Coloring Graphs in Oriented Coloring of Cubic Graphs

Abstract

Abstract Oriented coloring of an oriented graph G is an arc-preserving homomorphism from G into a tournament H . We say that the graph H is universal for a family of oriented graphs $$\mathcal {C}$$ C if for every $$G\in \mathcal {C}$$ G ∈ C there exists a homomorphism from G into H . We are interested in finding a universal graph for the family of orientations of cubic graphs. In this paper we present constructive proof that: if there exists a universal graph H on 7 vertices for every orientation of cubic graphs, then minimum out-degree and minimum in-degree of H are equal to 2. That gives a negative answer to the question presented in Pinlou’s PHD thesis.