Irreducible triangulations of surfaces
The authors show that, for any surface \(S\) and any \(k\), there are at most finitely many triangulations of \(S\) such that each edge is in a noncontractible cycle of length \(k\), but is in no shorter noncontractible cycle; such a triangulation is said to be \(k\)-irreducible. If \(S\) is the sphere with \(h\) handles (respectively the sphere with \(k\) crosscaps), then the Euler genus of \(S\) is \(2h\) (respectively \(k)\). Specifically, the authors show that if \(T\) is a \(k\)-irreducible triangulation of \(S\), of Euler genus \(g\), and if \(k\geq 2\), then \(|E(T)|\leq 3k\) \(k!(6k)^kg^2\).
scientific article; zbMATH DE number 954823