On the pebbling numbers of Flower, Blanuša, and Watkins 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 $π(G)$ 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. In this paper, we provide the first results on the pebbling numbers of snarks. Until now, only the Petersen graph had its pebbling number correctly established, although attempts had been made for the Flower and Watkins snarks.

Figures (5)

• Figure 1: The graph $J_{5}$ and its (green) $v_0$-unsolvable configuration $C$ of size $22$, which equals the configuration $C^*$ with an extra pebble on $z_{-1}$.
• Figure 2: The graph $J_{11}$ and its three $z_0$-strategies ${\boldsymbol{T}}_0$ (in red), ${\boldsymbol{T}}_1$ (in blue), and ${\boldsymbol{T}}_{-1}$ (in green).
• Figure 3: Blanuša 2 and its labelling.
• Figure 4: The Watkins graph, shown in its traditional drawing.
• Figure 5: The Watkins graph, organized by distance from target $a_1$, along with an $a_1$-unsolvable configuration $C$ of size $182$.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1
• Corollary 1
• Corollary 2: cranston
• Lemma 2: Retract Lemma Chung
• Lemma 3: HurlGeneral
• Lemma 4: Weight Function Lemma HurlbertWFL