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 1.5) construct a graph with choice number that is not -choosable. Thus

Gu and Xu (2026, Theorem 1.7) also proved that for every sufficiently large integer there is a graph of choice number that is not -choosable for some integer , where is an absolute constant. Thus the gap is unbounded already when two colors are chosen at each graph vertex, and

where denotes big-omega notation.

See also

Choice Number, List Coloring

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References

Gu, Y. and Xu, R. "The Strong Fractional Choice Number Can Exceed the Choice Number by an Arbitrarily Large Amount." 10 Nov 2026. https://arxiv.org/abs/2611.03236v3.Li, X. and Zhu, X. "The Strong Fractional Choice Number of Series-Parallel Graphs." Disc. Math. 343, 111796, 2020. https://doi.org/10.1016/j.disc.2019.111796.

Cite this as:

Weisstein, Eric W. "Strong Fractional Choice Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StrongFractionalChoiceNumber.html

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