Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: stat