List coloring of a finite simple graph is a proper coloring in which the color of each graph vertex is chosen from a prescribed finite set of available colors, called its list. The lists may differ from one graph vertex to another (Dvorák and Postle 2018).
For a nonnegative integer 
, a graph is 
-choosable iff every assignment of
lists with 
colors in each list admits a list coloring. The least such 
is its choice number, also
called its list chromatic number (Gu and Xu 2026, p. 1).
The proportional choice number additionally controls how often each available color is used. Mudrock (2026) bounded it by 
, where 
is the edge chromatic
number.
Wang (2026) proved that, for a 3-connected graph 
that is neither a complete
graph nor the triangular prism graph, lists with
admit proper list colorings,
and all such colorings are connected by list-valid Kempe
chain changes. Here 
is the vertex degree. Wang (2026) reports AI assistance
with the proof.
In the multiple-color version, an 
-choosable graph admits, for
every assignment of lists of 
colors, a choice of 
colors at each graph vertex
such that the chosen sets at adjacent
vertices are disjoint. The strong
fractional choice number requires this property for every pair of positive
integers 
above a specified ratio (Li and Zhu 2020).
-Minor-Free Graphs." 1 Oct 2026.