On the existence of specific stars in planar graphs

Given a graph G, a (k;a,b,c)-star in G is a subgraph isomorphic to a star K-1,K-3 with a central vertex of degree k and three leaves of degrees a, b and c in G. The main result of the paper is: Every planar graph G of minimum degree at least 3 contains a (k;a,b,c)-star with a <= b <= c and (i) k = 3, <= n 10, or (ii) k = 4, a = 4, 4 <= b <= 10, or (iii) k = 4, a = 5, 5 <= b <= 9, or (iv) k = 4, 6 <= a <= 7, 6 <= b <= 8, or (v) k = 5, 4 <= a <= 5, 5 <= b <= 6 and 5 <= c <= 7, or (vi) k = 5 and a = b = c = 6.