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.