We consider normal plane maps with minimum degree at least 4 and no adjacent 4-vertices. The height of a star is the maximum degree of its vertices. By and with we denote the minimum height of arbitrary -stars and -stars centered at vertices of degree at most 5, respectively, in a given .
Mohar, Škrekovski, and Voss proved (2003) that every satisfies . We improve this result by proving that and construct an with . On the other hand, we show that .
Also, we prove that every satisfies and , where both 10 and 11 are sharp.