University of IsfahanTransactions on Combinatorics2251-865711420221201Bounds for the pebbling number of product graphs3173262599710.22108/toc.2021.128705.1855ENNopparatPleanmaniDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand0000-0002-1889-3826NuttawootNupoDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandSomnuekWorawisetDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandJournal Article20210515Let $G$ be a connected graph. Given a configuration of a fixed number of pebbles on the vertex set of $G$, a pebbling move on $G$ is the process of removing two pebbles from a vertex and adding one pebble on an adjacent vertex. The pebbling number of $G$, denoted by $\pi(G)$, is defined to be the least number of pebbles to guarantee that there is a sequence of pebbling movement that places at least one pebble on each vertex $v$, for any configuration of pebbles on $G$. In this paper, we improve the upper bound of $\pi(G\square H)$ from $2\pi(G)\pi(H)$ to $\left(2-\frac{1}{\min\{\pi(G),\pi(H)\}}\right)\pi(G)\pi(H)$ where $\pi(G)$, $\pi(H)$ and $\pi(G\square H)$ are the pebbling number of graphs $G$, $H$ and the Cartesian product graph $G\square H$, respectively. Moreover, we also discuss such bound for strong product graphs, cross product graphs and coronas.https://toc.ui.ac.ir/article_25997_501663d1ee8baceb41a23ea159ff00d0.pdf