Finding paths of length $n$ on a directed graph

Suppose I have the following directed graph

g = Graph[{DirectedEdge[v1, v1, a], DirectedEdge[v1, v1, OverBar[a]],     DirectedEdge[v1, v2, b], DirectedEdge[v2, v2, c],     DirectedEdge[v1, v3, OverBar[b]],     DirectedEdge[v3, v3, OverBar[c]]}, VertexLabels -> "Name",    EdgeLabels -> "Index"] 

First of all, I would prefer to label the edges with the tag of the directed edge, but I’m not sure how to do this.

The main question is, can Mathematica give me all possible paths of length $ n$ on this graph starting from, say, $ v1$ ?