A conjecture about polygonal curves in the plane; let's play in ``Burattiland"!

Marco Buratti
Universit\`a di Perugia

Simone Costa
Universit\`a Roma 3



Content: Let $V$ be the set of vertices of a regular polygon in the plane and let $L(V)$ be the set of all lengths of all open polygonal curves with vertex-set $V$. Which is the size of $L(V)$? I conjecture that if $|V|=2n+1$ is a prime, then the answer is ${3n-1\choose n-1}$. I will show that this conjecture is equivalent to another conjecture - that bears my name even though I never worked on it - about Hamiltonian paths in the complete graph on a prime number of vertices. The conjecture can be also formulated by saying that a ``perfect player" always wins in the game of ``Burattiland". \vspace{5mm} \noindent{\bf References: } \begin{itemize} \item[{[1]}] J. H. Dinitz and S. R. Janiszewski, {\it On Hamiltonian paths with prescribed edge lengths in the complete graph}, Bull. Inst. Combin. Appl. {\bf57} (2009), 42--52. \item[{[2]}] P. Horak and A. Rosa, {\it On a problem of Marco Buratti}, Electron. J. Combin. {\bf16} (2009) $\sharp$R20. \item[{[3]}] A. Pasotti and M. A. Pellegrini, {\it A new result on the problem of Buratti, Horak and Rosa}, Discrete Math. {\bf319} (2014), 1--14. \item[{[4]}] A. Pasotti and M. A. Pellegrini, {\it On the Buratti-Horak-Rosa conjecture about Hamiltonian paths in complete graphs}, Electron. J. Combin. {\bf21} (2014) Paper $\sharp$P2.30 \item[{[5]}] D. West, http://www.math.uiuc.edu/~west/regs/buratti.html \end{itemize}

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