The fullerene graphs with a perfect star packing - Shi, Lingjuan (Ars mathematica contemporanea, 2023)
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 ?$K_{1,3}$?. For an independent set ?$D \subseteq V(G)$?, if each vertex in ?$V(G) \setminus D$? 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 of paper of T. Došlić et al. J. Math. Chem. 58, No. 10, 2223-2244 (2020) and list some subgraphs that preclude the existence of a perfect star packing of type ?$P0$?.
Keywords: efficient dominating set; fulerenski graf; fullerene graph; perfect star packing; popolno zvezdno pakiranje; učinkovita dominantna množica