Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond

The \emph{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 colouring 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 $|V(G)|\leq 2χ(G)+1$. 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.

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