#StackBounty: #algorithms #graphs #colorings #topological-ordering How to edge-color a directed acyclic graph so that every path visits…

Bounty: 100

Given a directed acyclic graph $G$ and a start vertex $s$ and an end vertex $e$, consider a coloring of the edges valid if, for every path from $s$ to $e$ and every color $c$, either $c$ is never encountered along that path, or every edge that is colored $c$ is visited by that path.

Given $G,s,e$, I would like to find a valid coloring that uses the minimal number of colors. Is there an efficient algorithm for this problem?

I show below an example graph and a sample solution. The circle on the left is the starting vertex, the filled circle on the right is the end vertex.

Example Graph

Get this bounty!!!

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