On the crossing number of K_13

TL;DR

This paper narrows down the crossing number of K_{13} from a set of five possible values to four by eliminating one candidate through specific graph analysis.

Contribution

It proves that the crossing number of K_{13} is not 217, refining the known possible values for this graph.

Findings

Crossing number of K_{13} is not 217.

Remaining possible crossing numbers are 219, 221, 223, 225.

Abstract

Since the crossing number of K_{12} is now known to be 150, it is well-known that simple counting arguments and Kleitman's parity theorem for the crossing number of K_{2n+1} combine with a specific drawing of K_{13} to show that the crossing number of K_{13} is one of the numbers in {217,219,221,223,225}. We show that the crossing number is not 217.