本論文探討了有關圖形的著色、圓環列表著色以及相適對局著色。圖形的著色上，本論文提供了一個充分條件，使得一個平面圖可以由三個顏色給予著色。假設G是一個平面圖。令H_G為另一個相對於G的圖，其中H_G的頂點集合為V (H_G) ={C : C是圖形G的圈且長度為4或6或7}，而邊集合為E(H_G) = {C_iC_j ： 若C_i與C_j在圖G內有共同的邊}。我們證明了，對於一個平面圖G，若所有長度為3的圈和長度為5的圈皆不與長度為3至7的圈含有共同的邊，且圖形H_G是一個森林(不包含任何的圈)，則圖形G是可以由三個顏色給予著色。
對於圓環列表著色數，針對任意的有限圖G，我們考慮其列表僅為一連續區段，並將之稱為圓環連續列表著色數。本論文探討了圓環連續列表著色數、一般著色數以及圓環著色數的基本關係。對於任意的有限圖G，其關係為：一般著色數-1 &#8804; 圓環連續列表著色數 < 2倍的圓環著色數。我們同時判定了完全圖、樹、圈、平衡完全二部圖以及某些完全多部圖的圓環連續列表著色數。此外，對於某些特殊類型圖，我們也提供了其圓環連續列表著色數的上界以及下界。
在相適對局著色數的議題上, 本論文證明了樹的最大相適對局著色數為3；外平面圖的最大相適對局著色數為5；局部k樹的最大相適對局著色數介在k+2至2k+1之間；平面圖的最大相適對局著色數介在6至11之間。對於一些特定類型的Cartesian乘積圖，例如：有關局部k樹、外平面圖以及平面圖的Cartesian乘積圖，我們也提供了其相適對局著色數的上界。

This thesis discusses colouring, circular list colouring and adapted game colouring of graphs. For colouring, this thesis obtains a sufficient condition for a planar graph to be 3-colourable. Suppose G is a planar graph. Let H_G be the graph with vertex set V (H_G) = {C : C is a cycle of G with |C| &#8712; {4, 6, 7}} and edge set E(H_G) = {CiCj : Ci and Cj have edges in common}. We prove that if any 3-cycles and 5-cycles are not adjacent to i-cycles for 3 &#8804; i &#8804; 7, and H_G is a forest, then G is 3-colourable.
For circular consecutive choosability, this thesis obtains a basic relation among chcc(G), X(G) and Xc(G) for any finite graph G. We show that for any
finite graph G, X(G) &#8722; 1 &#8804; chcc(G) < 2 Xc(G). We also determine the value of chcc(G) for complete graphs, trees, cycles, balanced complete bipartite graphs and some complete multi-partite graphs. Upper and lower bounds for chcc(G) are given for some other classes of graphs.
For adapted game chromatic number, this thesis studies the adapted game chromatic number of various classes of graphs. We prove that the maximum
adapted game chromatic number of trees is 3; the maximum adapted game chromatic number of outerplanar graphs is 5; the maximum adapted game
chromatic number of partial k-trees is between k + 2 and 2k + 1; and the
maximum adapted game chromatic number of planar graphs is between 6 and 11. We also give upper bounds for the Cartesian product of special classes of graphs, such as the Cartesian product of partial k-trees and outerplanar graphs, or planar graphs.

1 Introduction 1
1.1 Some basic notations . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 3-colourability of planar graphs . . . . . . . . . . . . . . . . . 4
1.3 Circular chromatic number . . . . . . . . . . . . . . . . . . . . 5
1.4 Circular consecutive choosability . . . . . . . . . . . . . . . . 6
1.5 Adapted chromatic number . . . . . . . . . . . . . . . . . . . 8
1.6 Game chromatic number . . . . . . . . . . . . . . . . . . . . . 9
1.7 Adapted game chromatic number . . . . . . . . . . . . . . . . 10
1.8 Review of results of this thesis . . . . . . . . . . . . . . . . . . 11

2 Cycle adjacency of planar graphs and 3-colourability 14
2.1 3-colour problem and known results . . . . . . . . . . . . . . . 14
2.2 Configuration on HG and some notations . . . . . . . . . . . . 19
2.3 Proof of Theorem 2.2.1 . . . . . . . . . . . . . . . . . . . . . . 22
3 Circular consecutive choosability of graphs 28
3.1 Circular chromatic number and circular choosability . . . . . . 28
3.2 Circular consecutive choosability . . . . . . . . . . . . . . . . 32
3.3 Equivalent definitions . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Some general bounds on chcc(G) . . . . . . . . . . . . . . . . . 39
3.5 Trees, cycles and complete graphs . . . . . . . . . . . . . . . . 41
3.6 Bounds on chcc(G) for special graphs . . . . . . . . . . . . . . 44

4 Adapted game colouring of graphs 50
4.1 Basic definitions and related concepts . . . . . . . . . . . . . . 50
4.2 Partial k-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Cartesian product graphs . . . . . . . . . . . . . . . . . . . . . 70

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