Colorações distintas nos vértices adjacentes em potências de caminho

Abstract

Two elements of a graph are adjacent if they are the vertices of an edge, or if they are two edges which share a common vertex, or if they are an edge and one of its vertices. A coloring of a graph is an assignment of colors to its elements (vertices, or edges, or both). Given a colored graph, the set of colors of a vertex 𝑣, denoted by 𝐶(𝑣) is composed by the color of 𝑣 and the colors of the edges incident to 𝑣. The Adjacent Vertex Distinguishing Edge Coloring Problem is to find out an edge coloring using the minimum number of colors, such that any two adjacent edges have distinct colors and for each edge 𝑢𝑣, 𝐶(𝑢) ̸= 𝐶(𝑣). The Adjacent Vertex Distinguishing Total Coloring Problem is to find out a total coloring using the minimum number of colors such that any two adjacent elements have distinct colors and for each edge 𝑢𝑣, 𝐶(𝑢) ̸= 𝐶(𝑣). This work presents the solution of the adjacent vertex distinguishing edge coloring problem and the adjacent vertex distinguishing total coloring problem when restricted to the class of the powers of paths.