On the P3 Coloring of Graphs.

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 P-2 coloring, where P-2 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 P-3 coloring. A graph is called P-3-colorable if we can assign colors to the vertices of the graph such that the vertices of every P-3 path are distinct. The minimum number of colors required for a graph to have P(3 )coloring is called the P-3 chromatic number. The aim of this article is, in general, to prove some basic results concerning this coloring, and, in particular, to compute the P-3 chromatic number for different symmetric families of graphs.