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].

翻译：一个曲面的三角剖分被称为k-不可约的，如果每条非可缩曲线的长度至少为k，且任何边收缩操作都会破坏这一性质。等价地，每条边都属于某条长度为k的非可缩曲线，且不存在更短的非可缩曲线。我们证明，亏格为g的可定向曲面上的k-不可约三角剖分至多包含$O(k^2g)$个三角形，这一界是最优的。这改进了Gao、Richter和Seymour之前得到的最佳界$k^{O(k)} g^2$ [Journal of Combinatorial Theory, Series B, 1996]。