Further progress on the Buratti-Horak-Rosa conjecture

Anita Pasotti, Marco Antonio Pellegrini

Published 2019-12-13Version 1

The conjecture posed by Marco Buratti, Peter Horak and Alex Rosa states that a multiset $L$ of $v-1$ positive integers not exceeding $\left\lfloor \frac{v}{2}\right\rfloor$ is the list of \emph{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 $v-d$. Here, we prove the validity of this conjecture for the lists $\{1^a,x^b,(x+1)^c\}$, with $x\geq 3$ and $a,b,c\geq 0$, in each of the following cases: (i) $x$ is odd and $a\geq \min\{3x-3, b+2x-3 \}$; (ii) $x$ is odd, $a\geq 2x-2$ and $c\geq \frac{4}{3} b$; (iii) $x$ is even and $a\geq \min\{3x-1, c+2x-1\}$; (iv) $x$ is even, $a\geq 2x-1$ and $b\geq c$. As a consequence we also obtain some results on lists of the form $\{y^a,z^b,(y+z)^c\}$.