論文摘要內容：

本論文在於探討哈林圖的對局色數. 我們將證明除了兩個哈林圖以外的其他哈林圖的對局色數都等於4. 這個遊戲是有兩個人在玩, 分別是愛麗絲和鮑伯, 給一個圖和顏色集, 由愛麗絲和鮑伯從顏色集裡找一個顏色在圖上的頂點著色, 使得相鄰兩點不可著同一個顏色. 順序由愛麗絲先動, 再換鮑伯, 輪流在圖上著色. 一直玩到最後, 如果每一個點都被著色了, 則為愛麗絲臝. 反之, 則為鮑伯贏.
對局色數為給最少的顏色使得愛麗絲又能獲勝的顏色數.
哈林圖為一種特別的平面圖, 包含了一個樹和一個圈,
其中圈上的點都是頂點都是樹上的葉, 並且除了葉以外,
其餘樹上的頂點的度都大於或等於3.

This thesis discusses the game chromatic number of Halin graphs. We shall
prove that with a few exceptions, all Halin graphs have game chromatic
number 4.

1 Introduction 1
1.1 Definition of ˜g and colg . . . . . . . . . . . . . . . . . . . . . 1
1.2 Halin graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Proof of the main result 6
2.1 Game coloring number and proof of the upper bound . . . . . 6
2.2 Some preliminary lemmas . . . . . . . . . . . . . . . . . . . . 8
2.3 First colored vertex lies on C . . . . . . . . . . . . . . . . . . 12
2.4 First colored vertex does not lie on C . . . . . . . . . . . . . . 19

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