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

TL;DR

This paper verifies the Buratti-Horak-Rosa Conjecture for certain three-element underlying sets using growable realizations, extending known cases up to maximum element 7 or product 24, with some exceptions.

Contribution

It introduces the growable realizations method to prove the conjecture for new classes of three-element sets, expanding the known validity range.

Findings

Conjecture holds for all three-element sets with max element ≤ 7.

Conjecture holds when the product of the set elements ≤ 24.

Validity for sets {1,2,x} with even x depends on finitely many cases.

Abstract

The Buratti-Horak-Rosa Conjecture concerns the possible multisets of edge-labels of a Hamiltonian path in the complete graph with vertex labels $0, 1, \ldots, {v-1}$ 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.