一步圓石移動(pebbling move)定義為將兩個圓石從一個點上拿起，丟掉一個，並將另一個放到相鄰的點上。一個圖G的配置或分配(configuration or distribution) C是一個從V(G)映至非負整數的函數。我們讓C(v)是點v所被分配的圓石數，所以G上的圓石總數為v∈V(G)C(v)。若一個配置C讓我們藉由重覆的(若有需要)圓石移動移動至少一個圓石到任意一點上，則C被稱為一個G的圓石分配(pebbling)。G的最優圓石分配數(optimal pebbling number) f’(G)是一個使用最少圓石的G的圓石分配之圓石總數。
　　有許多方法能處理這個問題，例如機率[6]、錯誤更正碼[9]或特殊類型的控制集[6]。此篇論文主要關心兩個圖相乘的最優圓石分配數。我們得到了f’(P3 × Pn)、f’(Q3) = f’(P2 × Q2) = f’(P2 × C4)、f’(Q4) = f’(P2 × Q3)和f’(Q5) = f’(P2 × Q4)的值。對於一般化的圖，給了一些上界。我們也提供了不同的方法證明f’(Pn)及f’(P2 × Pn)的值。

Abstract (in Chinese) . . . . . . . . . . . i

Abstract (in English) . . . . . . . . . . . ii

Acknowledgement . . . . . . . . . . . . . . iii

Contents . . . . . . .. . . . . . . . . . . iv

1 Introduction and Preliminaries 1

1.1 Basic notations . . . . . . . . . . . . 2

1.2 Preliminaries . . . . . . . . . . . . . 3

2 Known Results and Conjectures 6

2.1 Known results . . . . . . . . . . . . . 6

2.2 Conjectures . . . . . . . . . . . . . . 13

3 Main Results 14

3.1 On hypercubes . . . . . . . . . . . . . 14

3.2 On Pm x Pn . . . . . . . . . . . . . . 16

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