這篇論文討論圖的標示對局與著色對局。假設G是一個圖。圖G的標示對局是一個二人對局。最初的時候，圖上的頂點都是沒有標示的。甲乙二人輪流在圖上標示尚未標示的頂點。甲先行。當所有的頂點都標示了，對局就結束了。對所有的頂點v，s(v)記為在v著色之前v已經被標示的鄰居的個數。定義s是對局的價值為$s= 1+ max_{v in V}s(v)$。對局的價值為乙的獲利，也是甲的付出。故甲的目標是使對局的價值最小，乙的目標是使對局的價值最大。圖G的對局著色數colg(G)，是指最小的s使得甲在圖G上對局時，有一個策略保證對局的價值不超過s。
假設整數r大於等於1，d大於等於零。圖G的(r, d)-鬆弛著色對局，也是一個二人對局。甲乙二人輪流在圖G上的頂點著色。甲先行。顏色是從一共有r個顏色的顏色集X中取出。我們說顏色i對一個還沒著色的頂點x是合法的顏色，是指當點x著了顏色i之後，由所有著i的點所生成的子圖中，任一頂點最多有d個鄰居。甲乙二人均只能用合法的顏色著尚未著色的頂點。甲的目標是讓圖上所有的頂點都著色，而乙的目標則是相反：迫使一個沒著色的頂點沒有合法的顏色可著。圖G的d-鬆弛對局色數記作$chi_g^{(d)}(G)$，是指最少的顏色r使得當甲在圖G的(r, d)-鬆弛著色對局中，有一個贏的策略。當d等於零時，這個值叫做圖G的對局色數，又記作$chi_g(G)$。

這篇論文求出一些種類的圖的對局著色數和鬆弛對局色數的上界和下界。我們證明了局部k樹的最大的對局著色數恰好就等於3k+2。平面圖的最大的對局著色數大於等於11。對於鬆弛對局色數，這篇論文證明了如果$G$是一個外平面圖，則當t= 2, 3, 4且當 d geq t時，$chi_g^{(d)}(G) leq 7-t$。當 $d geq 6$時，$chi_g^{(d)}(G) leq 2$ 。特別地，平面圖的最大4-鬆弛對局色數恰好等於3。如果G 是一個樹，則當 $d geq 2$時，$chi_g^d(G) leq 2$。

This thesis discusses graph marking game and graph colouring game.

Suppose G=(V, E) is a graph. A marking game on G is played by two players, Alice and Bob, with Alice playing first. At the start of the game all vertices are unmarked. A play by
either player consists of marking an unmarked vertex. The game ends when all vertices are marked. For each vertex v of G, write t(v)=j if v is marked at the jth step. Let s(v)
denote the number of neighbours u of v for which t(u) < t(v), i.e., u is marked before v. The score of the game is $$s = 1+ max_{v in V} s(v).$$ Alice's goal is to minimize the score, while Bob's goal is to maximize it. The game colouring number colg(G) of G is the least s such that Alice has a strategy that results in a score at most s. Suppose r geq 1, d geq 0 are integers. In an (r, d)-relaxed colouring game of G, two players, Alice and Bob, take turns colouring the vertices of G with colours from a set X of r colours, with Alice having the first move. A colour i is legal for an uncoloured vertex x (at a certain step) if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Each player can only colour an uncoloured vertex with a legal colour. Alice's goal is to have all the vertices coloured, and Bob's goal is the opposite: to have an uncoloured vertex without legal colour. The d-relaxed game chromatic number of a graph G, denoted by $chi_g^{(d)}(G)$ is the least number r so that when playing the (r, d)-relaxed colouring game on G, Alice has a winning strategy. If d=0, then the parameter is called the game chromatic number of G and is also denoted by $chi_g(G)$. This thesis obtains upper and lower bounds for the game colouring
number and relaxed game chromatic number of various classes of graphs. Let colg(PT_k) and colg(P) denote the maximum game colouring number of partial k trees and the maximum game colouring number of planar graphs, respectively. In this thesis, we prove that colg(PT_k) = 3k+2 and colg(P) geq 11. We also prove that the game colouring number colg(G) of a graph is a monotone parameter, i.e., if H is a subgraph of G, then colg(H) leq colg(G). For relaxed game chromatic number of graphs, this thesis proves that if G is an outerplanar graph, then $chi_g^{(d)}(G) leq 7-t$ for $t= 2, 3, 4$, for $d geq t$, and $chi_g^{(d)}(G) leq 2$ for $d geq 6$. In particular, the maximum $4$-relaxed game chromatic number of outerplanar graphs is equal to $3$. If $G$ is a tree then $chi_ g^{(d)}(G) leq 2$ for $d geq 2$.

1 Introduction 3
1.1 Some basic notation . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Game chromatic number . . . . . . . . . . . . . . . . . . . . . 6
1.3 Game colouring number . . . . . . . . . . . . . . . . . . . . . 7
1.4 Relaxed game chromatic number . . . . . . . . . . . . . . . . 9
1.5 Game chromatic number of oriented graph . . . . . . . . . . . 10
1.6 Review of results of this thesis and related known results . . . 11
2 Game colouring number of partial k-trees and planar graphs 16
2.1 Partial k-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Relaxed colouring game on forests 24
3.1 Relaxed colouring game on forests with at least 3 colours . . . 24
3.2 Relaxed colouring game on forests with 2 colours . . . . . . . 26
4 Relaxed colouring game on outerplanar graph with at least
3 colours 33
4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Strategy A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 (7, 0)-relaxed colouring game and (6, 1)-relaxed colouring
game on outerplanar graphs . . . . . . . . . . . . . . . . . . . 37
4.4 (5, 2)-relaxed colouring game on outerplanar graphs . . . . . . 38
4.5 (3, 4)-relaxed colouring game and (4, 3)-relaxed colouring game on outerplanar graphs . . . . . . . . . . . . . . . . . . . 40
5 Relaxed colouring game on outerplanar graphs with 2 colours 44
5.1 Strategy B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 (2, 6)-relaxed colouring game on outerplanar graphs . . . . . . 46
5.3 (2, 4)-relaxed colouring game on outerplanar graphs . . . . . . 53

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