Irreducible Triangulations are Small

TL;DR

This paper proves that irreducible triangulations of surfaces with Euler genus g have at most 13g-4 vertices, significantly improving previous bounds and advancing understanding of surface triangulations.

Contribution

The authors establish a new linear upper bound on the number of vertices in irreducible triangulations of surfaces, improving prior exponential bounds.

Findings

Irreducible triangulations have at most 13g-4 vertices for genus g surfaces.

Previous bounds were much larger, at 171g-72.

The result tightens the understanding of minimal triangulations of surfaces.

Abstract

A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$ vertices. The best previous bound was $171g-72$.