The only Class 0 Flower snark is the smallest

TL;DR

This paper proves that among Flower snark graphs, only the smallest one, J_3, is Class 0, meaning it has the minimal pebbling number equal to its number of vertices, highlighting its unique property.

Contribution

The paper establishes that J_3 is the only Class 0 Flower snark, resolving a recent open question about pebbling numbers in snark graphs.

Findings

J_3 is Class 0

J_3 is the only Class 0 Flower snark

Provides a complete characterization of Class 0 Flower snarks

Abstract

Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. The pebbling number is the smallest $t$ so that from any initial configuration of $t$ pebbles it is possible, after a sequence of pebbling moves, to place a pebble on any given 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 $J_3$ is in fact the only Class~$0$ Flower snark.