On the P3 Coloring of Graphs

Abstract

The vertex coloring of graphs is a well-known coloring of graphs. In this coloring, all of the vertices are assigned colors in such a way that no two adjacent vertices have the same color. We can call this type of coloring P2 coloring, where P2 is a path graph. However, there are situations in which this type of coloring cannot give us the solution to the problem at hand. To answer such questions, in this article, we introduce a novel graph coloring called P3 coloring. A graph is called P3-colorable if we can assign colors to the vertices of the graph such that the vertices of every P3 path are distinct. The minimum number of colors required for a graph to have P3 coloring is called the P3 chromatic number. The aim of this article is, in general, to prove some basic results concerning this coloring, and, in particular, to compute the P3 chromatic number for different symmetric families of graphs.