  We show that no cubic graphs of order 26 have crossing number larger than 9,
which proves a conjecture of Ed Pegg Jr and Geoffrey Exoo that the smallest
cubic graphs with crossing number 11 have 28 vertices. This result is achieved
by first eliminating all girth 3 graphs from consideration, and then using the
recently developed QuickCross heuristic to find good embeddings of each
remaining graph. In the cases where the embedding found has 10 or more
crossings, the heuristic is re-run with a different settings of parameters
until an embedding with fewer than 10 crossings is found. We provide a minimal
example of a cubic graph on 28 vertices with crossing number 10, and also
exhibit for the first time a cubic graph on 30 vertices with crossing number
12, which we conjecture is minimal.
