Super Vertex Magic and E-Super Vertex Magic. Total Labelling

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1 Proceedigs of the Iteratioal Coferece o Applied Mathematics ad Theoretical Computer Sciece Super Vertex Magic ad E-Super Vertex Magic Total Labellig C.J. Deei ad D. Atoy Xavier Abstract--- For a fiite simple graph G let V = p ad E =. I this paper we fid out some graphs which posses super vertex ad E-super vertex magic total labellig ad also some graphs do ot posses. Also I this paper we use the duality of regular graphs to fid out the super vertex ad E- super vertex magic total labellig. Keywords--- Duality, E-Super Vertex Magic Total Labellig, Super Vertex Magic Total Labellig, Vertex Magic Total Labellig I I. INTRODUCTION N this paper cosider oly fiite simple udirected graph. The set vertices ad edges of a graph G will be deoted by V (G) ad E(G) respectively p = V ad = E. The set of vertices adjacet to a vertex u of G is deoted by N(u) MacDougall et.al [0] itroduces the idea of vertex magic total labellig. This is a assigmet of itegers from to p+ to the vertices ad edges of G. So that at each vertex, the vertex label ad the labels o the edges icidet at that vertex, adds to a fixed costat. f ( u) f ( uv) u N ( u) Where the sums rus over all vertices v adjacet to u. MacDougall, Miller ad Sugeg [8] called a vertex magic total labellig is super if f(v(g)) = (,, 3, p). i.e. the smallest labels are assiged to the vertices. Swamiatha ad Jeyathi [9] called a vertex magic total labellig is super if f(e(g)) = (,, 3, ). But Marimuthu ad Balarisha [] called the above type of labellig as E-super vertex magic total labellig. To avoid cofusio i this paper we use, i super vertex magic total labellig the smallest labels are assiged to the vertices ad the E-super vertex magic total labellig the smallest labels are assiged to the edges. I [8] they proved that a r-regular graph of order p has a super vertex magic total labellig the p ad r have opposite parity ad if p 0(mod 8) the 0(mod 4). If p 4(mod 8) the (mod 4). The cycle C has a super vertex magic total labellig if ad oly if is odd. They also cojectured that if 0(mod 4); > 4, the K has a super vertex magic total labellig. But this cojecture was proved by J.Gomez i [5] also tree, wheel, fa, ladder, or friedship graph has o super vertex magic total labellig. If G has a vertex of degree oe, the G is ot super vertex magic C.J. Deei, Research Scholar, Departmet of Mathematics, Loyola College, Cheai, Tamiladu. srdeeicj@gmail.com D. Atoy Xavier, Assistat Professor, Departmet of Mathematics, Loyola College, Cheai, Tamiladu. axloyola@gmail.com total labellig. Swamiatha ad Jeyathi [9] showed that the path P has a E-super vertex magic total labellig if ad oly if is odd ad if ad oly if =. mc is E-super vertex magic total labellig if ad oly if m ad are odd. Marimuthu ad Balarisha [] proved that, for a coected graph G ad G has a E-super vertex magic total labellig with magic costat the 5p 3. Also proved for a (p, ) graph, with eve p ad = p - or p, the the graph is ot E-super vertex magic total labellig. Geeralized Peterse graph P(,m) is ot E- super vertex magic total labellig if is odd. They also discussed about the E-super vertex magicess of m coected graph H m,. A graph with odd order ca be decomposed ito two Hamiltoia cycles, the G is E-super vertex magic total labellig. A graph G ca be decomposed ito two spaig sub graphs G ad G where G is E-super vertex magic ad G is magic ad regular the G E-super vertex magic. Also they proved as the two spaig sub graphs are E-super vertex magic ad oe is regular the the graph will E-super vertex magic. II. Theorem. [8] SUPER VETEX AND E-SUPER VERTEX MAGIC TOTAL LABELLINGS OF REGULAR GRAPHS If G has a super vertex magic total labellig, the ( p )( p ) p p Theorem. [9] If G has a E-super vertex magic total labellig the p ( ) p Corollary. [8] If G has a super vertex magic total labellig, the p divides ( +) if p is odd, ad p divides ( + ) if p is eve. Corollary. [8] If G is a graph of eve order, havig a super vertex magic total labellig the either p 0(mod 8) ad 0 or 3(mod 4) or p 4(mod 4) ad or (mod 4) Defiitio. Let f is a assigmet of itegers from E V to f,, 3, p ad defie f as follows: f (vi) = p f(vi) ad f (uv) = p f(uv). ISBN Bofrig

2 Proceedigs of the Iteratioal Coferece o Applied Mathematics ad Theoretical Computer Sciece clearly f is a oe-to-oe map ad we call f is the dual of f.. Prism Graph For a prism graph D the umber of vertices are ad the umber of edges is 3. Theorem.3 The prism graph D has a super vertex magic total labellig. Suppose D has a super vertex magic total labellig, the by theorem. ( p )( p ) p p ( 3 )( 3 ) 3 4 This is a iteger if ad oly if eve ad 4. D6. Suppose T, has a super vertex magic total labellig, the by theorem. ( p )( p ) p ( )( ) 7 5 This is a iteger if ad oly if odd. Here = 3 ad = 79. Figure 3 To fid E-super vertex magic total labellig, use defiitio : 4 Here = 6 ad = 7. Figure To fid E-super vertex magic total labellig, use defiitio :. D6 Here = 6 ad = 63. Figure. Torus For a torus the umber of vertices is ad umber of edges is. Theorem.4 The T, has a super vertex magic total labellig. Here = 3 ad = 6. Figure 4.3 Hypercubes For a hypercube the umber of vertices is ad umber of edges ( - ). Theorem.5 The hypercube Q has a super vertex magic total labellig. Suppose Q has a super vertex magic total labellig, the by theorem. ( p )( p ) p ( ( ))( ( ) ) ( ) ( ) ( ) ISBN Bofrig

3 Proceedigs of the Iteratioal Coferece o Applied Mathematics ad Theoretical Computer Sciece This is a iteger if ad oly if is odd ad 3 Figure 5 Result. Hm, is ot super vertex magic if both m ad are eve. Result. Hm, is ot super vertex magic if m odd ad is ot a multiple of 4. Result.3 Hm, is ot super vertex magic if both m ad are odd. Result.4 H4, is super vertex magic if ad oly of is odd. H 4, Here = 4 ad = 48. To fid E-super vertex magic total labellig, use defiitio : Here = 4 ad = 36. Figure 6.4 m-coected Graph Hm, I this sectio we discuss about the m-coeceted graph Hm, described i the boo of Body ad Murty []. The structure of Hm, depeds o the parities of m ad. There are three cases: CASE : m eve, eve or odd Let m = r the H r, is costructed as follows. It has vertices 0,,,, - ad i ad j are joied if i - r j i + r(where additio is tae modulo ). CASE : m odd, eve Let m = r + The H r+, is costructed by first drawig Hr, ad the addig edges joiig vertex i to i + for i CASE 3 : m odd, odd. Let m = r + The Hr+, is costructed by first drawig Hr, ad the addig edges joiig vertex 0 to vertex i + for I I [8] marimuthu ad Balarisha discussed about the various cases of super vertex magicess of Hm, regardig to the parity of m ad. So we apply the defiitio. we get the followig results. Figure 7 Here m = 4; = ad = 74. To fid super vertex magic total labellig, use defiitio : H 4, III. Figure 8 Here m = 4; = ad = 96. SUPER VERTEX AND E-SUPER VERTEX MAGIC TOTAL LABELLINGS OF SOME OTHER GRAPHS 3. Parachute Graph For parachute graph the umber of vertices is ad the umber of edges is 3-. Theorem 3. The parachute graph has a E-super vertex magic total labellig. Suppose there exist a magic costat. The by theorem. ISBN Bofrig

4 Proceedigs of the Iteratioal Coferece o Applied Mathematics ad Theoretical Computer Sciece p ( ) p (3 )(3 ) (3 ) 7 4 This is a iteger if ad oly if is eve ad 4. Here = 4 ad = 3. Figure 9 Accordig to Theorem. the magic costat for super vertex magic total labellig is Web Graph Defiitio 3. The web graph W,r is a graph cosistig of r cocetric copies of the cycle graph C with the correspodig vertices coected by spoes. For a web graph W,r the umber of vertices are r ad the umber of edges is (r - ). Theorem 3. The web graph W,r has a super vertex magic total labellig. Suppose W,r has a super vertex magic total labellig, the by theorem. ( p )( p ) p [ r (r )][ r r ] r r 7r 5 6 r Now, two cases arise: If r <, the is a iteger if ad oly if both are eve. If r >, the is a iteger if ad oly if both are eve ad - = r. Theorem 3.3 The web graph W,r has a E-super vertex magic total labellig. Suppose W,r has a E-super vertex magic total labellig, the by theorem. p ( ) p r r r r r r 3r 5 5 r Now, two cases arise: If r <, 4 4 the is a iteger if ad oly if both are eve. If r >, the is a iteger if ad oly if both are eve ad - = r. 3.3 Gear Graph Defiitio 3. Gear graph is obtaied by isertig a extra vertex betwee each pair of adjacet vertices o the perimeter of a wheel graph W The umber of vertices of G is + ad the umber of edges 3. Theorem 3.4 The gear graph G has o super vertex magic total labellig. Suppose G has a super vertex magic total labellig, the by theorem. ( p )( p ) p ( 3 )( 3 ] 3 This is ot a iteger for ay. Therefore the Gear graph has o super vertex magic total labellig. Theorem 3.5 The gear graph G has o E-super vertex magic total labellig. Proof. Suppose G has a E-super vertex magic total labellig, Suppose there exist a magic costat. The by theorem. ISBN Bofrig

5 Proceedigs of the Iteratioal Coferece o Applied Mathematics ad Theoretical Computer Sciece p ( ) p 3 (3 ) 7 9 This magic costat is ot a iteger for ay i,e for a gear graph o E-super vertex magic total labellig exist. 3.4 Helm Graph Defiitio 3.3 A Helm graph is deoted by H is a graph obtaied by attachig a sigle edge ad vertex to each vertex of the outer circuit of a wheel graph W The umber of vertices of H is + ad the umber of edges 3. Theorem 3.6 The helm graph H has o super vertex magic total labellig or E-super vertex magic total labellig. Proof. I [5] Macdougal et.al proved that a graph has a vertex with degree oe has o super vertex magic total labellig. Therefore helm graph has o such labellig. OBSERVATION For a graph with + vertices ad 3 edges has o super vertex or E-super vertex magic total labellig. REFERENCES [] G. Marimuthu, M. Balarisha. E-Super vertex Magic Labeligs of Graphs. Discrete Applied Mathematics.60, Pp , 0. [] J.A. Gallia. A dyamic survey of graph labelig. The Electroic Joural of Combiatorics 8, 0. [3] A.A.G. Ngurah,A.N.M. Salma,L.Susilowati.H-Super Magic Labeligs of Graphs.Discrete Mathematics,30,Pp , 00. [4] J. Gomez.Two ew methods to obtai super vertex magic total labelig of graphs. Discrete Math.308, Pp , 008. [5] J. Gomez, Solutio of the cojecture :if 0(mod4); > 4 the K has super vertex magic total labelig Discrete Math.307,Pp , 007. [6] C.Balbuea,E.Barer,K.C.Das,Y.Li,M.Miller,J.Rya,Slami,K.Sugeg,M.Tac.O the Degrees of a Strogly Vertex Magic Graph,Discrete Mathematics,306,Pp , 006. [7] V. Swamiatha, P. Jayathi. O Super vertex Magic Labelig. J.Discrete Math.Sciece ad Cryptography8, Pp.7-4, 005. [8] J.A. MacDougall, M. Miller, K.A Sugeg. Super Vertex Magic Total Labelig of Graphs. I Proc. of the 5th Australia worshop o Combiatorial Algorithms Pp. -9, 004. [9] V. Swamiatha, P. Jayathi Super vertex Magic Labelig. Idia Joural of pure Applied Math 34(6), Pp , 003. [0] J.A MacDougall, M. Miller, Slami, W.D. Willis Vertex Magic Total Labelig of Graphs. util. Math, 6, Pp. 3-, 00. [] J.A.Body, U.S.R. Murty, Graph Theory with Applicatios, North Hollad, New Yor.Amsterdam.Oxford, 976. [] D. Atoy Xavier, C.J. Deei, Adrew Aroiaraj. Modular Super Vertex Magic Total Labelig. Commuicated. ISBN Bofrig