Brendan McKay uploaded a short-but-sweet article to the arXiv titled Hypohamiltonian planar cubic graphs with girth five. Let’s quickly break down all of the words of this title:
- A graph is hypohamiltonian if it does not contain a hamilton cycle, but the deletion of any vertex does create a hamilton cycle.
- A planar graph is a graph that can be embedded in the plane with no edge crossings.
- A cubic graph is a 3-regular graph.
- The girth of a graph is the length of the shortest cycle.
McKay details the search for the smallest (and first known) graphs of this type, which have order 76. There are three non-isomorphic examples. The data for these graphs (and more) is available online [Note: at this time, it appears the data is not yet online].