Oriented colorings of triangle-free planar graphs

Abstract

Oriented graphs are directed graphs without opposite arcs. In other words an oriented graph is an orientation of an undirected graph, obtained by assigning to every edge one of the two possible orientations. If G is a graph, V (G) denotes its vertex set, E(G) denotes its set of edges (or arcs if G is an oriented graph) and F(G) denotes its set of faces if G is planar. A homomorphism from an oriented graph G to an oriented graph H is a mapping ' from V (G) to V (H) which preserves the arcs, that is (x,y) ∈ E(G) =⇒ ('(x),'(y)) ∈ E(H). We say that H is a target graph of G if there exists a homomorphism from G to H. ✷