Capacity restricted optimal pebbling in graphs

Abstract

A distribution δ of pebbles on a graph G is a mapping of V(G) onto the set N∪{0}. For each vertex v∈V(G), δ(v) denotes the number of pebbles distributed to v. A pebbling move consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex. A distribution of a given graph G is t-solvable if, whenever we choose any target vertex v of G, we can move t pebbles on v by using pebbling moves. The optimal t-pebbling number of the graph G, denoted by πt∗(G), is the minimum number of pebbles needed so that there is a t-solvable distribution of G. A distribution δ of G is called a c-restricted pebbling configuration (abbreviated c-RPC) if δ(v)≤c for each v∈V(G). The c-restricted optimal t-pebbling number, denoted by πtc(G), is the minimum number of pebbles needed so that there is a t-solvable c-RPC of G. When t=1, we use πc(G) to denote π1c(G). In the article titled “Restricted optimal pebbling and domination in graphs”, Chellali et al. showed that π2(T)≤⌈5n∕7⌉ and posed an open problem that characterizes the graphs G for which π2(G)=5. In this paper, we first prove that π2(T)≤⌈2n∕3⌉ for every tree T of order n. Then, we characterize the graphs G for which π2(G)=5.