The topic of this treatise is a combinatorial technique called Graph Pebbling. We investigate pebbling numbers, weight functions, flow networks, hypercubes, and the zero-sum conjecture of Erdős and Lemke. This investigation is a comprehensive starting point for anyone who wants to study graph pebbling for the first time, and also includes some new ideas and results for those who have studied it before.
For an introductory video, see: https://www.youtube.com/watch?v=yPzyQ4M1f3c
Graph pebbling models the transportation of consumable resources. As two pebbles move across an edge, one reaches its destination while the other is consumed. The $t$-pebbling number is the smallest integer $m$ so that any initially distributed supply of $m$ pebbles can place $t$ pebbles on any target vertex via pebbling moves. The 1-pebbling number of diameter two graphs is well-studied. Here we investigate the $t$-pebbling number of diameter two graphs under the lense of connectivity.
A new graph invariant called the secure vertex cover pebbling number, which is a combination of two graph invariants, namely secure vertex cover and cover pebbling number, is introduced in this paper. The secure vertex cover pebbling number of a graph G is the minimum number m so that every distribution of m pebbles can reach some secure vertex cover of G by a sequence of pebbling moves. In this paper, the complexity of the secure vertex cover problem and secure vertex cover pebbling problem are discussed. Also, we obtain some basic results and the secure vertex cover pebbling number for compl…
A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The goal of graph pebbling is: Given an initial distribution of pebbles, use pebbling moves to reach a specified goal vertex called the root. The pebbling number of a graph $π(G)$ is the minimum number of pebbles needed so every distribution of $π(G)$ pebbles can reach every choice of the root. We introduce a new variant of graph pebbling, a game between two players. One player aims to move a pebble to the root and the other player aims to prevent this. We show configurations…
Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number $π^*(G)$ is the smallest number of pebbles which we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most $\frac{4n}{δ+1}$, where $δ$ is the minimum degree of the graph. We strengthen this bound by showing that …
Pebbling on graphs is a two-player game which involves repeatedly moving a pebble from one vertex to another by removing another pebble from the first vertex. The pebbling number $π(G)$ is the least number of pebbles required so that, regardless of the initial configuration of pebbles, a pebble can reach any vertex. Graham conjectured that the pebbling number for the cartesian product, $G \hspace{1mm}\square\hspace{1mm} H$, is bounded above by $π(G) π(H)$. We show that $π(G\hspace{1mm}\square\hspace{1mm} H) \le 2π(G) π(H)$ and, more sharply, that $π(G \hspace{1mm}\square\hspace{1mm} H) \le (π(…
Graph pebbling is a network optimization model for satisfying vertex demands with vertex supplies (called pebbles), with partial loss of pebbles in transit. The pebbling number of a demand in a graph is the smallest number for which every placement of that many supply pebbles satisfies the demand. The Target Conjecture (Herscovici-Hester-Hurlbert, 2009) posits that the largest pebbling number of a demand of fixed size $t$ occurs when the demand is entirely stacked on one vertex. This truth of this conjecture could be useful for attacking many open problems in graph pebbling, including the famo…
A pebbling move on a graph consists of taking two pebbles off from one vertex and add one pebble on an adjacent vertex, the $t$-pebbling number of a graph $G$ is the minimum number of pebbles so that we can move $t$ pebbles on any vertex on $G$ regardless the original distribution of pebbles. Let $ω$ be a positive function on $V(G)$, the $ω$-cover pebbling number of a graph $G$ is the minimum number of pebbles so that we can reach a distribution with at least $ω(v)$ pebbles on $v$ for all $v\in V(G)$. In this paper, we give the $ω$-cover pebbling number of trees for nonnegative function $ω$, w…
Given a distribution of pebbles on the vertices of a graph, say that we can pebble a vertex if a pebble is left on it after some sequence of moves, each of which takes two pebbles from some vertex and places one on an adjacent vertex. A distribution is solvable if all vertices are pebblable; the pebbling threshold of a sequence of graphs is, roughly speaking, the total number of pebbles for which random distributions with that number of pebbles on a graph in the sequence change from being almost never solvable to being almost always solvable. We show that any sequence of connected graphs with …
In this paper, we extend the ideas of graph pebbling to oriented graphs and find a classification for all graphs with fully traversable pebbling assignments that are isomorphic to their assignment graph. We then give some cases in which a graph with a non-fully traversable pebbling assignment is isomorphic to its assignment graph.
On the Relative Strength of Pebbling and Resolution
在过去的十年里，在证明复杂性的背景下，人们对卵石博弈的兴趣又恢复了。Pebbling已经被证明是一个有用的工具，用于研究基于分辨率的证明系统，比较不同子系统的强度，显示证明空间的界限，并建立大小空间的权衡。典型的方法是将图上的卵石博弈编码为CNF公式，然后认为这个公式的证明必须继承底层图的卵石性质（的各个方面）。不幸的是，这里使用的缩减并不严格。要通过pebbling模拟分辨率证明，需要非确定性黑白pebbling的全部强度，而分辨率仅已知能够模拟确定性黑色pebbling。因此，为了得到强有力的结果，需要找到特定的图族，这些图族要么对黑色和黑白卵石具有基本相同的性质（一般来说根本不成立），要么在分辨率上允许模拟黑白卵石。本文有助于这两种方法。首先，我们设计了一个限制形式的黑白卵石，可以模拟的分辨率，并表明，有图形的家庭，这种限制卵石可以渐近优于黑色卵石。这证明了，也许有点出乎意料，分辨率可以严格击败黑色的鹅卵石，特别是在[Ben-Sasson和Nordstrom 2008]中鹅卵石公式的空间下限是紧的。其次，我们提出了一个多功能的参数化图形家庭基本上相同的属性为黑色和黑白卵石，这使得尖锐的同时权衡黑色和黑白卵石的各种参数设置。我们的两个贡献都有助于获得基于分辨率的证明系统的时空权衡结果[Ben-Sasson and Nordstrom 2009]。
Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placing one at an adjacent vertex to eventually have one pebble reach a predetermined vertex. The pebbling number, $π(G)$, is the minimum number of pebbles such that regardless of their exact configuration, the player can use pebbling moves to have a pebble reach any predetermined vertex. Previous work has related $π(G)$ to the diameter of $G$. Clarke, Hochberg, and Hurlbert demonstrated that every connected undirected gra…
Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling number $π(G)$ is the smallest number such that, for any distribution of $π(G)$ pebbles to the vertices of $G$ and choice of root vertex $r$ of $G$, there exists a sequence of pebbling moves that places a pebble on $r$. Computing $π(G)$ is provably difficult, and recent methods for bounding $π(G)$ have proved computationally intractable, even for moderately sized graphs. Graham conjectured that $π(G ~\square~ H) \leq …
A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number $π_{opt}$ is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph $P_n\square P_m$ was investigated in several papers. In this paper, we present a new method using some recent ideas to give a lower bound on $π_{opt}$. We apply this te…
