Tight and Untight Triangulations of Surfaces by Complete Graphs

Triangular embeddings of complete graphs into surfaces are studied through the notion of tightness which is a natural combinatorial generalization of connectedness for graphs. By means of a construction which "couples" two such surfaces to produce a new one, the existence of untight complete triangular embeddings is proved and the known archive of tight ones is broadened. In particular, K30 admits a tight and an untight triangular embedding into the same surface. Therefore, complete graphs may triangulate the same surface in nonisomorphic ways.