The crossing number of the Grötzsch graph is 5.

Crossing numbers are believed to be difficult to compute in general. (The corresponding decision problem is NP-hard.) However, for small graphs and small crossing numbers, it is possible to find an optimal planar drawing. For example, see Markus Chimani's thesis "Computing Crossing Numbers" for more information.

The Open Graph Drawing Framework (OGDF) can compute crossing numbers. After compiling the program on the linked page and entering the Grötzsch graph, my computer computed that the optimal planar drawing has 5 crossings.

The crossing number of the Grötzsch graph is 5.

Crossing numbers are believed to be difficult to compute in general. (The corresponding decision problem is NP-hard.) However, for small graphs and small crossing numbers, it is possible to find an optimal planar drawing. For example, see Markus Chimani's thesis "Computing Crossing Numbers" for more information.

The Open Graph Drawing Framework (OGDF) can compute crossing numbers. After compiling the program on the linked page and entering the Grötzsch graph, my computer computed that the optimal planar drawing has 5 crossings. Let me emphasize that this technique is exact, not heuristic.