The cover pebbling number of the join of some graphs

Abstract

Given a graph G and a configuration C of pebbles on the vertices of G, a pebbling step or move [u, v] consists of removing two pebbles off of one vertex u, and then placing one pebble on an adjacent vertex v. In a pebbling step [u, v], u is the support vertex while v is a target vertex. A graph is said to be cover-pebbled if every vertex has a pebble on it after a series of pebbling steps. The cover pebbling number γ(G) of a graph G is the minimum number of pebbles such that however the pebbles are initially placed on the vertices of G we can eventually put a pebble on every vertex simultaneously by a pebbling step. In this paper, the cover pebbling number of graphs resulting from the join of two graphs G and H are determined via a key vertex of the graph. In particular, this paper determines the cover pebbling number of the wheels W n , the fans F n , and the join of any graph G with P n and C n , respectively