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Lower Bound on the Maximum Denominator of Fractional Chromatic Numbers

Year: 2026

Type: article

Abstract: What is the least integer $\text{sd}(n)$ such that every graph on $n$ vertices has fractional chromatic number $p /q$, where $p$ and $q$ are positive integers and $q \le \text{sd}(n)$? An upper bound on the determinants of Hadamard matrices implies that $\text{sd}(n)\le 2^{-n}(n+1)^{(n+1) /2}$. The only known lower bound on $\text{sd}(n)$ that is exponential in $n$ (asymptotically, roughly $1.346^n/\sqrt{\log n}$) was obtained using an iterated Mycielski construction [D. C. Fisher, J. Graph Theory 20 (1995), 403-409]. We improve on this bound by constructing a family of graphs which shows that $\text{sd}(n) \geq 2^{n/2}$.

Source: The Electronic Journal of Combinatorics

DOI: 10.37236/14524

Authors: Marthe Bonamy, Karolína Hylasová, Tomáš Kaiser, Jean-Sébastien Sereni

Institutions: Laboratoire Bordelais de Recherche en Informatique, University of West Bohemia in Pilsen, Laboratoire des Sciences de l'Ingénieur, de l'Informatique et de l'Imagerie

Language: English

Topic: Advanced Graph Theory Research

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: diamond

OpenAlex ID: W4414761003

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