arXiv:2505.22941 · v3
The only Class 0 Flower snark is the smallest
摘要
In graph pebbling, pebbles are distributed across the vertices of a graph. A pebbling move consists of removing two pebbles from one vertex and adding one pebble to an adjacent vertex. The pebbling number is the minimum number $t$ such that, regardless of how the $t$ pebbles are initially placed, it is always possible to perform a sequence of moves that results in at least one pebble placed on any specified target vertex. Graphs whose pebbling number is equal to the number of vertices are called Class~$0$ and provide a challenging set of graphs that resist being characterized. In this note, we answer a question recently proposed by the pioneering study on the pebbling number of snark graphs: we prove that the smallest Flower snark $J_3$ is Class~$0$, establishing that $J_3$ in fact is the only Class~$0$ Flower snark. that systematically reduce the space of pebble configurations. Our bound-based approach shows particular promise for diameter $3$ graphs, where the neighborhood structure remains sufficiently constrained to enable effective analysis.
作者备注：10 pages, 6 figures. Proof reorganized and discussion improved. Supplementary software available at https://github.com/gabridi/pebbling_unsolvability