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作者:黃皜文
作者(英文):Huang, Hau-wen
論文名稱(中文):從代數觀點研究亮點西格瑪遊戲
論文名稱(英文):Lit-only sigma-game from the view of algebra
指導教授(中文):翁志文
指導教授(英文):Weng, Chih-wen
學位類別:博士
校院名稱:國立交通大學
系所名稱:應用數學系所
學號:9422537
出版年(民國):100
畢業學年度:99
語文別:中文
論文頁數:63
中文關鍵詞:亮點西格瑪遊戲
外文關鍵詞:Lit-only sigma-game
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亮點西格瑪遊戲是一個在有限簡圖上的單人益智遊戲。已知亮點西格瑪遊戲可視為群作用。在這篇論文裡,我們展示此遊戲和考斯特群的關係。我們並由代數的技巧推廣一些此遊戲已知的成果。
The lit-only σ-game is a one-player game played on a finite simple graph. It is known that this game can be view as a group action. In this thesis we show how this game is related to Coxeter groups. Moreover we use algebraic techniques to generalize some known results on the game.
1 Introduction 1
2 Lit-only sigma-game and simply-laced Coxeter groups 3
2.1 The flipping group of a graph 3
2.2 A representation of the Coxeter group of type Γ 5
2.3 The center of the flipping group W of type Γ 6
2.4 Lit-only σ-game on the Dynkin diagram of type An 7
2.5 Lit-only σ-game on the Dynkin diagram of type Dn 9
2.6 Lit-only σ-game on Γ and its induced subgraph 11
2.7 Lit-only σ-game on the Dynkin diagram of type En 12
2.8 Summary 16
3 Lit-only sigma-game on a graph with a long induced path 19
3.1 The sets Π, Π0 and Π1 20
3.2 The simple basis Δ of FS2 22
3.3 The case |Π1| is odd 22
3.4 The case |Π1| is even 25
3.5 Summary 28
3.6 Remarks 29
4 One-lit trees for lit-only sigma-game 31
4.1 The degenerate and nondegenerate graphs 31
4.2 Some combinatorial properties of nondegenerate trees 32
4.3 The Reeder’s game 32
4.4 Reeder’s game on a nondegenerate tree 34
4.5 Lit-only σ-game on a nondegenerate tree 36
4.6 A homomorphism between simply-laced Coxeter groups 38
4.7 More one-lit trees for lit-only σ-game 40
4.8 Combinatorial statements of Theorems 4.5.7 and 4.7.7 43
5 The edge-version of lit-only sigma-game 45
5.1 The edge space and the bond space 45
5.2 The edge-flipping group of Γ 46
5.3 The structure of WR in the case Γ is a tree 47
5.4 The WR-orbits of R 48
5.5 The minimum light number for e-lit-only σ-game on Γ 50
5.6 The structure of WR 51
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