Steiner Triple Systems, Pinched Surfaces, and Complete Multigraphs.

We consider complete multigraphs $${K_n^m}$$ on n vertices with every pair joined by m edges. We embed these graphs to triangulate $${S_n^k}$$ , a pinched surface with n pinch points each having k sheets. These embeddings have a vertex at each pinch point and any two sheets at a pinch point have the same number of edges. Moreover, we want to 2 m-color the faces such that each color class is a Steiner triple system. These embeddings generalize in two ways biembeddings of Steiner triple systems, the case m = 1, k = 1 of simple graphs in surfaces without pinch points. [ABSTRACT FROM AUTHOR]