Lollipop and cubic weight functions for graph pebbling

Abstract Given a configuration of pebbles on the vertices of a graph G , a pebbling move removes two pebbles from a vertex and puts one pebble on an adjacent vertex. The pebbling number of a graph G is the smallest number of pebbles required such that, given an arbitrary initial configuration of pebbles, one pebble can be moved to any vertex of G through some sequence of pebbling moves. Through constructing a non-tree weight function for $$Q_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> , we improve the weight function technique, introduced by Hurlbert and extended by Cranston et al., that gives an upper bound for the pebbling number of graphs. Then, we propose a conjecture on weight functions for the n -dimensional cube. We also construct a set of valid weight functions for variations of lollipop graphs, extending previously known constructions.