Some work on a problem of Marco Buratti

TL;DR

This paper explores a generalization of Marco Buratti's conjecture, proving it for specific multisets and relating the problem to tree structures with certain diameter properties.

Contribution

It extends Buratti's conjecture to trees, establishes realizability conditions based on multiset properties, and proves the conjecture for specific multiset forms.

Findings

Proves the conjecture for multisets of the form {φ_k(1)^a, φ_k(2)^b, φ_k(3)^c}

Shows multisets can be realized as trees with diameter at least one more than the number of distinct elements

Generalizes the problem from Hamiltonian paths to tree structures

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 $\phi_k(i)=\min\{ki \pmod p, -ki \pmod p\}$.