當我們給定了一張圖G = (V, E), 由兩位玩家Alice和Bob輪流選擇一條尚未著色的邊進行著色。每當玩家選定一條尚未著色的邊時,僅能使用不會產生單色圈的最小編號顏色來著它。而這場對局進行時,Alice的目標是希望在對局結束後，所使用的顏色數越少越好;但是Bob的目標是希望在對局結束後,所使用的顏色數越多越好。
我們將圖G的對局格蘭迪蔭度定義為,當兩位玩家皆使用最佳策略 時,遊戲結束後所使用的總顏色數。
我們證明了,若是圖G的蔭度為k時,則該圖的對局格蘭迪蔭度最多為3k − 1。我們也證明了,若是圖G有一個最大出度至多為k的無圈定向時, 則該圖的對局 格蘭迪蔭度最多為3k − 2。

Given a graph G = (V, E), two players, Alice and Bob, alternate their turns to choose uncolored edges to be colored. Whenever an uncolored edge is chosen, it is colored by the least positive integer so that no monochromatic cycle is created. Alice’s goal is to minimize the total number of colors used in the game, while Bob’s goal is to maximize it. The game Grundy arboricity of G is the number of colors used in the game when both players use optimal strategies. This thesis discusses the game Grundy arboricity of graphs. It is proved that if a graph G has arboricity k, then the game Grundy arboricity of G is at most 3k − 1. If a graph G has an acyclic orientation D with maximum out-degree at most k, then the game Grundy arboricity of G is at most 3k − 2.

致謝 i
摘要 ii
Abstract iii
1 Introduction 1
1.1 Somebasicnotation............................. 1
1.2 Coloringgames............................... 4
1.3 Arboricityandgamearboricity....................... 8
1.4 GameGrundyarboricity .......................... 17
1.5 Resultsofthisthesis ............................ 18
2 Game Grundy arboricity 19
2.1 SomeresultsongameGrundyarboricity.................. 19
2.2 AnupperboundforΛg(G)intermofA(G) ................ 21
2.3 UpperboundforΛg(G)onacyclicorientation . . . . . . . . . . . . . . . 24
2.4 GameGrundyarboricityofwheels..................... 25
2.5 GameGrundyarboricityoffans ...................... 29
3 Conclusions 31
Bibliography 32

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