The fractional chromatic number, the Hall ratio, and the lexicographic product

Let χ f denote the fractional chromatic number and ρ the Hall ratio, and let the lexicographic product of G and H be denoted G l e x H . Main results: (i) ρ ( G l e x H ) ≤ χ f ( G ) ρ ( H ) ; (ii) if ρ ( G ) = χ f ( G ) then ρ ( G l e x H ) = ρ ( G ) ρ ( H ) for all H ; (iii) χ f − ρ is unbounded. In addition, the question of how big χ f / ρ can be is discussed.