χ*-Extremal Graphs and the Lexicographic Product

The star-chromatic number and the fractional-chromatic number are two generalizations of the ordinary chromatic number of a graph. We say a graph G is χ*-extremal if its star-chromatic number is equal to its fractional-chromatic number. We prove that χ*-extremal graphs G have the following nice property: For an arbitrary graph H the starchromatic number χ*(G[H]) of the lexicographic product G[H] is equal to the product of χ*(G) and χ(H). Then we show that several classes of circulant graphs are χ*-extremal. Thus for these circulant graphs G and arbitrary graphs H , if χ*(G) and χ(H) are known then we can easily determine the star-chromatic number (hence the ordinary chromatic number) of the lexicographic product G[H]