Here is our paper.
Corrections, additions & developments:
CORRECTION: I misspelled the name of Edward Reingold as "Rheingold", possibly being confused by the Wagner opera.
ADDITION to page 93: "when recursing, always prioritize a higher degree vertex over a lower degree vertex.". It should be understood "a higher degree vertex" means "a vertex which is not already in the path, that has the largest number of edges to unvisited vertices". In other words: the priority list of available vertices is determined and sorted during each recursion call.
The interactive graphs for Hamiltonian cycles can be found here.
Source Data for Interactive Graphs
Download our results here.
Source Code for the Algorithms
And here is the source code of the algorithms.