I recently constructed the graph shown below in the process of investigating some problems regarding line graphs and homomorphisms, and then happened to see it on wikipedia. I was wondering if anyone knew of this graph coming up anywhere else.

If it helps, I was partly inspired by this paper which shows that any graph with maximum degree 3 and circular chromatic index 4 must contain $K_4$ with one edge subdivided as a subgraph. Note that the graph in the link above is three copies of $K_4$ with one edge subdivided plus another vertex which is adjacent to the vertices that are subdividing the edges.

Thanks

smallestsimple cubic graph with no perfect matching. – Andrew D. King Oct 18 '11 at 13:40