On the size of k-irreducible triangulations
Computational Geometry
2026-05-18 v2 Discrete Mathematics
Combinatorics
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 triangles, which is optimal. This is an improvement over the previous best bound of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].
Cite
@article{arxiv.2603.20030,
title = {On the size of k-irreducible triangulations},
author = {Vincent Delecroix and Oscar Fontaine and Arnaud de Mesmay},
journal= {arXiv preprint arXiv:2603.20030},
year = {2026}
}
Comments
v2: Corrected an error in the treatment of non-orientable surfaces; we no longer claim a bound in that setting