### Chromatic Number of Amalgamation of Wheel Graph-Star Graph and Amalgamation of Wheel Graph-Sikel Graph

#### Abstract

A graph denoted by is a pair of where is a non-empty set of vertices in G, and E is a set of edges in G. In graph theory, there are various types of graphs including star graphs, cycle graph, and wheel graph. Graph operations on two or more types of graphs can produce new graphs. Amalgamation is one of the operations on graphs. Suppose and are two connected graphs, amalgamating the vertices of graph 1 and graph 2 by joining vertices and vertices denoted by is the graph obtained by joins vertex of graph and vertex of graph into one vertex , where is the common vertex of the graph resulting from , while amalgamating the edges of graph and graph by joining edges and edge , denoted by are graphs obtained by combining edge of graph 1 and edge of graph 2 into one edge g, where g is common side of graph of (𝐺1,𝐺2;𝑒,𝑓). Vertex coloring is one of the fastest-growing topics in graphs. The coloring of the vertices, that is, assigning a color to each vertex of the graph so that each neighboring vertex has a different color. The minimum number of colors used to color the vertices of a graph is called the chromatic number. In this article, we discuss the chromatic number in the amalgamated graph of wheel graph and star graph and the chromatic number on the amalgamation of wheel graph and cycle graph. The results obtained are as follows: (1) the chromatic number of the amalgamated graph of the wheel graph () and the star graph () for n even is 3 and for n odd is 4, (2) the chromatic number of the graph resulting from the operation amalgamation of the wheel graph () and the cycle graph () for n even is 3 and for odd is 4.

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Afriantini, Helmi. & F. Fran., (2019). Pewarnaan Simpul, Sisi, Wilayah Pada Graf Dan Penerapannya. Bimaster Ilmiah. Stat. dan Terapannya (Bimaster). Volume 08. No. 4, pp. 773-782. Chartrand, G. & Lesniak, L., (1986). Graph and Digraph 2^nd Edition. California: Wadswoeth. Inc. Risti, D. R. & Yemi, K., (2018). Dekomposisi Graf Helm. Journal of Mathematics and Mathematics Education. Volume 8, No. 1. pp 31 – 45. https://dx.doi.org/10.20961/jmme.v8i1.25822 Rinovia, S. & Danang, T. M., (2014). Metric Dimension of Amalgamation of Regular Graphs. ArXiv. https://doi.org/10.48550/arXiv.1401.5164 Fangfang Wu., Shenggui, Z., Binlong Li., Tingting, & H., (2022). Color neighborhood union conditions for proper. Discrete Applied Mathematics, Volume 307, pp. 145-152. https://doi.org/10.1016/j.dam.2021.10.016 Ian, B., George, M. S., Erik, G. B., Karen, D. D., & Sisvasankaran, R., (2022). Parallel graph coloring algorithms for distributed GPU environments. Parallel Computing, pp. 1-11. https://doi.org/10.1016/j.parco.2022.102896 Leonardo, M. & Luis, M., (2015). Fractional Turan’s theorem and bounds for the Chromatic Number. Electronic Notes in Discrete Mathematics, Volume 50, pp. 415-420. http://dx.doi.org/10.1016/j.endm.2015.07.069 M. Rajesh Kannan, S. P. H. W., 2022. On the construction of cospectral nonisomorphic bipartite graphs. Discrete Mathematics, Volume 345. https://doi.org/10.1016/j.disc.2022.112916 Yangyan Gu., Hao Qi., Yeong-Nah, Y., & Xuding, Z., (2022). Generalized signed graphs of large girth and large chromatic. Discrete Mathematics, Volume 345. https://doi.org/10.1016/j.disc.2022.112980 Yijun, X., Huajun, W., Mustafa, H., & Muhammad, A. U., (2019). Amalgamations and Cycle-Antimagicness. IEEE Access, Volume 7, pp. 147345 -147349.

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