Trails and Circuits, Part I
In the last section we used some longwinded phrases like "a path that crosses every edge exactly once". In this lesson we'll develop some shorthand phrases that make describing paths within a graph simpler. Then we'll look at another application that leads into another classic problem, in this case one that still hasn't been solved.
What You Should Learn

If u and v are vertices in a graph then a uv walk is a series of vertices, starting with u and ending with v, such that every vertex in the series is connected by an edge to the vertices before and after it in the series. 
A walk is just what the name implies: a path between two points. If you think of the graph as a set of walkways connecting play areas in a park then a "walk" exists between two areas if there's a series of walkways connecting them. 
Look at the graph to the right. (Think of it as the map of a park if you like.) In that graph there exists a v_{1}v_{4} walk: v_{1}, v_{2}, v_{4}. Notice that the definition doesn't say anything about walks being unique. You can easily come up with several more ways to get from v_{1} to v_{4}. The definition also doesn't say anything about repetition. For example, v_{1}, v_{3}, v_{2}, v_{1}, v_{3}, v_{4} is a perfectly good walk even though it crosses the v_{1}v_{3} edge twice. If we want to simplify the language we used to discuss the Konigsberg Bridge Problem we're going to need something more limited. 
If u and v are vertices in a graph then a uv trail is a uv walk that doesn't repeat any edges. A uv circuit is a uv trail that contains at least three edges and that has u = v. A circuit that doesn't repeat any vertices, except the first/last, is called a cycle. 
Both of these definitions narrow down our definition of a walk. A trail is a special case of a walk where none of the edges is crossed twice. A circuit is a special kind of a trail that finishes at the same place that it started. A cycle is a cicuit that doesn't repeat any vertices. Take special note of that. We're talking about vertices this time, not edges. This is a much more limiting case  if you're looking at the drawing of a graph, a cycle looks like a circular path. 