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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.