Resolution of a conjecture about linking ring structures

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge -set into cycles. LR-structures were introduced in a paper by P. Potocnik and S. Wilson (2014) [12], as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper.& COPY; 2023 Elsevier Inc. All rights reserved.