Some Results On Vertex Odd Divisor Cordial Labeling of Graphs
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1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Some Results On Vertex Odd Divisor Cordial Labeling of Graphs A. Sugumaran 1 and K. Suresh 2 1,2 Department of Mathematics, Government Arts College, Tiruvannamalai , Tamil Nadu, India. 1 sugumaranaruna@gmail.com 2 dhivasuresh@gmail.com February 10, 2017 Abstract In this paper we prove that the graphs Globe Glpnq, Jewel J n, G W n, G Cpn, n 3q, and Wheel W n, are vertex odd divisor cordial graphs. AMS Subject Classification: 05C78 Key Words and Phrases: Labeling, Cordial labeling, Divisor cordial labeling, Vertex odd divisor cordial labeling. 1 Introduction Graph theory has several interesting applications in system analysis, operations research and economics. Since most of the time the aspects of graph problems are uncertain, it is nice to deal with these aspects via the methods of labeling. The concept of labeling of 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. A graph labeling, is a process in which each vertex is assigned a value from the given set of numbers, the labeling of edges depends on the labels of its end vertices. An excellent survey of ijpam.eu
2 various graph labeling problems, we refer to Gallian [2]. Two well known graph labeling methods are graceful labeling and harmonious labeling. These labelings are studied by Cahit [1]. Cordial labeling was introduced by Cahit [1]. Many labeling schemes were introduced with slight variations in cordial such as prime cordial labeling, divisor cordial labeling. Varatharajan et al. [7] have analyzed the divisor cordial labeling.the divisor cordial labeling of various types of graph is presented in [5],[6]and [8] Muthaiyan et al. [4] introduce the concept of vertex odd divisor In section 2, we summarize the necessary definitions and basic results. In section 3, we proved that some standard graphs are vertex odd divisor We conclude in section 4. 2 Basic Definitions In this section, we provide a brief summary of the definitions and other results which are prerequisites for the present work. All the graphs considered here are simple finite, undirected without loops and multiple edges. Let G pv, Eq be a graph and as usual we denote p V and q E. For terminology and notations not specifically defined here, we refer to Harary [3]. We recall the following definition from Harary [3]. Definition 1. Let G pv, Eq be a graph. A mapping f : V Ñ t0, 1u is called the binary vertex labeling of G and fpvq is called the label of the vertex v P V of G under f. The induced edge labeling f : E Ñ t0, 1u is given by f peq fpuq fpvq, for all e uv P E. We denote v f piq is the number of vertices of G having label i under f and e f piq is the number of edges of G having label i under f, where i 0, 1. Now we define cordial labeling of a graph. Definition 2. Let G pv, Eq be a graph and f : V Ñ t0, 1u be a binary vertex labeling of G. The map f is called a cordial labeling if v f p0q v f p1q ď 1 and e f p0q e f p1q ď 1. Definition 3. cordial labeling. A graph G is called cordial graph if it admits ijpam.eu
3 Definition 4. A divisor cordial labeling of a graph G pv, Eq is a bijection f : V Ñ t1, 2, 3,... V u such that if each edge uv is assigned the label 1 if fpuq{fpvq or fpvq{fpuq and the label 0 if fpuq ffl fpvq, then e f p0q e f p1q ď 1. Definition 5. A vertex odd divisor cordial labeling of a graph G pv, Eq is a bijection f : V Ñ t1, 2, 3,...2n 1u such that if each edge uv is assigned the label 1 if fpuq{fpvq or fpvq{fpuq and the label 0 if fpuq ffl fpvq, then e f p0q e f p1q ď 1. A graph which admits odd divisor cordial labeling is called a vertex odd divisor The concept of vertex odd divisor cordial labeling was introduced by Muthaiyan et al.[] Definition 6. The Jewel graph J n is a graph with vertex set V pj n q tu, v, x, y, u i : 1 ď i ď nu and edge set EpJ n q tux, uy, xy, xv, yv, uu i, vu i : 1 ď i ď nu. Definition 7. A Globe is a graph obtained from two isolated vertices which are connected by n paths of length two. It is denoted by Glpnq. The graph Glpnq contains n ` 2 vertices and 2n edges. Definition 8. The Shell S n is a graph obtained by taking n 3 concurrent chords in cycle C n. The vertex at which all the chords are concurrent is called the apex vertex. The shell S n is also called the Fan F n 1. Definition 9. Let G be any graph. The graph denoted by G W n is obtained by adjoining the the central vertex of wheel W n with that any one of the vertices of G. Definition 10. Let G be any graph. The graph denoted by G S n is obtained by adjoining the apex of shell graph S n with that any one of the vertices of G. 3 Main Results Theorem 11. graph. The Globe Glpnq is vertex odd divisor cordial ijpam.eu
4 Proof. Let G be the the graph Glpnq. Let V pgq tu, v, y i : 1 ď i ď nu and let EpGq tuy i : 1 ď i ď nuytvy i : 1 ď i ď nu. Then V pgq n ` 2, and EpGq 2n. To define the vertex labeling f : V Ñ t1, 3, 5,..2n ` 1, 2n ` 3nu, we consider the following two cases. Case (1) When 2n ` 3 is a prime number fpuq 1, fpy i q 2i ` 1, 1 ď i ď n, fpvq 2n ` 3. Case (2) When 2n ` 3 is not a prime number. We define fpuq 1, fpvq p where p is the largest prime number such that 3 ď p ď 2n ` 1. The remaining vertices y i, p1 ď i ď nq are assigned the labels 3, 5, 7,..., 2n ` 3 in any order except p. From these two cases, we obtain e f p0q e f p1q n. Thus we have e f p0q e f p1q ď 1. Hence G is a vertex odd divisor cordial graph. Example 12. Vertex odd divisor cordial labeling of the graphs Glp8q and Glp11q are shown in Figures 1 and 2. Fig.1 ijpam.eu
5 Theorem 13. Fig.2 The Jewel graph J n is a vertex odd divisor Proof. Let G be a Jewel graph J n. V pgq tq, x, r, y, q i : 1 ď i ď nu and EpGq tqx, rx, qy, ry, qq i, rq i : 1 ď i ď nu. Then V pgq n ` 4, and EpGq 2n ` 5. We define f : V Ñ t1, 3, 5...2n ` 7u as follows. fpxq 3, fpyq 5, fpqq 1, fprq p, where p is the largest prime number such that p ď 2n ` 7. Now we label the remaining vertices q 1, q 2,...q n with numbers from 7 to 2n ` 7 in any order except p. We observe that, from the above labeling pattern, we have e f p1q n ` 2 and e f p0q n ` 3. Therefore e f p0q e f p1q ď 1. Hence G is a vertex odd divisor Example 14. Vertex odd divisor cordial labeling for J 7 is shown in Figure 3. Fig.3 ijpam.eu
6 Theorem 15. graph. The G W n graph is vertex odd divisor cordial Proof. Let f be any vertex odd divisor cordial labeling of G, where V pgq p and EpGq q. Then e f p0q e f p1q ď 1. Let v 1, v 2, v 3,, v n be the vertices of wheel W n K 1 ` C n 1 and the central vertex of the wheel v 1 is labeled as 1. Extend f to G W n by assign the labels as follows. Note that the graph G W n contains p ` n 1 vertices and q ` 2n edges. fpv i q 2i ` 5, 2 ď i ď n. From the above labeling if fpv 2 q{fpv n q then interchange the labels fpv n q and fpv n 1 q so that fpv 2 q ffl fpv n q. In view of above defined labeling pattern, we have e f p0q n ` 2 and e f p1q n ` 3. Therefore e f p0q e f p1q ď 1. Hence G is a vertex odd divisor Example 16. Vertex odd divisor cordial labeling for G W 8 is shown in Figure 4. Fig.4 Theorem 17. graph. The graph G S n is vertex odd divisor cordial Proof. Let G be any vertex odd divisor Let f be a vertex odd divisor cordial labeling of G. Then e f p0q e f p1q ď 1. Let v 1, v 2, v 3,, v n be the vertex of a shell graph as a cycle C n with n 3 chords sharing a common end point called the apex vertex v 1. We denote G 1 G S n. To define vertex labeling f : V pg 1 q Ñ t1, 3, 5,...2n ` 7u as follows. fpv 1 q 1, fpv i q 2i ` 7, 2 ď i ď n. In view of above defined labeling pattern, we have e f p0q n ` 2 and e f p1q n ` 1. ijpam.eu
7 Therefore e f p0q e f p1q ď 1. Hence G 1 is vertex odd divisor cordial labeling graph. Example 18. shown in Figure 5. Vertex odd divisor cordial labeling for G S 6 is Fig.5 Theorem 19. The wheel graph W n K 1 `C n 1 is vertex odd divisor Proof. Let v 0 be the apex vertex of wheel W n and v 1, v 2, v 3,, v n be the rim vertices. We denote G be the wheel graph W n. Then Then V pgq n ` 1, and EpGq 2n. To define f : V pgq Ñ t1, 3, 5,..., 2n ` 1u as follows. fpv 0 q 1, fpv i q 1 i, 1 ď i ď n. From the above labeling if fpv 1 q{fpv n q, then we interchange the labels fpv n 1 q and fpv n q so that fpv 1 q ffl fpv n q. Thus e f p0q e f p1q n. Therefore e f p0q e f p1q ď 1. Hence G is a vertex odd divisor Example 20. shown in Figure 6. Vertex odd divisor cordial labeling for W 10 is Fig.6 ijpam.eu
8 4 Conclusion In this paper we proved that the graphs Globe Glpnq, Jewel J n, G W n, G Cpn, n 3q and Wheel W n, are vertex odd divisor cordial graphs. References [1] Cahit, I.,1987, Cordial Graphs: A Weaker Version of Graceful and Harmonious Graphs, Ars Combinatoria, 23, pp [2] Gallian, J.A., 2015, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, 16, # DS6. [3] Harary, F., 1972, Graph Theory, Addison-Wesley, Massachusetts. [4] Muthaiyan, A., and Pugalenthi, P., 2015 Vertex Odd Divisor Cordial Graphs, International Journal of Innovative Science, Engineering and Technology, ISSN Vol.2 Issue 10. [5] Sugumaran, A., Suresh, K., 2015, Some Results On Divisor Cordial Graphs, International Journal of Applied Engineering Research, ISSN Vol.10 No.72 [6] Vaidya, S.K., and Shah,.N.H., 2013, Further Results On Divisor Cordial labeling, Annals of Pure and Applied Mathematics Vol.4 No. 2, pp [7] Varatharajan, R., Navaneethakrishnan, S., and Nagarajan, K., 2011, Divisor Cordial Graphs, International Journal of Mathematical Combinatorics, 4, pp [8] Varatharajan, R., Navaneethakrishnan, S., and Nagarajan, K., 2012, Special Classes of Divisor Cordial Graphs, International Mathematical Forum, Vol. 7 No. 35, pp ijpam.eu
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