Star-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 star-extremal if its star-chromatic number is equal to its fractional-chromatic number. We prove that star-extremal graphs G have the following interesting property: For an arbitrary graph H the star-chromatic number chi(star)(G[H]) of the Lexicographic product G[H] is equal to the product of chi(star)(G) and chi(H). Then we show that several classes of circulant graphs are star-extremal. Thus for these circulant graphs G and arbitrary graphs H, if chi(star)(G) and chi(H) are known then we can easily determine the star-chromatic number (hence the ordinary chromatic number) of the lexicographic product G[H]. For these star-extremal circulant graphs, we also derive polynomial-time anti-clique-finding and coloring algorithms.