A Lower Bound on the Diameter of the Flip Graph

TL;DR

This paper establishes a new lower bound on the diameter of the flip graph, showing it is at least approximately 7n/3, which improves previous bounds and advances understanding of triangulation transformations.

Contribution

The paper provides a tighter lower bound on the flip graph's diameter, enhancing theoretical understanding of triangulation flip sequences.

Findings

Lower bound on flip graph diameter is at least 7n/3 + Θ(1).

Improves upon previous lower bound of 2n + Θ(1).

Advances theoretical bounds in combinatorial triangulation studies.

Abstract

The flip graph is the graph whose nodes correspond to non-isomorphic combinatorial triangulations and whose edges connect pairs of triangulations that can be obtained one from the other by flipping a single edge. In this note we show that the diameter of the flip graph is at least $\frac{7n}{3} + \Theta(1)$, improving upon the previous $2n + \Theta(1)$ lower bound.