The fullerene graphs with a perfect star packing

Fullerene graph G is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the molecular graph of carbon fullerene. A spanning subgraph of G is called a perfect star packing in G if its each component is isomorphic to K1,3. For an independent set D ⊆ V (G), if each vertex in V (G) \D has exactly one neighbor in D, then D is called an efficient dominating set of G. In this paper we show that the number of vertices of a fullerene graph admitting a perfect star packing must be divisible by 8. This answers an open problem asked by Došlić et al. and also shows that a fullerene graph with an efficient dominating set has 8n vertices. In addition, we find some counterexamples for the necessity of Theorem 14 in [14] and list some subgraphs that preclude the existence of a perfect star packing of type P0.