Improved Spanning on Theta-5

TL;DR

This paper improves the upper bound on the spanning ratio of Theta-5 graphs from 9.96 to approximately 5.70, demonstrating a tighter bound on their efficiency as geometric spanners.

Contribution

The authors establish a new, significantly lower upper bound for the spanning ratio of Theta-5 graphs, advancing understanding of their geometric properties.

Findings

Upper bound of approximately 5.70 on the spanning ratio

Improved the previous bound of 9.96

Enhanced understanding of Theta-5 graph efficiency

Abstract

We show an upper bound of $\frac{ \sin\left(\frac{3\pi}{10}\right) }{ \sin\left(\frac{2\pi}{5}\right)-\sin\left(\frac{3\pi}{10}\right) } <5.70$ on the spanning ratio of $\Theta_5$-graphs, improving on the previous best known upper bound of $9.96$ [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]