How to solve the Shortest Hamiltonian Path problem on Sparse Graphs?
Indeed: Eppstein has shown that the TSP can be solved in time $O(1.26^n)$ if all vertices are of degree at most 3.


  David Eppstein:
  The Traveling Salesman Problem for Cubic Graphs.
  J. Graph Algorithms Appl. 11(1): 61-81 (2007)


Some tiny improvements of this result are also known, for instance by Liśkiewicz and Schuster (2014) to $O(1.255^n)$.