New methods to attack the Buratti-Horak-Rosa conjecture

TL;DR

This paper introduces new linear realization-based methods to verify the Buratti-Horak-Rosa conjecture for various list types, expanding the cases where the conjecture can be proven true.

Contribution

The authors develop novel techniques based on linear realizations to prove the conjecture for a wide range of list configurations.

Findings

Methods successfully verify the conjecture for lists with specific structures

Applicable to lists with sets like {x,y,x+y} and {1,2,4,...,2x,2x+1}

Effective for lists containing many consecutive elements

Abstract

The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list $L$ of $v-1$ positive integers not exceeding $\left\lfloor \frac{v}{2}\right\rfloor$ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set $\{0,1,\ldots,v-1\}$ if and only if, for every divisor $d$ of $v$, the number of multiples of $d$ appearing in $L$ is at most $v-d$. In this paper we present new methods that are based on linear realizations and can be applied to prove the validity of this conjecture for a vast choice of lists. As example of their flexibility, we consider lists whose underlying set is one of the following: $\{x,y,x+y\}$, $\{1,2,3,4\}$, $\{1,2,4,\ldots,2x\}$, $\{1,2,4,\ldots,2x,2x+1\}$. We also consider lists with many consecutive elements.