Graham's Pebbling Conjecture Holds for the Product of a Graph and a Sufficiently Large Complete Graph

Nopparat Pleanmani

doi:10.20429/tag.2020.070101

Abstract

For connected graphs $G$ and $H$, Graham conjectured that $\pi(G\square H)\leq\pi(G)\pi(H)$ where $\pi(G), \pi(H)$, and $\pi(G\square H)$ are the pebbling numbers of $G$, $H$, and the Cartesian product $G\square H$, respectively. In this paper, we show that the inequality holds when $H$ is a complete graph of sufficiently large order in terms of graph parameters of $G$.

Highlights

Throughout this paper, all graphs are considered to be finite and simple
For connected graphs G and H, Graham conjectured that π(G H) ≤ π(G)π(H) where π(G), π(H), and π(G H) are the pebbling numbers of G, H, and the Cartesian product G H, respectively
We show that the inequality holds when H is a complete graph of sufficiently large order in terms of graph parameters of G

Summary

Introduction

Throughout this paper, all graphs are considered to be finite and simple. For a graph G, we denote the order of G by |G|. For connected graphs G and H, Graham conjectured that π(G H) ≤ π(G)π(H) where π(G), π(H), and π(G H) are the pebbling numbers of G, H, and the Cartesian product G H, respectively. For a positive integer n, we denote Kn to be a complete graph of n vertices. For a moveable configuration D on a graph G and adjacent vertices u and v with D(v) ≥ 2, the (pebbling) move from u to v in G is defined to be the triple (D, u, v) and we denote it by D(u → v) for convenience.

Results

Conclusion