The chromatic number of the product of two graphs is at least half the minimum of the fractional chromatic numbers of the factors

summary:One consequence of Hedetniemi's conjecture on the chromatic number of the product of graphs is that the bound $\chi(G \times H) \geq \min \{ \chi_f(G), \chi_f(H) \}$ should always hold. We prove that $\chi(G \times H) \geq \frac{1}{2} \min \{ \chi_f(G), \chi_f(H) \}$