Abstract
Given a set P of n points in the plane, the order-k Gabriel graph on P, denoted by k-GG, has an edge between two points p and q if and only if the closed disk with diameter pq contains at most k points of P, excluding p and q. It is known that 10-GG contains a Euclidean bottleneck matching of P, while 8-GG may not contain such a matching. We answer the following question in the affirmative: does 9-GG contain any Euclidean bottleneck matching of P? Thereby, we close the gap for the containment problem of Euclidean bottleneck matchings in Gabriel graphs.It is also known that 10-GG contains a Euclidean bottleneck Hamiltonian cycle of P, while 5-GG may not contain such a cycle. We improve the lower bound and show that 7-GG may not contain any Euclidean bottleneck Hamiltonian cycle of P.
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