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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?

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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.

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    If you mean draw a pentagon then I'm having trouble deciding which city goes on which vertex... – 2017-02-24
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    I would aim to get cities a little closer to a normal map position. If you have a left-pointing pentagon, then from top-right *clockwise* NY, M, D, LA, C would be quite close. But the concept looks good to me. – 2017-02-24
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    One last question (sorry): What is the minimal spanning tree for the graph? – 2017-02-24
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    Is CD, DM, CNY, DLA correct? – 2017-02-25
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    That looks pretty much the best spanning tree (by inspection). – 2017-02-25
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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.

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    What is the best approach? Should I list all 24 possibilities and work out the weights? – 2017-02-24
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    Where do you get $24$? There are $10$ edges in the graph, Each represents a direct flight between two cities. The weight is how long the flight takes. – 2017-02-24