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

The Buratti-Horak-Rosa Conjecture concerns the possible multisets of edge-labels of a Hamiltonian path in the complete graph with vertex labels \(0, 1, \ldots, {v-1}\) 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 \(\max(U) \leq 7\) or when \(xyz \leq 24\), 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.