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**1 - 1**of**1**### Choosability of Graphs with Bounded Order: Ohba’s Conjecture and Beyond

, 2013

"... c©Jonathan A. Noel, 2013 The choice number of a graph G, denoted ch(G), is the minimum integer k such that for any assignment of lists of size k to the vertices of G, there is a proper colour-ing of G such that every vertex is mapped to a colour in its list. For general graphs, the choice number is ..."

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c©Jonathan A. Noel, 2013 The choice number of a graph G, denoted ch(G), is the minimum integer k such that for any assignment of lists of size k to the vertices of G, there is a proper colour-ing of G such that every vertex is mapped to a colour in its list. For general graphs, the choice number is not bounded above by a function of the chromatic number. In this thesis, we prove a conjecture of Ohba which asserts that ch(G) = χ(G) whenever |V (G) | ≤ 2χ(G) + 1. We also prove a strengthening of Ohba’s Conjecture which is best possible for graphs on at most 3χ(G) vertices, and pose several conjectures related to our work. ii Abrégé Le nombre de choix d’un graphe G, note ́ ch(G), est le plus petit entier k tel que pour toute affectation de listes de taille k au sommets de G, il y a une coloration de G tel que chaque sommet de G est colore ́ par une couleur de sa liste. En général, le nom-bre de choix n’est pas borne ́ supérieurement par une fonction du nombre chromatique. Dans cette thèse, nous démontrons une conjecture de Ohba qui affirme que ch(G) = χ(G) dès que |V (G) | ≤ 2χ(G) + 1. Nous démontrons aussi une version plus forte de la conjecture de Ohba qui est optimale pour les graphes ayant au plus 3χ(G) sommets, et énonçons plusieurs conjectures par rapport a ̀ nos travaux. iii Declaration This thesis contains no material which has been accepted in whole, or in part, for any other degree or diploma. Chapters 4 and 6 of this thesis contain new contributions to knowledge. The results of these chapters have been, or will be, submitted for publication in peer-reviewed journals. The result of Chapter 4 is based on joint work with Bruce A. Reed and Hehui Wu. The result of Chapter 6 is based on joint work with Douglas B. West, Hehui Wu, and Xuding Zhu. iv