The 2-pebbling property of squares of paths and Graham’s conjecture

A pebbling move on a graph G consists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graph G, denoted by f(G), is the least n such that any distribution of n pebbles on G allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we determine the 2-pebbling property of squares of paths and Graham’s conjecture on $\begin{array}{} P_{2n}^2 \end{array} $.

Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

A graph pebbling is a network optimization model for the transmission of consumable resources. A pebbling move on a connected graph [Formula: see text] is the process of removing two pebbles from a vertex and placing one of them on an adjacent vertex after configuration of a fixed number of pebbles on the vertex set of [Formula: see text]. The pebbling number of [Formula: see text], denoted by [Formula: see text], is defined to be the least number of pebbles to guarantee that for any configuration of pebbles on [Formula: see text] and arbitrary vertex [Formula: see text], there is a sequence of pebbling movement that places at least one pebble on [Formula: see text]. For connected graphs [Formula: see text] and [Formula: see text], Graham’s conjecture asserted that [Formula: see text]. In this paper, we show that such conjecture holds when [Formula: see text] is a complete bipartite graph with sufficiently large order in terms of [Formula: see text] and the order of [Formula: see text].

A Generalization of Graham’s Conjecture

Let $G$ be a finite Abelian group of order $|G|=n$, and let $S=g_1\cdot\ldots\cdot g_{n-1}$ be a sequence over $G$ such that all nonempty zero-sum subsequences of $S$ have the same length. In this paper, we completely determine the structure of these sequences.