The optimal t-pebbling number of a certain complete m-ary tree

Abstract

A distribution of a given graph G is t-fold 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 $$\pi ^*_{t}(G)$$ , is the minimum number of pebbles necessary so that there is a t-fold solvable distribution of G. Let $$T^m_h$$ denote the complete m-ary tree with height h. In this paper, we first determine that $$\begin{aligned} \pi ^{*}_{t}(T^m_{h})=t\cdot 2^{h}\quad (m\ge 3; \; t\ge 1). \end{aligned}$$ We then show that $$\begin{aligned} \pi ^{*}_{2}(T^2_{h})=\pi ^{*}(T^2_{h+1})\quad \text {and} \quad \pi ^{*}_{4}(T^2_{h})=\pi ^{*}(T^2_{h+2}). \end{aligned}$$ Finally, we determine that $$\begin{aligned} \pi ^{*}_{3}(T^2_{h})=2^{h+1}+1. \end{aligned}$$