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

Let χfχf denote the fractional chromatic number and ρρ the Hall ratio, and let the lexicographic product of GG and HH be denoted GlexHGlexH. Main results: (i) ρ(GlexH)≤χf(G)ρ(H)ρ(GlexH)≤χf(G)ρ(H); (ii) if ρ(G)=χf(G)ρ(G)=χf(G) then ρ(GlexH)=ρ(G)ρ(H)ρ(GlexH)=ρ(G)ρ(H) for all HH; (iii) χf−ρχf−ρ is unbounded. In addition, the question of how big χf/ρχf/ρ can be is discussed.