Light stars in large polyhedral maps on surfaces

It is well known that every polyhedral map with large enough number of vertices contains a vertex of degree at most 6. In this paper the existence of stars having low degree sum of their vertices in polyhedral maps is investigated. We will prove: if G   is a polyhedral map on compact 2-manifold MM with non-positive Euler characteristic χ(M)χ(M) and G   has more than 149|χ(M)|149|χ(M)| vertices then G contains an edge of weight at most 15, or a path of weight at most 20 on three vertices with a central 4-vertex, or a 3-star of weight at most 24 with a central 5-vertex, or a 4-star of weight at most 32 with a central 6-vertex.