The Optimal Pebbling Number of Square of Paths and Cycles

Abstract

A pebbling move involves removing two pebbles from one vertex and placing one on an adjacent vertex. The optimal pebbling number of a graph \(G\), denoted by \(f_{opt}(G)\), is the least positive integer \(n\) such that \(n\) pebbles are placed suitably on vertices of \(G\) and, for any specified vertex \(v\) of \(G\), one pebble can be moved to \(v\) through a sequence of pebbling moves. In this paper, we determine the optimal pebbling number of the square of paths and cycles.