Scenic Graphs I: Traceable Graphs

A path of a graph is maximal if it is not a proper subpath of any other path of the graph. The path spectrum is the set of lengths of all maximal paths in the graph. A graph is scenic if its path spectrum is a singleton set. In this paper, we give a new proof characterizing all scenic graphs with a Hamiltonian path; this result was first proven by Thomassen in \(1974\). The proof strategy used here is also applied in a companion paper in which we characterize scenic graphs with no Hamiltonian path.