Geodesic
Hamiltonian paths in the complete graph with edge-lengths 1, 2, 3
The electronic journal of combinatorics, Tome 17 (2010)
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Marco Buratti has conjectured that, given an odd prime $p$ and a multiset $L$ containing $p-1$ integers taken from $\{1,\ldots,(p-1)/2\}$, there exists a Hamiltonian path in the complete graph with $p$ vertices whose multiset of edge-lengths is equal to $L$ modulo $p$. We give a positive answer to this conjecture in the case of multisets of the type $\{1^a,2^b,3^c\}$ by completely classifying such multisets that are linearly or cyclically realizable.
DOI : 10.37236/316
Classification : 05C38 
@article{10_37236_316,
     author = {Stefano Capparelli and Alberto Del Fra},
     title = {Hamiltonian paths in the complete graph with edge-lengths 1, 2, 3},
     journal = {The electronic journal of combinatorics},
     year = {2010},
     volume = {17},
     doi = {10.37236/316},
     zbl = {1215.05095},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/316/}
}
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Stefano Capparelli; Alberto Del Fra. Hamiltonian paths in the complete graph with edge-lengths 1, 2, 3. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/316

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