Homomorphisms and oriented colorings of equivalence classes of oriented graphs Original Research Article

We combine two topics in directed graphs which have been studied separately, vertex pushing and homomorphisms, by studying homomorphisms of equivalence classes of oriented graphs under the push operation. Some theory of these mappings is developed and the complexity of the associated decision problems is determined. These results are then related to oriented colorings. Informally, the pushable chromatic number of an oriented graph G is the minimum value of the oriented chromatic number of any digraph obtainable from G using the push operation. The pushable chromatic number is used to give tight upper and lower bounds on the oriented chromatic number. The complexity of deciding if the pushable chromatic number of a given oriented graph is at most a fixed positive integer k is determined. It is proved that the pushable chromatic number of a partial 2-tree is at most four. Finally, the complexity of deciding if the oriented chromatic number of a given oriented graph is at most a fixed positive integer k is determined.