Some Star Related Vertex Odd Divisor Cordial Labeling of Graphs

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1 Annals o Pure and Applied Mathematics Vol. 6, No., 08, ISSN: X (P), (online) Published on 6 February 08 DOI: Annals o Some Star Related Vertex Odd Divisor Cordial Labeling o Graphs A. Sugumaran and K. Suresh Department o Mathematics, Government Arts College Tiruvannamalai , Tamil Nadu, India. sugumaranaruna@gmail.com, dhivasuresh@gmail.com Corresponding author Received February 08; accepted February 08 Abstract. In this paper we prove that the graphs D ( K, n ), K, n u ( K) K, n K, n, B n, n : w and G K, n * P n +, are vertex odd divisor cordial graphs. Also, investigated lots o properties. Keywords: labeling, cordial labeling, divisor cordial labeling, vertex odd divisor cordial labeling. AMS Mathematics Subject Classiication (00): 05C78. Introduction Graph theory has several interesting applications in system analysis, operations research and economics. Since most o the time the aspects o graph problems are uncertain, it is nice to deal with these aspects via the methods o labeling. The concept o labeling o graphs is an active research area and it has been widely studied by several researchers. In a wide area network (WAN), several systems are connected to the main server, the labeling technique plays a vital role to label the cables. The labeling o graphs has been applied in the ields such as circuit design, communication network, coding theory, and crystallography. A graph labeling is a process in which each vertex is assigned a value rom the given set o numbers, the labeling o edges depends on the labels o its end vertices. An excellent survey o various graph labeling problems, we reer to Gallian []. Two well-known graph labeling methods are graceul labeling and harmonious labeling. These labelings are studied by Cahit []. Graph colouring is an active research area in graph theory. Saha et al. [5] have discussed one nice application o graph colouring. Cordial labeling was introduced by Cahit []. Many labeling schemes were introduced with slight variations in cordial such as prime cordial labeling, divisor cordial labeling. Varatharajan et al. [] have analyzed the divisor cordial labeling.the divisor cordial labeling o various types o the graph is presented in [6,7,8,9,]. Muthaiyan et al. [4] have introduced the concept o vertex odd divisor cordial graph. In this paper, we summarize the necessary deinitions and basic results in section. In section 3, we proved that some standard graphs are vertex odd divisor cordial graph. 385

2 We conclude in section 4. A. Sugumaran and K. Suresh. Basic deinitions In this section, we provide a brie summary o the deinitions and other results which are prerequisites or the present work. All the graphs considered here are simple inite, undirected without loops and multiple edges. Let G ( V, E) be a graph and as usual, we denote p V and q E. For terminology and notations not speciically deined here, we reer to Harary [3].We recall the ollowing deinition rom Harary [3]. Deinition.. Let G ( V, E) be a graph. A mapping : V {0,} is called the binary vertex labeling o G and (v) is called the label o the vertex v V o G * under. The induced edge labeling : E {0,} is given by * ( e) ( u) ( v), or all e uv E. We denote v (i) is the number o vertices o G having a label i under and e (i) is the number o edges o G having a label i under, where i 0,. Now we deine cordial labeling o a graph. Deinition.. [] Let G ( V, E) be a graph and : V {0,} be a binary vertex labeling o G. The map is called a cordial labeling i v (0) v () and e (0) e (). A graph G is called cordial graph i it admits cordial labeling. Deinition.3. [] A divisor cordial labeling o a graph G ( V, E) is a bijection : V {,,3,..., V } such that i each edge uv is assigned the label i ( u)/ ( v) or ( v)/ ( u) and the label 0 i ( u ) \ ( v ), then e (0) e (). A graph which admits divisor cordial labeling is called a vertex divisor cordial graph. Deinition.4. [4] A vertex odd divisor cordial labeling o a graph G ( V, E) is a bijection : V {,,3,...,n } such that i each edge uv is assigned the label i ( u)/ ( v) or ( v)/ ( u) and the label 0 i ( u) ( v), then e (0) e (). A graph which admits odd divisor cordial labeling is called a vertex odd divisor cordial graph. Deinition.5. The shadow graph D ( G) o a connected graph G is constructed by taking two copies og say G and G. Join each vertex in to the neighbors o the corresponding vertex v in G. Deinition.6. [] Consider the graph bistar B m, n. Let u, v be the central vertices. We subdivide the edge uv as a path o length by adding a new vertex w is called the 386

3 Some Star Related Vertex Odd Divisor Cordial Labeling o Graphs subdivision o the bistar. The subdivision o the bistar B m, n is denoted by Deinition.7. The graph K, n * P n+ is a graph obtained rom the initial vertex o the path P n+ is attached with the apex vertex o K, n. n vertex to the vertex u or some i. i Deinition.9. [0] Consider two copies o graph G namely G and G. Then the graph G G G is the graph obtained by joining the apex vertices o G and G by an edge as well as to a new vertex v. 3. Main results Theorem 3.. D K ) is a vertex odd divisor cordial graph. (, n Proo: Consider the two copies o K, n. Let v, v, v3... vn be the pendant vertices o the irst copy o K, n and u, u, u3,..., u n be the pendant vertices o a second copy o K, n with u and v are respective apex vertices. Let G D ( K ). Then VV ( G) n +, and E( G) 4n.We deine : V ( G) {,3,5,...,4n + 3} as ollows. ( v), ( vi ) i +, i n, ( u) p where p is the largest prime number such that n + 3 p 4n + 3. The remaining vertices u i ( i n ) are assigned the labels n + 3,...,4n + 3 in any order except p. From the above labeling, we obtain e (0) e () n. K ( m, n K, n w B m, n :. Deinition.8. [] Let ( X, Y ) be the bipartition o K m,n, where X { u, u,..., um} and Y { v, v,..., v }. The graph u ) is deined by attaching a pendant Thus we have e (0) e (). Hence G is a vertex odd divisor cordial graph. Example 3.. Vertex odd divisor cordial labeling o the graph D ( K,7 ) is shown in Figure. Figure : Vertex odd divisor cordial labeling o the graph ( K D (,7 ) 387

4 Theorem 3.3. G B n, n : w is a vertex odd divisor cordial graph. Proo: Let u u, u,..., be the pendant vertices attached to the vertex u and, 3 v, v, v3,..., v n be the pendant vertices attached to the vertex v. Let G B, w. Then V ( G) n + 3, and E ( G) n +. n n : A. Sugumaran and K. Suresh We deine : V ( G) {,3,5,...,4n + 5} as ollows ( u), ( ui ) i +, i n, ( w) n + 3, ( v) p, where p is the largest prime number such that n + 5 p 4n + 5. The remaining vertices v i ( i n) are assigned the labels n + 5,...,4 n + 5 in any order except p. We observe that rom the above labeling pattern, we have, Thereore e (0) e (). Hence G is a vertex odd divisor cordial graph. u n Example 3.4. Vertex oddd divisor cordial labeling or. G B6,6 : w is shown in Figure Figure : Vertex odd divisor cordial labeling or Theorem 3.5. G K, n * P n+ is a vertex odd divisor cordial graph. Proo: Let G K, n * P n+. Let u be the apex vertex o K, n, and the remaining vertices o G are labeled as u, u, u3,..., un respectively. Then V ( G) n + 3, and E ( G) n +.We deine : V ( G) {,3,5,...,4n + 5} as ollows ( u), ( u ) i + ( i n + ). In view o above-deined labeling pattern, we have, Thereore (0) e i e G B : w 6,6 ().Hence G is a vertex odd divisor cordial graph. Example 3.6. Vertex oddd divisor cordial labeling or G K * P,5 7 is shown in Figure 3. Figure 3: Vertex odd divisor cordial labeling or G K,5 * P 388 7

5 Some Star Related Vertex Odd Divisor Cordial Labeling o Graphs Theorem 3.7. G K u, ( K ) n is a vertex odd divisor cordial graph. Proo: Let V V V be the bipartition o K (, n u K) such that V u, u and a pendant vertex w is attached with u. Then V ( G) n + 3, and E( G ) n +. We deine : V ( G) {,3,5,...,n + 5}, as ollows ( u), ( ui ) i +, i n. ( u ) p where p is the largest prime number such that p n + 5 and the remaining label is assigned to the vertex w. In view o above-deined labeling pattern, we have e (0) n + and e () n. Thereore e (0) e ().Hence G is vertex odd divisor cordial labeling graph. Example 3.8. Vertex odd divisor cordial labeling or G K u ( K is shown in Figure 4.,7 K ) Figure 4: Vertex odd divisor cordial labeling or G K,7 u ( K ) Theorem 3.9. G K, n K,n is vertex odd divisor cordial graph. Proo: Let be the graph G G. Let u, u, u3,..., un be the vertices o G and let v, v, v3,..., v n be the vertices o G. Let u, v be the apex vertices o G and G. Let w be the new vertex joining the apex vertices o G and G in G. Then V ( G) n + 3, and E ( G) n + 3. We deine : V ( G) {,3,5,...,4n + 5} as ollows ( u), ( ui ) i + ( i n), ( v) p where p is the largest prime number such that n + 3 p 4 n + 5. The remaining vertices v i ( i n) and w are assigned the labels n + 3,...,4n + 5 in any order except p. From the above labeling pattern, Thus e (0) n + and () n +.. Thereore e (0) e (). Hence G is a vertex odd divisor cordial graph. e 389

6 A. Sugumaran and K. Suresh Example 3.0. Vertex odd divisor cordial labeling or G K,8 K,8 is shown in Figure 5. Figure 5: Vertex odd divisor cordial labeling or G K,8 K,8 4. Conclusion The vertex odd divisor cordial labeling is a variation o cordial labeling. It is very interesting to study graph or amilies o graph which are vertex odd divisor cordial as all the graphs do not admit vertex odd divisor cordial labeling. In this paper, we proved that the graphs D ( K, n ), K ( ), n u K, K, n K, n, B n, n : w and K, n * P n +, are vertex odd divisor cordial graphs. Acknowledgement. The authors are highly grateul to the anonymous reeree or his valuable suggestions and comments to improve this paper rom its earlierr version. REFERENCES. I.Cahit, Cordial graphs: a weaker version o graceul and Harmonious graphs, Ars Combinatoria, 3 (987) J.A.Gallian, A dynamic survey o graph labeling, The Electronic Journal o Combinatorics, 6, # DS6 (05). 3. F.Harary, Graph Theory, Addison-Wesley, Massachusetts, A.Muthaiyan and P.Pugalenthi, Vertex odd divisor cordial graphs, International Journal o Innovative Science, Engineering and Technology, (0) (05) A.Saha, M.Pal and T.K.Pal, Selection o programme slots o television channels or giving advertisement: A graph theoretic approach, Inormation Sciences, 77 () (007) A.Sugumaran and K.Rajesh, Some new results on sum divisor cordial graphs, Annals o Pure and Applied Mathematics, 4() (07) A.Sugumaran and K. Rajesh, Sum divisor cordial labeling o theta graph, Annals o Pure and Applied Mathematics, 4() (07) S.K.Vaidya and N.H.Shah, Further results on divisor cordial labeling, Annals o Pure and Applied Mathematics, 4() (03)

7 Some Star Related Vertex Odd Divisor Cordial Labeling o Graphs 4. S.K.Vaidya and C.M.Barasara, Product cordial graphs in the context o some graph operations, International Journal o Mathematics and Scientiic Computing, () (0) R.Varatharajan, S.Navaneethakrishnan and K.Nagarajan, Divisor cordial graphs, International Journal o Mathematical Combinatorics, 4 (0) R.Varatharajan, S.Navaneethakrishnan and K.Nagarajan, Special classes o divisor cordial graphs, International Mathematical Forum, 7(35) (0)