Final answer:
A complete bipartite graph where the sets have unequal numbers of vertices (e.g., Km,n with m ≠ n) is an example of a graph that does not have a Hamiltonian path, despite no vertex having degree one.
Step-by-step explanation:
A graph that does not have a Hamiltonian path even though no vertex has degree one is the graph of a bipartite structure where the two sets of vertices have different cardinality. An example of such graph is the complete bipartite graph Km,n where m \\eq n. In a bipartite graph, vertices are divided into two disjoint sets and every edge connects a vertex from one set to a vertex from the other set. If one set has more vertices than the other, there cannot be a Hamiltonian path because there would be no way to visit all the vertices in the larger set without either repeating vertices or missing some edges, both of which are not allowed in a Hamiltonian path.