The cut-partition characterization of fullerenes without generalized Stone–Wales paths

Let $T_n$ be a dual fullerene graph and let $T_n^6$ be its hexagonal subgraph, for feasible $n$. A cut-partition of $T_n^6$ is the set of components generated by the stated cut-partition construction; $t$-triangles and $(t,(r_1,r_2,r_3))$-triangles are the triangle types defined there, and a gSW path is a generalized Stone–Wales path. Cut-partition characterization conjecture. The cut-partition of $T_n^6$ consists solely of elements that are either $t$-triangles or $(t,(r_1,r_2,r_3))$-triangles if and only if $T_n$ contains no gSW path, with the exception that the cut-partition contains only $0$-triangles.

This conjecture proposes a structural characterization of the absence of generalized Stone–Wales paths using the cut-partition. It is presented as a numerical conjecture, with no resolution stated in the source.