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Suppose we have a complete graph with 13 vertices. Can we assign a different positive integer length to each edge such that all Hamiltonian circuits in the graph are equal in length, which is exactly 2017.

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  • $\begingroup$ @bof oh,i forgot to mention that all edges are different $\endgroup$ – Takanashi Mar 15 '17 at 10:28
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Let $a_1=1$ and $a_2=3;$ let $a_n=a_{n-1}+a_{n-2}+1$ for $3\le n\le12;$ and let $a_{13}=a_{12}+a_{11}+61.$ Thus we get the $13$ numbers $$1,\ 3,\ 5,\ 9,\ 15,\ 25,\ 41,\ 67,\ 109,\ 177,\ 287,\ 465,\ 813.$$ Observe that these are odd numbers, their $\binom{13}2$ pairwise sums are distinct, and they add up to $2017.$ Assign these $13$ numbers to the vertices of the complete graph, and assign to each edge the average of the numbers assigned to its endpoints.

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