Extending Kotzig's theorem

The weight of a graph G is the minimum sum of the two degrees of the end points of edges of G. Kotzig proved that every graph triangulating the sphere has weight at most 13, and Grünbaum and Shephard proved that every graph triangulating the torus has weight at most 15. We extend these results for graphs, multigraphs and pseudographs "triangulating" the sphere with g handles S _g, g≧1, showing that the corresponding weights are at most about {Mathematical expression} and 24 g-9, respectively; if a (multi, pseudo) graph triangulates S _g and it is big enough, then its weight is at most 15.