Oriented colorings of 2-outerplanar graphs

Abstract

A graph G is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the outer face is outerplanar. The oriented chromatic number of an oriented graph H is defined as the minimum order of an oriented graph H ′ such that H has a homomorphism to H ′ . In this paper, we prove that 2-outerplanar graphs are 4-degenerate. We also show that oriented 2-outerplanar graphs have a homomorphism to the Paley tournament QR 67 , which implies that their (strong) oriented chromatic number is at most 67.