I'm reading Combinatorics and Graph Theory, 2nd Ed., and am beginning to think the terms used in the book might be outdated. Check out the following passage:
If the vertices in a walk are distinct, then the walk is called a path. If the edges in a walk are distinct, then the walk is called a trail. In this way, every path is a trail, but not every trail is a path. Got it?
On the other hand, Wikipedia's glossary of graph theory terms defines trails and paths in the following manner:
A trail is a walk in which all the edges are distinct. A closed trail has been called a tour or circuit, but these are not universal, and the latter is often reserved for a regular subgraph of degree two.
Traditionally, a path referred to what is now usually known as an open walk. Nowadays, when stated without any qualification, a path is usually understood to be simple, meaning that no vertices (and thus no edges) are repeated.
Am I to understand that Combinatorics and Graph Theory, 2nd Ed. is using a now outdated definition of path, referring to what is now referred to as an open walk? What are the canonical definitions for the terms "walk", "path", and "trail"?
Best Answer
You seem to have misunderstood something, probably the definitions in the book: they’re actually the same as the definitions that Wikipedia describes as the current ones.