Partial triangulations of surfaces with girth constraints

Abstract

Barnette and Edelson have shown that there are finitely many minimal triangulations of a connected compact 2-manifold M. Similar finiteness results are obtained for cellular partial triangulations that satisfy various girth inequality constraints for embedded cycles. A characterisation of various M-embedded sparse graphs is given in terms of the satisfaction of higher genus girth inequalities. With this it is shown that there are finitely many contraction-minimal M-embedded graphs that are (3,6)-tight or (3,3)-tight.