10-Gabriel graphs are Hamiltonian

TL;DR

This paper investigates the minimum k for which the k-Gabriel graph of any point set in the plane always contains a Hamiltonian cycle, establishing bounds between 2 and 10.

Contribution

The paper improves the bounds on the minimum k needed for Hamiltonian cycles in k-Gabriel graphs from 15-1 to 10-2.

Findings

Upper bound of 10 for k

Lower bound of 2 for k

Previous bounds were 15 and 1

Abstract

Given a set $S$ of points in the plane, the $k$-Gabriel graph of $S$ is the geometric graph with vertex set $S$, where $p_i,p_j\in S$ are connected by an edge if and only if the closed disk having segment $\bar{p_ip_j}$ as diameter contains at most $k$ points of $S \setminus \{p_i,p_j\}$. We consider the following question: What is the minimum value of $k$ such that the $k$-Gabriel graph of every point set $S$ contains a Hamiltonian cycle? For this value, we give an upper bound of 10 and a lower bound of 2. The best previously known values were 15 and 1, respectively.