Construction Techniques for Linear Realizations of Multisets with Small Support
Onur A\u{g}{\i}rseven, M. A. Ollis

TL;DR
This paper develops construction techniques for linear realizations of specific multisets, advancing understanding of the Buratti-Horak-Rosa Conjecture and providing new methods for supports of size 3.
Contribution
It introduces new construction methods for linear realizations of multisets with small support, especially for cases where the multiset has a specific form and parameters.
Findings
Constructed linear realizations for multisets with support size 2 when certain divisibility conditions hold.
Established conditions under which linear realizations exist for multisets with support size 2, including the case when k=2.
Provided partial results supporting the coprime version of the BHR Conjecture for y ≤ 16.
Abstract
A Hamiltonian path in the complete graph whose vertices are labeled with the integers is a linear realization for the multiset of the linear edge-lengths (given by for the edge between vertices and ) of the edges in the path. A linear realization is standard if an end-vertex is 0 and perfect if the end-vertices are 0 and . Linear realizations are useful in the study of the Buratti-Horak-Rosa (BHR) Conjecture on the existence of cyclic realizations (where cyclic edge-lengths are given by distance modulo ) for given multisets. In this paper, we focus on multisets of the form . Using core perfect linear realizations for supports of size 2 (which have the forms whenever ), we construct standard linear realizations (with , , ) when or .…
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Taxonomy
TopicsTopology Optimization in Engineering
