Low stars in normal plane maps with minimum degree 4 and no adjacent 4-vertices

We consider normal plane maps M4∗ with minimum degree at least 4 and no adjacent 4-vertices. The height of a star is the maximum degree of its vertices. By h(Sk)h(Sk) and h(Sk(m)) with 1≤k≤41≤k≤4 we denote the minimum height of arbitrary kk-stars and kk-stars centered at vertices of degree at most 5, respectively, in a given M4∗.Mohar, Škrekovski, and Voss proved (2003) that every M4∗ satisfies h(S4)≤107h(S4)≤107. We improve this result by proving that h(S4)≤23h(S4)≤23 and construct an M4∗ with h(S4)=18h(S4)=18. On the other hand, we show that h(S4(m))=∞.Also, we prove that every M4∗ satisfies h(S3)≤10h(S3)≤10 and h(S3(m))≤11, where both 10 and 11 are sharp.