Acyclic Chromatic Number Of Central Graph

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1 Volume 118 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v118i10.60 ijpam.eu Acyclic Chromatic Number Of Central Graph S.Bhaskaran 1 and G.Menaka 2 1 Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai , Tamilnadu, India. 2 Department of Mathematics, S.A.Engineering College, Chennai , Tamilnadu, India. 1 bhaskaran.s@hotmail.com, 2 gksmenaka@gmail.com. December 28, 2017 Abstract In this article, we examined acyclic chromatic number concerning central graph of Gear graphs G m and inferred the accurate value regarding its very own acyclic chromatic number. Key Words and Phrases: Vertex coloring, Gear graph, Central graph. 1 Introduction Assuming the graph G is undirected, accompanying vertices of finite numbers, which never contains multiple edges and loops. The central graph [10], C(G) is realized by partitioning any edge of the graph G precisely one time and adjoining the remaining dissociated vertices of G. Its representation is K C(G) = l + m. Specifically (l, m) graph that contains finitely l vertices with degree l 1 and m vertices consisting degree 2 in C(G)

2 2 Definitions Definition 1. The lowest count of colors required to color the vertices continually in G is said to be acyclic chromatic number. Definition 2. Let G m be the Gear graph, arrived from wheel graph W m, after inducting one vertex in the midst of any pair of neighboring vertices concerning the outlying m cycle. Therefore, G m contains 2m + 1 vertices and 3m edges. Assuming the root vertex be v and v 1, v 2,..., v 2m are the vertices of outer 2m-cycle. Assuming v 1,v 3,...,v 2m 1 are vertices joined with v and v 2,v 4,..v 2m, the intermediary vertices of outer cycle. 3 Structural Algorithim of C(G m ) Input: C(G m ) V v, v 1, v 2,..., v 2m, e 1, e 3,..., e 2m 1, f 1, f 2,..., f 2m E e 1, e 2,..., e 2m, e 1,..., e 2m 1, f 1, f 2,..., f 2m, f 1,..., f 2m, h 13, h 14,..., h 2m 1, h 1m (l = 2 to 2m 1, m = l + 2 to m) for p = 1 to m for p = 1 to 2m 1 ve 2p 1 e 2p 1; vv 2p e 2p ; v 2p 1 e 2p 1 e 2p 1; v p f p f p; 2 184

3 for p = 1 to 2m 1 for p = 3 to 2m 1 for l = 2 to 2m 1 f p v p+1 f p ; v 1 v p n 1p; for p = l + 2 to m v 1 v p h lp; end procedure 4 Coloring Algorithim concering C(G m ), m > 3. Input: C(G m ) V v, v 1, v 2,..., v 2m, e 1, e 3,..., e 2m 1, f 1, f 2,..., f 2m E e 1, e 2,..., e 2m, e 1,..., e 2m 1, f 1, f 2,..., f 2m, f 1,..., f 2m, h 13, h 14,..., h 2m 1, h lm (l = 2 to 2m 1, m = l + 2 to m) 3 185

4 v 1; for p = 1 to 2m v 1 2; v 3 3; for p = 3 to m for p = 1 to m f p 1; v 2p 1 m + p 1; s p + 2 if s < m + 2, e 2p 1 s; else e 2p 1 s m; Theorem 3. The central graph of Gear graph possess an acyclic chromatic sum 2m 1, where m > 3. Proof. We employ the effective coloring algorithm, in order to prove the coloring in the theorem is acyclic. When an edge e 2p 1 has two neighbors of distinct colors, ve 2p 1 v 2p 1. Case(i): Identifying colors 1 together with p, where p = 2, 3. The color classes of 1,2 and 3 are v, f p ; p=1 to 2m, v 1,v 2,e 2m 1 and v 3,v 4,e 1, respectively. The resulting subgraph contains the color classes with the paths f 2 v 3 f 3 v 4 and f 2m v 1 f 1 v 2. Hence C(G m ) is (1,p)-acyclic graph

5 Case(ii): Identifying colors 1 together with p, where 5 < p < m + 2. Though bicolored cycle may have the color class of k and minimum 4 vertices is v 2(p 1), hence C(G m ) is (1,p)-acyclic graph. Case(iii): Identifying colors 1 together with k, where m+3<p<2m- 2. Strictly following the procedures in case(i) as well as (ii). C(G m ) remain (1,p)-acyclic graph. Case(iv): Identifying colors l and p where l =2,3 and 3<p<m+2. The resulting subgraph will have two colored path v 1 v 2(p 1) v 2 (if l=2) and v 3 v 2(p 1) v 4 (if l=3). Case(v): Identifying colors l and p, where l =2,3 and n+1<p<2n. Similar argument in the above case illustrates C(G m ) is (l,p)-acyclic graph. Case(vi): Identifying colors l and p, 3<l <p<n+2. The resulting subgraph will have two colored edge v 2(l 1) v 2(p 1). Therefore C(G m ) do not have cycle. Case(vii): Identifying colors l and p, n+3<l <p<2n. The resulting subgraph will have two colored edge v 2l 1 v 2p 1. Therefore C(G m ) do not have cycle. Case(viii): Identifying colors l and p, 5 < l < p < (2m 1). The resulting subgraph will have either a standalone vertix (or) two colored edges. Therefore C(G m ) do not have cycle. Hence, coloring defined and described in algorithm is acyclic. Therefore the central graph of Gear graph has an acyclic chromatic number 2m 1, where m >

6 Fig1. Acyclic C(G 4 ) = 7 5 Conclusion The accurate estimate of acyclic chromatic sum concerning central graph of gear graphs is estimated as 2m 1, where m > 3. References [1] J.A. Bondy and U.S.R. Murty, Graph theory with Applications, McMillan, London, 976. [2] N. Alon, C. McDiarmid, and B. Reed, Acyclic colourings of graphs, Random Structures and Algorithms, 2, , [3] C.B. Boyer, A history of mathematics, Wiley, New York, [4] Thilagavathi, D.Vijayalakshmi and Roopesh, B-Coloring of central Graphs, International Journal of computer applications, 3(11)(2010) [5] A.E. Brouwer, A.M. Cohen and A.Neumaier, Distance-Regular Graphs, Springer-Verlag, New Distance-Regular Graphs, Springer-Verlag, New York,

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