Some work on a problem of Marco Buratti

Abstract

Marco Buratti's conjecture states that if $p$ is a prime and $L$ a multiset containing $p-1$ non-zero elements from the integers modulo $p$, then there exists a Hamiltonian path in the complete graph of order $p$ with edge lengths in $L$. Say that a multiset satisfying the above conjecture is realizable. We generalize the problem for trees, show that multisets can be realized as trees with diameter at least one more than the number of distinct elements in the multiset, and affirm the conjecture for multisets of the form $\{\phi_k(1)^a, \phi_k(2)^b, \phi_k(3)^c\}$ where $\phi_k(i)=\min\{ki \pmod p, -ki \pmod p\}$.