Gao, Z.; Richter, R. B.; Seymour, P. D. Irreducible triangulations of surfaces. (English) Zbl 0863.05029 J. Comb. Theory, Ser. B 68, No. 2, 206-217 (1996). 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 \(|E(T)|\leq 3k\) \(k!(6k)^kg^2\). Reviewer: A.T.White (Kalamazoo) Cited in 6 Documents MSC: 05C10 Planar graphs; geometric and topological aspects of graph theory Keywords:surface; triangulations; sphere; handles; crosscaps; Euler genus PDFBibTeX XMLCite \textit{Z. Gao} et al., J. Comb. Theory, Ser. B 68, No. 2, 206--217 (1996; Zbl 0863.05029) Full Text: DOI