On the size of k-irreducible triangulations

TL;DR

This paper establishes an optimal bound on the size of k-irreducible triangulations of orientable surfaces, improving previous bounds significantly.

Contribution

It proves that k-irreducible triangulations of genus g surfaces have O(k^2 g) triangles, refining earlier exponential bounds.

Findings

Bound of O(k^2 g) triangles for k-irreducible triangulations

Improved upon previous exponential bounds

Optimal bound matching the lower bound

Abstract

A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of an orientable surface of genus g has $O(k^2g)$ triangles, which is optimal. This is an improvement over the previous best bound $k^{O(k)} g^2$ of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].