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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.22941 |
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| _version_ | 1866909634523037696 |
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| author | Bridi, Guilherme Adamatti Martins, André Luis Alves Marquezino, Franklin de Lima de Figueiredo, Celina Miraglia Herrera |
| author_facet | Bridi, Guilherme Adamatti Martins, André Luis Alves Marquezino, Franklin de Lima de Figueiredo, Celina Miraglia Herrera |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22941 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The only Class 0 Flower snark is the smallest Bridi, Guilherme Adamatti Martins, André Luis Alves Marquezino, Franklin de Lima de Figueiredo, Celina Miraglia Herrera Combinatorics Discrete Mathematics 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. |
| title | The only Class 0 Flower snark is the smallest |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2505.22941 |