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

TL;DR

This paper proves Ohba's conjecture that the choice number equals the chromatic number for graphs with a limited number of vertices relative to their chromatic number, and explores related conjectures.

Contribution

It confirms Ohba's conjecture for graphs with up to 2χ(G)+1 vertices and strengthens it for graphs up to 3χ(G) vertices, advancing understanding of graph choosability.

Findings

Proved Ohba's conjecture for |V(G)| ≤ 2χ(G)+1

Established a strengthened version for |V(G)| ≤ 3χ(G)

Posed new conjectures related to graph choosability

Abstract

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 $\ch(G)=\chi(G)$ whenever $|V(G)|\leq 2\chi(G)+1$. We also prove a strengthening of Ohba's Conjecture which is best possible for graphs on at most $3\chi(G)$ vertices, and pose several conjectures related to our work.