On Chromatic Number and Edge-Chromatic Number of the Ottomar Graph
Mark Caay, Esperanza Blancaflor Arugay · American Scientific Research Journal for Engineering, Technology, and Sciences (Global Society of Scientific Research and Researchers) · 2016
The path graph , consists of the vertex set and the edge set . The cycle graph , is the path graph, with an additional edge Define the Ottomar Graph, denoted by , to be the graph , with a vertex connected by a path to a vertex of . is called the heart while is called a foot (feet for plural). Note that there are copies of . The chromatic number of a graph , denoted by , is the minimum number of colors the vertices of maybe colored such that any two adjacent vertices have different colors. The edge-chromatic number of a graph , denoted by , is the minimum number of colors the edges of maybe colored such that any two incident edges have different colors. The chromatic number and the edge-chromatic number of the ottomar graph are determined. When will the two invariants be equal or when will they be unequal? When the connecting path has order greater than 2, what happens to the value of and ? Also in the paper, the other coloring invariants are compared and investigated with chromatic number and edge-chromatic number.