Geodesic
The chromatic number of the product of two graphs is at least half the minimum of the fractional chromatic numbers of the factors
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 2, pp. 353-355.

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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 $\chi(G \times H) \geq \frac{1}{2} \min \{ \chi_f(G), \chi_f(H) \}$.
Classification : 05C15
Keywords: Hedetniemi's conjecture; (fractional) chromatic number 
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     title = {The chromatic number of the product of two graphs is at least half the minimum of the fractional chromatic numbers of the factors},
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Tardif, Claude. The chromatic number of the product of two graphs is at least half the minimum of the fractional chromatic numbers of the factors. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 2, pp. 353-355. http://geodesic.mathdoc.fr/item/CMUC_2001__42_2_a11/