Star-extremal graph

Star-extremal graph --- звездно-экстремальный граф.

A graph [math]\displaystyle{ G }[/math], for which the star-chromatic number is equal to the fractional-chromatic number, i.e., [math]\displaystyle{ \chi^{\star}(G) = chi_{f}(G) }[/math], is a star-extremal graph [math]\displaystyle{ G }[/math]. It has a very inte\-res\-ting property, namely, if [math]\displaystyle{ G }[/math] is a star-extremal graph and [math]\displaystyle{ H }[/math] is an arbitrary graph, then the star-chromatic number [math]\displaystyle{ \chi^{\star}(G[H]) }[/math] of the lexicographic product [math]\displaystyle{ G[H] }[/math] is equal to the product of [math]\displaystyle{ \chi^{\star}(G) }[/math] and the chromatic number [math]\displaystyle{ \chi(H) }[/math] of the graph [math]\displaystyle{ H }[/math].