Hamilton circuits between cities 
I am trying to construct a Hamilton circuit. I have the cities NY, C, LA, M and D. I know it will be in the shape of a pentagon.
So far I have: NY at the top. C on the left. M on the right. Then LA on the left below C and D on the right below M.
I am confused how I can add weights. Can I make any scale up like 30mins=1 cm?
 A: You have not read the question carefully.


*

*The cities should go on a map of the US (at least approximately).
That tells you the positions on the paper. That determines the scale     essentially some number of inches per mile, depending on the size of your paper.

*The question calls specifically for a complete graph: the pentagon
and all its diagonals.

*The weights are the travel times. Longer distances take more time, but the relationship is not direct proportion. The weights have no explicit relation to the scale of the drawing.
A: You're not constructing a Hamilton circuit (or not yet) according to the question, just a weighted graph. I suggest that you convert all the times to minutes and then just label the appropriate edges of the graph with the time in minutes. (Fortunately the time is the same in both directions). Don't try to scale the edge lengths to the weights. Given the correlation of distance and flight time, you might as well lay the vertices out roughly geographically, but also conveniently for reading the weights. You're not drawing a map: it's a graph.
Then later, if you are using this graph to find a Hamiltonian circuit, since this is a complete graph, you will have to choose an arbitrary start point (it's a circuit, so this doesn't increase the options), then $4$ choices, $3$ choices, $2$ choices, and two forced choices for $4!=24$ different circuits, or $4!/2=12$ if the reversed circuit counts as the same.
