The cheapest path in the graph I am supposed to decide, if the statement is true or false and use arguments for my answer.

In every weighted n-vertices graphs:


with no negative weighted edges,
with n>10,
in which every weighted edge appears constant number of times(e.g. 1,2,3.., but not n number of times), but graphs, in which every edge has the same value, does not satisfy this condition,


there exists between every two vertices at most 4*n^3.

I tried to draw some graphs and I conclude that all of them satisfy my conditions. But I do not have general explanation.

So is that true? If not, can you say me some counter example?

Thanks.
Hint: consider cases where every weight is very large.