Infinite series of triangulations on closed surfaces with vertex covers of given degree (Q1889845)

A finite graph embedded into a closed surface is called a triangulation on the surface if the graph realizes the 1-skelton of a triangulation of the surface. A triangulation on a closed surface is said to be \(d\)-covered if at least one end of each edge of the graph has degree \(d\). There exist only finitely many 3 and 4-covered triangulations on closed surfaces, actually they have been already classified in the concrete. On the other hand, Nakamoto and Negami showed in [\textit{S. Negami} and \textit{A. Nakamoto}, Graphs Comb. 17, 529--537 (2001; Zbl 1010.05023)] that given a closed surface there exist only finitely many \(d\)-covered triangulations on the surface if \(d \geq 13\), and the 2-sphere and the projective plane admit no \(d\)-covered triangulations for \(d \geq 11\). In the paper under review the author proves that given a closed surface of non-positive (resp. positive) Euler characteristic there exist infinitely many \(d\)-covered triangulations on the surface if \(5 \leq d \leq 12\) (resp. \(5 \leq d \leq 10\)).