## Tag: cubic graphs

### Hypohamiltonian Planar Graphs

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:

1. A graph is hypohamiltonian if it does not contain a hamilton cycle, but the deletion of any vertex does create a hamilton cycle.
2. A planar graph is a graph that can be embedded in the plane with no edge crossings.
3. A cubic graph is a 3-regular graph.
4. 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].

The three smallest examples of hypohamiltonian planar cubic graphs of girth 5. The numbers give the face lengths for faces other than 5-faces. Figures by B.D. McKay.

### Generating Cubic Graphs

Today, we return to the canonical deletion method to generate cubic graphs. Cubic graphs are 3-regular graphs, and since the degree is constant, there are many fewer cubic graphs of order $n$ than general graphs of order $n$. Also, since the structure of cubic graphs is very restricted, we can use custom augmentations to generate them more quickly than using an augmentation that works for general graphs.

This implementation of the method comes from Brinkmann, Goedgebeur, and McKay in their paper Generation of Cubic graphs from Discrete Mathematics and Theoretical Computer Science. By desigining custom augmentations, they are able to restrict their generation algorithm to create only connected cubic graphs. This augmenation is paired with a canonical deletion step which removes all isomorphic duplicates. Their implementation is given in the `snarkhunter` software, which is the current-fastest implementation to generate cubic graphs (and snarks; the generation snarks is investigated in a different paper).
Read the rest of this entry »