Low 5-stars in normal plane maps with minimum degree 5

In 1996, Jendrol’ and Madaras constructed a plane triangulation with minimum degree 5 in which the minimum vertex degree h(S5)h(S5) of 5-stars is arbitrarily large. This construction has minor (5,5,5,5)(5,5,5,5)-stars, that is 5-vertices with four 5-neighbors. It has been open if forbidding minor (5,5,5,5)(5,5,5,5)-stars makes h(S5)h(S5) finite.We prove that every normal plane map with minimum degree 5 and no minor (5,5,5,5)(5,5,5,5)-stars satisfies h(S5)≤13h(S5)≤13 and construct such a map with h(S5)=12h(S5)=12.