Lower Bound on the Maximum Denominator of Fractional Chromatic Numbers
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 $\text{sd}(n) \geq 2^{n/2}$.
Fields of Study & Subjects
Image Retrieval and Classification Techniques
Graph Labeling and Dimension Problems
Publication details
- Journal / source
- The Electronic Journal of Combinatorics
- Volume
- 33
- Issue
- 2
- Publication date
- 21 May 2026
- Language
- EN