On connectivities of edge-colored graphs

Abstract

Let ( G , c ) be an edge-colored graph where c : E ( G ) → N is an edge-coloring. For x , y ∈ V ( G ) , if there exists a sequence of properly colored cycles C 1 , C 2 , … , C l in G such that x ∈ V ( C 1 ) , y ∈ V ( C l ) and V ( C i ) ∩ V ( C i + 1 ) ≠ 0̸ for any 1 ≤ i < l , then we say the pair x , y is cyclically color-connected , denoted x ≈ y . The color of the edge incident to an end vertex x of a path P is denoted by c x ( P ) . If there exist two properly colored paths P , Q joining x and y such that c x ( P ) ≠ c x ( Q ) and c y ( P ) ≠ c y ( Q ) , then the pair x , y is called color-connected , denoted x ∼ y . If color-connectivity satisfies a transitive relation on V ( G ) , then we say ( G , c ) is convenient . It is straightforward to see that if x ≈ y , then always x ∼ y . However the converse does not hold always. An edge-colored graph ( G , c ) is called strongly convenient if the converse holds. Saad showed that 2-edge-colored complete graphs are convenient. Bang-Jensen and Gutin generalized it for some family of 2-edge-colored complete multipartite graphs. In this paper, we extend those results as follows: any k -edge-colored complete l -partite graph where k , l ∈ N is strongly convenient.