The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \chi _{f} ( G \times H ) \geqslant \frac{1} {4} \cdot \min \{ \chi _{f} ( G ),\chi _{f} ( H ) \} $$\end{document} holds for all directed graphs G and H. Similar bounds for the usual chromatic number seem to be much harder to obtain: It is still not known whether there exists a number n such that χ(G×H) ≥ 4 for all directed graphs G, H with χ(G) ≥ χ(H) ≥ n. In fact, we prove that for every integer n ≥ 4, there exist directed graphs Gn, Hn such that χ(Gn) = n, χ(Hn) = 4 and χ(Gn×Hn) = 3.