Show that this graph has no hamiltonian cycle. 
I wish to show that this graph does not have a Hamiltonian cycle. My thought process has just been to start at arbitrary vertices and to try and find a hamiltonian cycle, which is blunt and inefficient. I need a starting idea or hint to guide me. 

Notice that the cycles are even. You can redraw the graph as a bipartite graph with uneven parts ($5$ and $6$). This means that every step is between the two parts, and the fact that the two parts do not have the same number of nodes mean that the cycle cannot close - start and end must be on the larger part, which are therefore not connected.
Showing the bipartite nature by colouring:

