本篇論文討論了五種圖形上的著色：圖的圓環染色、圖的全染色、圖的(d,1)-全標號、圖的圓環(r,1)-全標號及無二色圈列表染色。

對於最大平均度較小的圖形的圓環染色數，我們給了一些上界。
已經被證實的有：對於最大平均度小於22/9的圖，此類圖有一個11/4-圓環染色；
對於最大平均度小於5/2的圖，此類圖有一個14/5-圓環染色。 由 Behzad
和 Vizing
給的猜測可導出對於最大度為Delta的圖，此類圖為Delta+2全染色。
對於平面圖來說，當圖的最大度是6時為唯一未解決的情況。令G為一平面圖且不存在頂點被包含於介於長度
3至8的圈。若G的最大度為6，則G為8全染色。若G的最大度為8，則G為9全染色。

Havet 和 Yu cite{havet02}
猜測對於不為四個點的完全圖及所有頂點的度皆不大於4的圖形，此類圖的(2,1)-全染色數不大於5。
對於最大平均度小於7/3的圖及對於{em flower
snarks}，我們證明了此猜測成立。

我們介紹圖的圓環(r,1)-全標號。由於放寬了上述猜想，我們猜測對所有頂點的度皆不大於4的圖形，
此類圖的圓環(2,1)-全標號數不大於7。對於最大平均度小於5/2的圖，我們證明了此猜測成立。

我們也證明了對於所有無4、7、8圈的平面圖，此類圖為無二色圈4選色圖。與最新的結果合併，
我們得到對於$5leq k<lleq
8$，所有無4,k,l圈的平面圖，此類圖為無二色圈4選色圖。

Abstract: This thesis studies five kinds of graph colorings: the circular coloring,
the total coloring, the (d; 1)-total labeling, the circular (r; 1)-total labeling, and the
acyclic list coloring.
We give upper bounds on the circular chromatic number of graphs with small
maximum average degree, mad for short. It is proved that if mad(G)<22=9 then G
has a 11=4-circular coloring, if mad(G) < 5=2 then G has a 14=5-circular coloring.
A conjecture by Behzad and Vizing implies that Δ+2 colors are always sufficient
for a total coloring of graphs with maximum degree Δ. The only open case for planar graphs is for Δ = 6. Let G be a planar in which no vertex is contained in cycles of all lengths between 3 and 8. If Δ(G) = 6, then G is total 8-colorable. If Δ(G) = 8, then G is total 9-colorable.
Havet and Yu [23] conjectured that every subcubic graph G &#824;=K4 has (2; 1)-total
number at most 5. We confirm the conjecture for graphs with maximum average
degree less than 7=3 and for flower snarks.
We introduce the circular (r; 1)-total labeling. As a relaxation of the aforementioned
conjecture, we conjecture that every subcubic graph has circular (2; 1)-total number at most 7. We confirm the conjecture for graphs with maximum average degree less than 5=2.
We prove that every planar graph with no cycles of lengths 4, 7 and 8 is acyclically
4-choosable. Combined with recent results, this implies that every planar
graph with no cycles of length 4;k; l with 5 6 k < l 6 8 is acyclically 4-choosable.

1 Introduction 6
1.1 Basic definitions . . . 6
1.2 Circular coloring . . 9
1.3 Total coloring 12
1.4 (d; 1)-total labeling . 13
1.5 Circular (r; 1)-total labeling . 16
1.6 Acyclic choosability 18
2 Circular coloring 20
2.1 Introduction . 20
2.2 Main tools . . 22
2.3 Graphs with maximum average degree less than 22=9 . . . 26
2.4 Graphs with maximum average degree less than 5=2 30
2.5 Conclusion . 53
3 Total coloring, (d;1)-total labeling and circular (r;1)-total labeling 55
3.1 Introduction . 55
3.2 Total coloring 57
3.2.1 Planar graphs with maximum degree 8 . . . 57
3.2.2 Planar graphs with maximum degree 6 . . . 64
3.3 (d; 1)-total labeling . 76
3.3.1 Graphs with maximum average degree less than 7=3 . . . 76
3.3.2 Flower snarks 82
3.4 Circular (r;1)-total labeling . 82
3.4.1 Graphs with maximum average degree less than 5=2 . . . 84
3.5 Conclusion . 90
4 Acyclic choosability 91
4.1 Introduction . 91
4.2 Planar graphs with no 4;7 and 8-cycles . . . 92
4.3 Conclusion . 110
5 Conclusion 112
5.1 Circular coloring . . 112
5.2 Total coloring, (d; 1)-total labeling and circular (r; 1)-total labeling 113
5.3 Acyclic choosability 114

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