List-colouring, introduced independently in the 1970s by Vizing and by Erdős, Rubin, and Taylor, generalizes ordinary vertex colouring by assigning each vertex a personalized palette of admissible colours. A recurring theme in this field is chromatic-choosability, the condition where a graph's list chromatic number equals its ordinary chromatic number. This talk provides a survey of the progress made in identifying chromatic-choosable graph classes and characterizing the ‘jump’ — the gap that can exist between these two parameters.
We delve into the principal methods that have shaped the field, providing technical insights into the Alon–Tarsi method, which utilizes graph polynomials and orientations to establish upper bounds; the Kernel method, famously applied by Galvin to settle the List Edge Colouring Conjecture for bipartite multigraphs; and Thomassen’s rigid boundary induction, which proved the 5-choosability of planar graphs among others.
Based on a survey co-authored with Somasundaram and Nandana Vasudevan.
