Oriented colorings of partial 2-trees

Abstract

A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping f from V ( G ) to V ( H ) , that is f ( x ) f ( y ) is an arc in H whenever xy is an arc in G. The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H. In this paper, we determine the oriented chromatic number of the class of partial 2-trees for every girth g ⩾ 3 . We also give an upper bound for the oriented chromatic number of planar graphs with girth at least 11.