Does the graph with only one vertex have an Eulerian path? And, does it have a Hamiltonian path?
Hamiltonian & Eulerian paths, one vertex graph
1
$\begingroup$
graph-theory
-
0@Andrew: You could post that as an answer so the question doesn't remain unanswered. – 2012-12-12
1 Answers
3
I'm not sure if all graph theory books treat degenerate cases the same way, but Diestel's Graph Theory, at least, allows a path to have length $0$, i.e., to consist of a single vertex with no edges. If a graph consists of a single vertex $v$, then the path consisting of $v$ is vacuously Eulerian. It is also a Hamiltonian path, since it contains all of the vertices of the graph.