The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three

Abstract

The Buratti-Horak-Rosa Conjecture concerns the possible multisets of edge-labels of a Hamiltonian path in the complete graph with vertex labels $0,1,\dots ,v-1$ under a particular induced edge-labeling. The conjecture has been shown to hold when the underlying set of the multiset has size at most~2, is a subset of $\{1,2,3,4\}$ or $\{1,2,3,5\}$, or is $\{1,2,6\}$, $\{1,2,8\}$ or $\{1,4,5\}$, as well as partial results for many other underlying sets. We use the method of growable realizations to show that the conjecture holds for each underlying set $U=\{x,y,z\}$ when $\max (U)\le 7$ or when $xyz\le 24$, with the possible exception of $U=\{1,2,11\}$. We also show that for any even $x$ the validity of the conjecture for the underlying set $\{1,2,x\}$ follows from the validity of the conjecture for finitely many multisets with this underlying set.

arXiv comment20 pages