The classical (proper) graph colouring problem asks for a colouring of the vertices of a graph with the minimum number of colours such that no two vertices with the same colour are adjacent. Equivalently the colouring is required to be such that the graph induced by the vertices coloured the same colour has the maximum degree equal to zero. The graph parameter associated with the minimum possible number of colours of a graph is called chromatic number of that graph. One generalization of this classical problem is to relax the requirement that the maximum degree of the graph induced by the vertices coloured the same colour be zero, and instead allow it to be some integer d. For d = 0, we are back at the classical proper colouring. For other values of d we say that the colouring has defect d. Another generalization of the classical graph colouring, is list colouring and its associated parameters: choosability and choice number. The main result of this thesis is to show that every graph G of Euler genus μ is ⌈2 + √(3μ + 3)⌉–choosable with defect 1 (equivalently, with clustering 2). Thus allowing any defect, even 1, reduces the choice number of surface embeddable graphs below the chromatic number of the surface. For example, the chromatic number of the family of toroidal graphs is known to be 7. The bound above implies that toroidal graphs are 5-choosable with defect 1. This strengthens the result of Cowen, Goddard and Jesurum (1997) who showed that toroidal graphs are 5-colourable with defect 1. In a graph embedded in a surface, two faces that share an edge are called adjacent. We improve the above bound for graphs that have embeddings without adjacent triangles. In particular, we show that every non-planar graph G that can be embedded in a surface of Euler genus μ without adjacent triangles, is ⌈(5+ √(24μ + 1)) /3⌉–choosable with defect 1. This result generalizes the result of Xu and Zhang (2007) to all the surfaces. They proved that toroidal graphs that have embeddings on the torus without two adjacent triangles are 4-choosable with defect 1.

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| ∈ {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 ≤ i ≤ 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) − 1 ≤ 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.

Thomassen proved that there are only finitely many 6-critical graphs embeddable on a fixed surface. He also showed that planar graphs are 5-list-colorable. This thesis develops new techniques to prove general theorems for 5-list-coloring graphs embedded in a fixed surface. Indeed, a general paradigm is established which improves a number of previous results while resolving several open conjectures. In addition, the proofs are almost entirely self-contained. In what follows, let S be a fixed surface, G be a graph embedded in S and L a list assignment such that, for every vertex v of G, L(v) has size at least five. First, the thesis provides an independent proof of a theorem of DeVos, Kawarabayashi and Mohar that says if G has large edge-width, then G is 5-list-colorable. Moreover, the bound on the edge-width is improved from exponential to logarithmic in the Euler genus of S, which is best possible up to a multiplicative constant. Second, the thesis proves that there exist only finitely many 6-list-critical graphs embeddable in S, solving a conjecture of Thomassen from 1994. Indeed, it is shown that the number of vertices in a 6-list-critical graph is at most linear in genus, which is best possible up to a multiplicative constant. As a corollary, there exists a linear-time algorithm for deciding 5-list-colorability of graphs embeddable in S. Furthermore, we prove that the number of L-colorings of an L-colorable graph embedded in S is exponential in the number of vertices of G, with a constant depending only on the Euler genus g of S. This resolves yet another conjecture of Thomassen from 2007. The thesis also proves that if X is a subset of the vertices of G that are pairwise distance Omega(log g) apart and the edge-width of G is Omega(log g), then any L-coloring of X extends to an L-coloring of G. For planar graphs, this was conjectured by Albertson and recently proved by Dvorak, Lidicky, Mohar, and Postle. For regular coloring, this was proved by Albertson and Hutchinson. Other related generalizations are examined.

Dans cette thèse, nous nous intéressons à la coloration circulaire des graphes planaires. Des bornes supérieures ont été données pour des graphes avec degré maximum borné, avec girth, la longueur de son plus petit cycle, bornée, avec des cycles manquants, etc. Ici nous donnerons de nouvelles bornes pour les graphes avec degré moyen maximum borné. Nous étudions également la coloration totale et la coloration (d,1)-totale de plusieurs familles infinies de graphes. Nous décrivons le nouveau concept de coloration (d,1)-totale circulaire. Enfin, nous discutons les conditions nécessaires pour qu'un graphe planaire admette une coloration acyclique par listes de taille 4. / In this thesis, we study the circular coloring of planar graphs. Upper bounds have been given for graphs with bounded maximum degree, with bounded girth, that is the length of its smallest cycle, with missing cycles, and so on. It has also been studied for graphs with bounded maximum average degree. Here we give new upper bounds for that latter case. We also study the total coloring and ($d,1$)-total labeling of a few infinite families of graphs and describe the new concept of circular ($d,1$)-total labeling of graphs. In the last part, we will discuss conditions for a planar graph to be acyclically $4$-choosable.

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 £G+2 colors are always sufficient for a total coloring of graphs with maximum degree £G. The only open case for planar graphs is for £G = 6. Let G be a planar in which no vertex is contained in cycles of all lengths between 3 and 8. If £G(G) = 6, then G is total 8-colorable. If £G(G) = 8, then G is total 9-colorable. Havet and Yu [23] conjectured that every subcubic graph G ̸=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.