International Journal of Pure and Applied Mathematics Volume 116 No. 16 2017, 467-470 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu ON STAR AND ACYCLIC COLORING OF GRAPHS 1N.Ramya, 2Muthukumar 1Asst.Professor, 2PG scholar, Department Of Mathematics BIST, BIHER, Bharath University, Chennai-73 [email protected] Abstract: The notion of Star coloring and Acyclic coloring of graphs are discussed below. We give the Case i : For n = 4, Algorithm is as follows exact value of the Star chromatic number and Acyclic Label the vertices with 1, 2, 3 and 4 respectively in chromatic number of different 3- Regular graphs clock wise direction.[18] including H-Graph. In this case The Star and Acyclic chromatic Number is χs 1. Introduction (G) = χa(G) = 4[16] In this section we gave the basic definitions on Star Case ii : and Acyclic coloring of graphs. In graph coloring, a Star Coloring of a graph G is Algorithm for the 3-Regular graph with n ≥ 6 is as a proper vertex coloring in which every path of four follows.[12] vertices uses atleast three distinct colors, i.e.[13] any path Label the vertices as 1,2,3 and 4 accordingly. of length 3 in G is not bi-colored.[14] Equivalently, in a f( v ) = 1, f(v ) = 2, f(v ) = 3,…….., f(v ) = 4. star coloring, the induced sub graphs formed by the 1 2 3 n In this case also Star and Acyclic Chromatic vertices of any two colors have connected components Number is χ (G) = χ (G) = 4[19] that are Star graphs. Star coloring has been introduced by s a Grunbaum (1973). The Star Chromatic number χ (G) is s 2.2 Theorem 2: the least number of colors needed to star color the graph G [1,2] AnH-Graph H (r)with ‘6r’ vertices and ‘9r’ edges admits In Graph theory, an Acyclic Coloring is a proper Star and Acyclic colorings and the Chromatic number is vertex coloring in which every 2-Chromatic sub graph of same for both the [10]colorings. That is χ (G) = χ (G) = G is acyclic, i.e. no cycle in G is bi-colored. The Acyclic s a 3. Chromatic Number χ (G) of a graph G is the least number a of colors needed[15] in any acyclic coloring of G. Proof: Acyclic edge coloring of a graph is the acyclic vertex Algorithm: (Star and Acyclic coloring of some 3- coloring of its line graph. regular Graphs) [21] 2. Main result Case i : 2.1 Theorem 1: For n = 1,4,7… the algorithm is as follows. The Right arm of the graphs H (1), H (4), H (7)… are The 3-Regular graphs with n ≥ 4, admits Star and labeled as 1, 3, 2 and[22] Acyclic colorings [11]and also, the coloring and The Left arm of those graphs is labeled as 3,2,1. algorithm is same for both Star and Acyclic colorings [3,4]. Case ii : Algorithm is as follows,[20] When r = 2, 5, 8… the algorithm is as follows. Proof: The Right arm of H (2), H (5), H (8)…are labeled as 2,1,3 and[8] Algorithm :( Star and Acyclic colorings 3-Regular The Left arm is labeled as 3, 2, and 1 graphs) respectively in clock wise direction. 467 International Journal of Pure and Applied Mathematics Special Issue [8] Brintha Rajakumari S., Nalini C., An efficient cost Case iii : model for data storage with horizontal layout in the cloud, Indian Journal of Science and Technology, v-7, i-, When r = 3,6,9,……, the algorithm is as follows. pp-45-46, 2014. [9] Brintha Rajakumari S., Nalini C., An efficient data The Right arm of H (3), H (6), H (9)…are mining dataset preparation using aggregation in relational labeled as 2, 3 and 1and database, Indian Journal of Science and Technology, v-7, The Left arm is labeled as 1,2 and 3 respectively i-, pp-44-46, 2014. in clock wise direction.[9] [10] Khanna V., Mohanta K., Saravanan T., Recovery The Star and Acyclic Chromatic Number of H of link quality degradation in wireless mesh networks, (1), H (4), H (7) ..., is χs (G) = χa(G) = 3. Indian Journal of Science and Technology, v-6, i- SUPPL.6, pp-4837-4843, 2013. 3. Conclusion [11] Khanaa V., Thooyamani K.P., Udayakumar R., A secure and efficient authentication system for distributed In the study of Proper, Star and Acyclic coloring of some wireless sensor network, World Applied Sciences classes of Graphs we arrive at following conclusion. Journal, v-29, i-14, pp-304-308, 2014. It concludes that, Proper coloring of 3-Regular [12] Udayakumar R., Khanaa V., Saravanan T., Saritha graph with n = 4 (n is even) have the chromatic number G., Retinal image analysis using curvelet transform and 4, for n = 6, 10, 14,... the Chromatic number is 2 and for multistructure elements morphology by reconstruction, n = 8, 12, 16, ... chromatic number is 3. The H-Graph Middle - East Journal of Scientific Research, v-16, i-12, which is also a 3-Regular graph has Chromatic number 2 pp-1781-1785, 2013. in all the cases. [13] Khanaa V., Mohanta K., Saravanan. T., Also we conclude that, Star and Acyclic Performance analysis of FTTH using GEPON in direct chromatic number of 3-Regular graph is 4 for n ≥ 4 and n and external modulation, Indian Journal of Science and is even. Star and Acyclic Chromatic number of H-Graph Technology, v-6, i-SUPPL.6, pp-4848-4852, 2013. is 3 in all cases [14] Kaliyamurthie K.P., Udayakumar R., Parameswari D., Mugunthan S.N., Highly secured online voting References system over network, Indian Journal of Science and Technology, v-6, i-SUPPL.6, pp-4831-4836, 2013. [1] Annamma. V.”Total Vertex Irregularity Strength of [15] Thooyamani K.P., Khanaa V., Udayakumar R., H-Graph and Gear graph”, Proceedings of International Efficiently measuring denial of service attacks using Conference on Mathematics and Computer Science appropriate metrics, Middle - East Journal of Scientific ICMCS, pp 237-239, 2011. Research, v-20, i-12, pp-2464-2470, 2014. [2] Appel. K.I. and Haken, W.”Every planar map is [16] R.Kalaiprasath, R.Elankavi, Dr.R.Udayakumar, Four colorable”, Bull.Amer.Math.Soc. No.82, pp.711- Cloud Information Accountability (Cia) Framework 712, 1976. Ensuring Accountability Of Data In Cloud And Security [3] Arundhathi. R. and Sattanathan, R.”Acyclic In End To End Process In Cloud Terminology, coloring of Central graph of path P n and Fan graph F m,n “, International Journal Of Civil Engineering And Proceeding of the International Conference on Technology (Ijciet) Mathematics in Engineering and Business management, Volume 8, Issue 4, Pp. 376–385, April 2017. pp.153-155, 2012. [17] R.Elankavi, R.Kalaiprasath, Dr.R.Udayakumar, A [4] Bondy. J.A. and Murthy, U.S.R.,”Graph theory fast clustering algorithm for high-dimensional data, with Applications”, Elsevier Science Publishing Co., International Journal Of Civil Engineering And Inc., 1985. Technology (Ijciet), Volume 8, Issue 5, Pp. 1220–1227, [5] Borodin. O.V.”On Acyclic coloring of Planar May 2017. Graphs”, Discrete Math., pp.211-236, 1979. [18] R. Kalaiprasath, R. Elankavi and Dr. R. [6] Udayakumar R., Kaliyamurthie K.P., Khanaa, Udayakumar. Cloud. Security and Compliance - A Thooyamani K.P., Data mining a boon: Predictive system Semantic Approach in End to End Security, International for university topper women in academia, World Applied Journal Of Mechanical Engineering And Technology Sciences Journal, v-29, i-14, pp-86-90, 2014. (Ijmet), Volume 8, Issue 5, pp-987-994, May 2017. [7] Kaliyamurthie K.P., Parameswari D., Udayakumar [19] Thooyamani K.P., Khanaa V., Udayakumar R., R., QOS aware privacy preserving location monitoring in Virtual instrumentation based process of agriculture by wireless sensor network, Indian Journal of Science and automation, Middle - East Journal of Scientific Research, Technology, v-6, i-SUPPL5, pp-4648-4652, 2013. v-20, i-12, pp-2604-2612, 2014. 468 International Journal of Pure and Applied Mathematics Special Issue [20] Udayakumar R., Thooyamani K.P., Khanaa, Random projection based data perturbation using geometric transformation, World Applied Sciences Journal, v-29, i-14, pp-19-24, 2014. 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