The strong fractional choice number of a finite simple graph 
with at least one graph vertex is the infimum
of the real numbers 
such that 
is 
-choosable for every pair of positive
integers 
with 
(Li and Zhu 2020, Gu and Xu 2026). Here 
-choosability means that every assignment of lists of 
colors to the vertices admits a choice of 
colors from each list such that the chosen sets
at adjacent vertices are disjoint.
The case 
is ordinary list coloring. The strong fractional
choice number is denoted 
. Its definition requires the stated coloring property
for every denominator 
, rather than for just a selected ratio 
.
Gu and Xu (2026, Theorem 1.4) proved that, for all integers 
and 
,
there is a planar graph with choice
number 3 that is not 
-choosable. In particular, their construction gives
a planar graph on 121 vertices
that is 3-choosable but not 
-choosable. Consequently,
 |
where 
denotes the supremum. For every choice of integers 
,
,
and 
,
Gu and Xu (2026, Theorem 1.6) construct a graph with
choice number 
that is not 
-choosable. Thus
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