The trio started by temporarily ignoring a subset of possible graphs: those containing negative cycles. These are paths that loop back to where they started after passing through a series of edges ...

Using this definition of path, OP, you must realize that there may be exponentially (in the number of vertices + edges in the graph) many paths between two vertices (even in simple graphs).

Sound familiar? This is exactly the kind of path that would solve the Bridges of Königsberg problem and is called an Eulerian cycle. As it visits all edges of the de Bruijn graph, which represent ...