[1] Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg.

However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull.

They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices.

[6] Ringel's conjecture asks if the complete graph K2n+1 can be decomposed into copies of any tree with n edges.

[13] Rectilinear Crossing numbers for Kn are A complete graph with n nodes represents the edges of an (n – 1)-simplex.

As part of the Petersen family, K6 plays a similar role as one of the forbidden minors for linkless embedding.