I was reading Allen Hatcher's Algebraic Topology, and he mentioned a 2-oriented graph while describing the covering spaces of S1∨S1. What is a 2-oriented graph? Can you give an examples of 2-oriented, 3-oriented, and 4-oriented graphs?
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A $2$-oriented graph is a $2$-labelled (each edge has one of two labels from some labelset $L_2=\{\alpha,\beta\}$), oriented graph, such that each vertex has degree four, with the indegree of $\alpha$ edges being one and the outdegree of $\alpha$ edges being one at each vertex, and similarly for $\beta$.
An $n$-oriented graph is similarly defined with the labelset $L_n$ having cardinality $n$, and such that each vertex has degree $2n$ and has exactly one ingoing and one outgoing edge of each label from the labelset $L_n$. (For the purposes of this definition, a loop counts as both an ingoing and outgoing edge of the same label).
It is an exercise to show that each graph $G$ with constant vertex degree $2n$ can be $n$-oriented (Hint: show that $G$ admits an Eulerian circuit)
