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

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