In a graph with a Hamiltonian path, the maximum number of vertices of degree one is one. Having more than one vertex of degree one would prevent the formation of a Hamiltonian path, as it requires a single endpoint to visit all vertices exactly once.
In a graph with a Hamiltonian path, the maximum number of vertices of degree one that the graph can have is one. This is because a Hamiltonian path is a path in a graph that visits each vertex exactly once. If there were more than one vertex of degree one, it would mean that there are multiple endpoints in the graph, making it impossible to form a Hamiltonian path, as the path would have to terminate at one of these vertices, leaving other vertices unvisited.
Therefore, in a graph with a Hamiltonian path, the maximum number of vertices of degree one is one.