The IUPAC nomenclature of fullerene derivatives is based on the vertex spiral, but not all fullerenes possess one. There are 1495 isomers with <or=20 vertices (66 isolated-pentagon isomers with <or=150 vertices) that are vertex-aspiral, but only four for which all vertex spiral starts succeed. Vertex-aspiral trivalent polyhedra are common (40 190 with <or=24 vertices); infinite series include truncates of all trivalents larger than the tetrahedron. An alternative and readily automated breadth-first-search numbering scheme is proposed to deal with all trivalent polyhedra including fullerenes and incidentally to give robust, efficient determination of the automorphism/point group. For fullerenes, the new scheme has two clear advantages over the vertex spiral nomenclature: it is exception-free, removing the need for complex prescriptions to deal with exceptional cases, and it is mathematically simpler in that the numbering can be found in linear rather than quadratic time (as all canonical numberings for a fullerene begin on a pentagon).