On the fractional chromatic number, the chromatic number, and graph products

Abstract

It is shown that the difference between the chromatic number χ and the fractional chromatic number χ f can be arbitrarily large in the class of uniquely colorable, vertex transitive graphs. For the lexicographic product G∘H it is shown that χ(G∘H)⩾χ f(G) χ(H) . This bound has several consequences, in particular, it unifies and extends several known lower bounds. Lower bounds of Stahl (for general graphs) and of Bollobás and Thomason (for uniquely colorable graphs) are also proved in a simple, elementary way.