On the oriented chromatic number of grids

Abstract

In this paper, we focus on the oriented coloring of graphs. Oriented coloring is a coloring of the vertices of an oriented graph G without symmetric arcs such that (i) no two neighbors in G are assigned the same color, and (ii) if two vertices u and v such that ( u, v)∈ A( G) are assigned colors c( u) and c( v), then for any ( z, t)∈ A( G), we cannot have simultaneously c( z)= c( v) and c( t)= c( u). The oriented chromatic number of an unoriented graph G is the smallest number k of colors for which any of the orientations of G can be colored with k colors. The main results we obtain in this paper are bounds on the oriented chromatic number of particular families of planar graphs, namely 2-dimensional grids, fat trees and fat fat trees.