Optimally t-Pebbling Graphs
博士 === 中原大學 === 應用數學研究所 === 106 === Let $G$ be a simple graph. A distribution $delta$ of $G$ is a mapping from $V(G)$ into the set of non-negative integers, where $delta(v)$ is the number of pebbles distributed to the vertex $v$ for each $vin V(G)$. A {it pebbling move} consists of removing two peb...
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ndltd-TW-106CYCU55070272019-09-26T03:28:08Z http://ndltd.ncl.edu.tw/handle/3tnr33 Optimally t-Pebbling Graphs 最優t-圓石分配圖 Hung-Hsing Chiang 姜宏興 博士 中原大學 應用數學研究所 106 Let $G$ be a simple graph. A distribution $delta$ of $G$ is a mapping from $V(G)$ into the set of non-negative integers, where $delta(v)$ is the number of pebbles distributed to the vertex $v$ for each $vin V(G)$. A {it 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 $t$-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 $G$. When $t=1$, the optimal $t$-pebbling number is the optimal pebbling number. In this thesis, we mainly study the optimal $t$-pebbling number of a graph. In Chapter $1$, we introduce the background of this study and give some basic definitions. For completeness, we also introduce the known results on the optimal pebbling number and the optimal $t$-pebbling number of a graph in Chapter $2$. In Chapter $3$, we study the optimal $t$-pebbling number of the cycle. As a consequence, we determine the exact value of $pi^*_{2}(C_{n})$ for each cycle $C_n$. We then derive an upper bound and a lower bound for $pi^*_{t}(C_{n})$ for $tgeq 3$. In Chapter $4$, let $T^m_h$ denote the complete $m$-ary tree with height $h$, we first show that $$pi^{ast}_{2}(T^2_{h})=pi^{ast}(T^2_{h+1})qquad ext{and} qquad pi^{ast}_{4}(T^2_{h})=pi^{ast}(T^2_{h+2})$$ and then determine the exact value of $pi^{ast}_{3}(T^2_{h})$. In Chapter $5$, we conclude our research results in this thesis. Mu-Ming Wong Chin-Lin Shiue 王牧民 史青林 2018 學位論文 ; thesis 60 en_US |
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博士 === 中原大學 === 應用數學研究所 === 106 === Let $G$ be a simple graph. A distribution $delta$ of $G$ is a mapping from $V(G)$ into the set of non-negative integers, where $delta(v)$ is the number of pebbles distributed to the vertex $v$ for each $vin V(G)$. A {it 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 $t$-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 $G$. When $t=1$, the optimal $t$-pebbling number is the optimal pebbling number.
In this thesis, we mainly study the optimal $t$-pebbling number of a graph. In Chapter $1$, we introduce the background of this study and give some basic definitions. For completeness, we also introduce the known results on the optimal pebbling number and the optimal $t$-pebbling number of a graph in Chapter $2$. In Chapter $3$, we study the optimal $t$-pebbling number of the cycle. As a consequence, we determine the exact value of $pi^*_{2}(C_{n})$ for each cycle $C_n$. We then derive an upper bound and a lower bound for $pi^*_{t}(C_{n})$ for $tgeq 3$. In Chapter $4$, let $T^m_h$ denote the complete $m$-ary tree with height $h$, we first show that $$pi^{ast}_{2}(T^2_{h})=pi^{ast}(T^2_{h+1})qquad ext{and} qquad
pi^{ast}_{4}(T^2_{h})=pi^{ast}(T^2_{h+2})$$ and then determine the exact value of $pi^{ast}_{3}(T^2_{h})$. In Chapter $5$, we conclude our research results in this thesis.
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author2 |
Mu-Ming Wong |
author_facet |
Mu-Ming Wong Hung-Hsing Chiang 姜宏興 |
author |
Hung-Hsing Chiang 姜宏興 |
spellingShingle |
Hung-Hsing Chiang 姜宏興 Optimally t-Pebbling Graphs |
author_sort |
Hung-Hsing Chiang |
title |
Optimally t-Pebbling Graphs |
title_short |
Optimally t-Pebbling Graphs |
title_full |
Optimally t-Pebbling Graphs |
title_fullStr |
Optimally t-Pebbling Graphs |
title_full_unstemmed |
Optimally t-Pebbling Graphs |
title_sort |
optimally t-pebbling graphs |
publishDate |
2018 |
url |
http://ndltd.ncl.edu.tw/handle/3tnr33 |
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