Abstract
We describe two computational methods for the construction of cubic graphs with given girth. These were used to produce two independent proofs that the (3,9)-cages, defined as the smallest cubic graphs of girth 9, have 58 vertices. There are exactly 18 such graphs. We also show that cubic graphs of girth 11 must have at least 106 vertices and cubic graphs of girth 13 must have at least 196 vertices.
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