A new result on the problem of Buratti, Horak and Rosa

TL;DR

This paper investigates a conjecture about Hamiltonian paths with specific edge-length conditions in complete graphs, proving it holds when all edge lengths are from the set {1,2,3,5}.

Contribution

The paper provides a proof confirming the conjecture for multisets with elements only in {1,2,3,5}, advancing understanding of the problem.

Findings

Conjecture holds for multisets with elements in {1,2,3,5}.

Preliminary discussions link the conjecture to graph decompositions.

The paper discusses the conjecture's broader implications.

Abstract

The conjecture of Peter Horak and Alex Rosa (generalizing that of Marco Buratti) states that a multiset L of v-1 positive integers not exceeding [v/2] is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set {0,1,...,v-1} if and only if the following condition (here reformulated in a slightly easier form) is satisfied: for every divisor d of v, the number of multiples of d appearing in L is at most v-d. In this paper we do some preliminary discussions on the conjecture, including its relationship with graph decompositions. Then we prove, as main result, that the conjecture is true whenever all the elements of L are in {1,2,3,5}.