ) = {1,0,1}, code(v ) = {,,0}, code(v 3 ) = {1,0,}, code(v 4 ) = {,1,0}, which implies that code(v 3 ) = code(v 4 ). Hece χ irs (P 4 ) 3. The color sequece of P 4 is 1,,3,. Clearly the color codes of the vertices are distict. Therefore χ irs (P 4 ) = 3. I this paper we determie the exact value of χ irs (G) for paths, cycles ad wheels.. Observatios The followig observatios are immediate. Observatio.1: For a irregular set colorig c of a graph G ad for ay two distict pedat vertices x,y of G, c(x) c(y). Observatio.: For a irregular set colorig c of a graph G, a color ca be assiged to at most two vertices of degree two. Observatio.3: For a irregular set colorig c of a graph G ad a vertex v

3 IRREGULAR SET COLORINGS OF GRAPHS 145 of G, we shall deote d(v) = {d 1,d,...,d k } where d i represets the umber of vertices colored i. Hece deg(v) = i=1 d i. Observatio.4: Let c be a irregular set colorig of a graph G. If u ad v are distict vertices of G with N(u) = N(v) the c(u) c(v). 3. Paths ad Cycles I this sectio we ivestigate the irregular set colorigs of paths ad cycles. Theorem 1. For paths P,, =,3, χ irs (P ) = 3, = 4,5, 4, = 6,7,8. Proof. Let V(P ) = {v 1, v,...,v k }. Oe ca easily observe that χ irs (P ) = whe =,3 ad χ irs (P ) = 3 whe = 4,5. Let c be a irregular set colorig of P 6. As a color ca be assiged to at most two vertices of degree two, χ irs (P 6 ) 3. Suppose χ irs (P 6 ) = 3. Sice the pedat vertices caot be assiged the same color, without loss of geerality, let c(v 1 ) = 1 ad c(v 6 ) =. Suppose c(v ) =, the either c(v 3 ) = 1 or 3. If c(v 3 ) = 1, the c(v 4 ). For otherwise, code (v 3 ) = {1,0,,0} = code (v 6 ), a cotradictio. Hece c(v 4 ) = 3 ad c(v 5 ) = 1 which implies code (v 3 ) = code (v 5 ) = {1,0,1,1}, a cotradictio. Suppose c(v 3 ) = 3, the either c(v 4 ) = 1 or. If c(v 4 ) = 1,the c(v 5 ) = 3 which implies code (v 3 ) = code (v 5 ) = {3,1,1,0}, a cotradictio. If c(v 4 ) =, the either c(v 5 ) = 1 or 3. If c(v 5 ) =1, code c(v 5 ) = c(v 3 ) = {3,0,,0}, a cotradictio. Similarly, if c(v 5 ) = 3, we get a cotradictio. Hece χ irs (P 6 ) 4. Now we defie the color sequece of P 6 with respect to c as s: 1,,1,,3,4. Now, code(v 1 ) = {1,0,1,0,0}, code(v ) = {,,0,0,0}, code(v 3 ) = {1,0,,0,0}, code(v 4 )={,1,0,1,0}, code(v 5 )={3,0,1,0,1}, code(v 6 )={4,0,0,1,0}. HececisairregularsetcolorigofP 6 Heceχ irs (P 6 ) 4.Therefore, χ irs (P 6 ) = 4. A similar argumet holds for P 7 ad P 8. Theorem. For paths P, > 8, χ irs (P ) = Proof. Let c be a irregular set colorig of P. Sice at most two vertices of degree two ca be assiged the same color, it is clear that χ irs (P )..

4 146 P. Roushii Leely Pushpam, N. Srilakshmi Let u ad v be o adjacet vertices of degree two i P such that they are ot adjacet to pedat vertices. Suppose these vertices are assiged the same color say 1, the d(u) ad d(v) are either {,0} or {1,1}. Without loss of geerality, let d(u) = {,0} ad d(v) = {1,1}. Now d(u) ={,0} implies that theieghbours of u amely u 1,u has to be assiged the same color, say. Now, agai d(u 1 ) ad d(u ) are either {,0} or {1,1}. Without loss of geerality, let d(u 1 ) = {,0} ad d(u ) = {1,1}. Hece the eighbour of u 1 other tha u has to be assiged the color 1 ad v has already bee assiged the color 1. Therefore u 1 ad v are adjacet. Hece 1,,1, is a subsequece of the colorig c i which all the four vertices are of degree two. Hece i geeral if the ith color has to be assiged to two vertices, a subsequece of c has to be i,j,i,j. Now we give the color sequece with respect to c as follows. 1), ( Whe 0(mod4), s: 1,( ),1,,1,,...,( ),( Whe 1(mod 4), s : 3,1,,1,,...,( 1), ( ( ). Whe (mod 4), s : 3,1,,1,,...,( ( ),1. Whe 3(mod 4), s : 3,1,,1,,...,( ( 1),( ),4. 1),( ),( ),( 1),( ),3. ), ( 1), ), ( 1), ), 1), ( I all the above cases we see that the codes of the vertices are distict. Hece χ irs (P ). Therefore χirs (P ) =. Theorem 3. For cycles C,, = 3,4, χ irs (C ) =, 0,1(mod 4), +1,,3(mod 4). Proof. Oe ca easily observe that χ irs (C ) =, whe = 3,4. Let 5 ad c be a irregular set colorig of C. As a color ca be assiged to at most two vertices of degree two, as discussed i Theorem 3., we have i,j,i,j as a subsequece of the color sequece c. Hece χ irs (C ). Now we ivestigate the followig cases. Case 1: 0,1(mod 4). We give the color sequece with respect to c as follows: Whe 0(mod 4), s: 1,,1,,...,( 1),( ),( 1),( ). Whe 1(mod 4), s: 1,,1,,...,( ),( 1), ( ), ( 1),( ).

5 IRREGULAR SET COLORINGS OF GRAPHS 147 We see that the codes are distict for distict vertices. Hece χ irs (C ). Therefore χ irs (C ) =. Case :,3(mod 4). Whe (mod 4), ( ) colors are assiged twice. Clearly the remaiig two vertices are assiged distict colors. Thus χ irs (C ) The color sequece with respect to c is: s: 1,,1,,...,( + 3)( + ),( + = ),( + ), ( + 1), ( + ). Whe 3(mod 4), ( 3 ) colors are assiged twice. Suppose that vertices v 1,v,...,v 3 have bee colored with ( 3 ) colors. We claim that the remaiig three vertices are assiged distict colors. Suppose c(v ) = c(v ) = x, the code (v ) = code (v ),a cotradictio. Therefore c (v ) c (v ). Thus χ irs (C ) We defie the color sequece with respect to c as s: 1,,1,,...,( + 4),( + 3),( + 4),( + 3),( + We see that the codes are distict for distict vertices. Thus χ irs (C ) +. Therefore χirs (C ) = +. ), ( + 1),( + ) 4. Wheels A wheel graph W is a graph formed by coectig a sigle vertex to all the vertices of a cycle. I this sectio we ivestigate the irregular set colorigs of wheels. Theorem 4. For wheels W,

6 148 P. Roushii Leely Pushpam, N. Srilakshmi χ irs (W ) =, = 4,5, ( +4 ), 0(mod 4), ( +1 ), 1(mod 4), ( + ), (mod 4), ), 3(mod 4), ( +3 Proof. Let v be the vertex of W with deg(v)= ( 1) ad let v 1,v,...,v 1 be the vertices of degree three. Whe = 4, W 4 is isomorphic to K 4. Therefore, χ irs (W 4 ) =4. Whe = 5, let c(v) =1. If c(v 1 ) = c(v 3 ) =adc(v ) =c(v 4 ) = 3, the code (v 1 ) = code (v 3 ) = {,1,0,} ad code (v ) = code (v 4 ) = {3,1,,0}, a cotradictio. Hece a color ca be used exactly oce. Thusχ irs (W 5 ) =5. Let cbeairregular set colorig ofg.thec(v) c(v i ), for 1 i 1. Without loss of geerality, let c(v) = 1. Now, the possiblities of d(v i ), 1 i 1 are {1,1,1,0,0,0,...,0}, {3,0,0,0,...,0}, {,1,0,0,0,...,0}. Sice c(v) = 1, {3,0,0,...,0} is ot possible. Hece a color ca be assiged to at most two vertices o the cycle. As discussed i the proof of Theorem 3., i,j,i,j is a subsequece of the colorig c. Hece χ irs (W ) Now we cosider the followig cases. Case 1: 0(mod 4). It is clear that at most two vertices o the cycle are assiged the same color. Supposec(v)=1adv 1,v,...,v 4 have beecolored with( 4 )colors. Cosider the remaiig vertices v 3,v,v 1. If c(v 3 ) = c(v 1 ) = x, the code (v 3 ) = code (v 1 ) = {x,1,1,1,0,0,...,0}, a cotradictio. Therefore χ irs (W ) The color sequece of W is s: 1,,3,,3,...,( 4 ),( 4 +1),( 4 ),( 4 +1),( 4 +), ( 4 +3), ( 4 +4). We see that the codes of all the vertices are distict. Therefore, χ irs (W ) +4. Hece, χ irs(w ) = +4. Case : 1(mod 4).

7 IRREGULAR SET COLORINGS OF GRAPHS 149 It is clear that χ irs (W ) The color sequece is s : 1,,3,,3,...,( 1 ),(+1 ),( 1 ),(+1 ). Thus, we see that the codes of all the vertices are distict. Therefore, χ irs (W ) +1. Hece χ irs(w ) = +1. Case 3: (mod 4). I this case it is clear that χ irs (W ) 1+ + The color sequece is s : 1,,3,,3,..., ( ( +). Thus the codes of distict vertices are distict. Therefore, χ irs (W ) +. Therefore, χ irs (W ) = +. Case 4: 3(mod 4) I this case it is clear that, ), ( χ irs (W ) ), ( ), ( + 1), The color sequece of the vertices are s : 1,,3,,3,...,( 3 ),( 3 +1),( 3 ),( 3 +1),( 3 +),( 3 +3). We see that the color codes are distict for distict vertices. Therefore χ irs (W ) +3. Thus, χ irs (W ) = Refereces [1] A. C. Burris, O graphs with irregular colorig umber. Cogr. Numer. 100 (1994) [] A. C. Burris, The irregular colorig umber of a tree. Discrete Math. 141 (1995) [3] G. Chartrad ad P. Zhag, Itroductio to graph theory. McGraw - Hill, Bosto (005). [4] F. Okamoto, M. Radcliffe ad P. Zhag, O the irregular chromatic umber of a graph. Cogr. Numer. 181 (006) [5] M. Radcliffe ad P. Zhag, O irregular colorigs of graphs. AKCE J Graphs. Combi.,3, No. (006)

8 150 P. Roushii Leely Pushpam, N. Srilakshmi [6] M. Radcliffe ad P. Zhag, O irregular colorigs of graphs. Bull. Ist. Combi. Appl.49 (007)