Group coloring via its geometric structures

Abstract

Let Γ be a graph with vertex set V (Γ) and edge set E(Γ). A vertex coloring of Γ is an assignment of colors to V (Γ), so that no any two adjacent vertices share the same color. Meanwhile an edge coloring of Γ is an assignment of colors to E(Γ), so that no any two incident edges share the same color. Let G be a finite group, an order product prime graph of G, is a graph Γopp(G), having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. In this paper, the general structure of the order product prime graph is used to investigate the vertex chromatic number, the dominated chromatic number, the locating chromatic number and the edge chromatic number of the order product prime graph on cyclic groups and dihedral groups.