Transitive coloring of graphs

Abstract

Let V={V1,…,Vk} be a partition of the vertices of a graph G. We say that an automorphism f of G is a transitive color automorphism associate to V if there exist vertices vi∈Vi and vj∈Vj with 1≤i,j≤k, such that (i) f(vi)=vj, (ii) f(Vℓ)=Vℓ for ℓ with 1≤ℓ≤k and ℓ∉{i,j}. We denote f by fi,j. We say that V is transitive k-coloring if, for every two distinct colors i and j with 1≤i,j≤k, there exists a transitive color automorphism fi,j. The transitive coloring number of G, denoted by T(G), is the greatest integer k that G has a transitive k-coloring.The main purpose of this paper is to study some basic properties of transitive coloring graphs and determine the transitive coloring number of certain graphs.