Describing -stars at -vertices, , in normal plane maps

We prove that every normal plane map has a ( 3 , 1 0 − ) -edge, or a ( 5 − , 4 , 9 − ) -path, or a ( 6 , 4 , 8 − ) -path, or a ( 7 , 4 , 7 ) -path, or a ( 5 ; 4 , 5 , 5 ) -star, or a ( 5 ; 5 , b , c ) -star with 5 ≤ b ≤ 6 and 5 ≤ c ≤ 7 , or a ( 5 ; 6 , 6 , 6 ) -star. Moreover, none of the above options can be strengthened or dropped. ticular, this extends or strengthens several known results and disproves a related conjecture of Harant and Jendrol’ (2007) [10].