On the Buratti-Horak-Rosa Conjecture about Hamiltonian Paths in Complete Graphs

Abstract

In this paper we investigate a problem proposed by Marco Buratti, Peter Horak and Alex Rosa (denoted by BHR-problem) concerning Hamiltonian paths in the complete graph with prescribed edge-lengths. In particular we solve BHR$(\{1^a, 2^b, t^c\})$ for any even integer $t \geq 4$,  provided that $a+b \geq t-1$. Furthermore, for $t=4, 6, 8$ we present a complete solution of BHR$(\{ 1^a,2^b,t^c \})$ for any positive integer $a,b,c$.