Pebbling number of polymers

Abstract

Let $G=(V,E)$ be a simple graph. A function $f:V\rightarrow \mathbb{N}\cup \{0\}$ is called a configuration of pebbles on the vertices of $G$ and the quantity $\vert f\vert=\sum_{u\in V}f(u)$ is called the weight of $f$ which is just the total number of pebbles assigned to vertices. A pebbling step from a vertex $u$ to one of its neighbors $v$ reduces $f(u)$ by two and increases $f(v)$ by one. A pebbling configuration $f$ is said to be solvable if for every vertex $ v $, there exists a sequence (possibly empty) of pebbling moves that results in a pebble on $v$. The pebbling number $ π(G) $ equals the minimum number $ k $ such that every pebbling configuration $ f $ with $ \vert f\vert = k $ is solvable. Let $ G $ be a connected graph constructed from pairwise disjoint connected graphs $ G_1,...,G_k $ by selecting a vertex of $ G_1 $, a vertex of $ G_2 $, and identifying these two vertices. Then continue in this manner inductively. We say that $ G $ is a polymer graph, obtained by point-attaching from monomer units $ G_1,...,G_k $. In this paper, we study the pebbling number of some polymers.

Figures (9)

• Figure 1: The friendship graphs $F_{2,3}$ and $F_{4,3}$
• Figure 2: The friendship graphs $F_{2,4}$, $F_{3,4}$ and $F_{4,4}$
• Figure 3: (a) An unsolvable configuration $f$ of size $19$ and a spanning tree of graph $T_{4}$, (b) Graph $T_{4}+e$ and an its spanning tree, (c) Graph $T_{4}+e'$ and an its spanning tree.
• Figure 4: The tree constraints of $Q_{3}$.
• Figure 5: The graphs $Q_{4}+e$.

Theorems & Definitions (28)

• Theorem 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• Theorem 2.6
• proof
• Theorem 3.1
• Corollary 3.2