The fullerenes with a perfect star packing

TL;DR

This paper characterizes when fullerene graphs contain perfect star packings, showing a necessary divisibility condition and providing counterexamples and forbidden configurations related to efficient dominating sets.

Contribution

It provides a new characterization for perfect star packings in fullerene graphs and proves a necessary divisibility condition, answering an open problem.

Findings

Fullerene graphs with perfect star packings must have order divisible by 8.

Counterexamples are found for previous theorems.

Forbidden configurations are identified to prevent perfect star packings.

Abstract

A spanning subgraph of a graph $G$ is called a perfect star packing in $G$ if every component of the spanning subgraph is isomorphic to the star graph $K_{1,3}$. An efficient dominating set of graph $G$ is a vertex subset $D$ of $G$ such that each vertex of $G$ not in $D$ is adjacent to exactly one vertex from $D$ and any two vertices of $D$ are not adjacent in $G$. Fullerene graph is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the molecular graph of carbon fullerene. Clearly, a perfect star packing in a fullerene graph $G$ on $n$ vertices will exist if and only if $G$ has an efficient dominating set of cardinality $\frac{n}{4}$. The problem of finding an efficient dominating set is algorithmically hard \cite{Alg_hard}. In this paper, we give a characterization for a fullerene graph to own a perfect star packing. And mainly show that it is necessary for a fullerene $G$ owning a perfect star packing to have order being divisible by $8$. This answers an open problem asked by Dosli\'{c} et. al. and also shows that a fullerene graph with an efficient dominating set has $8n$ vertices. By the way, we find some counterexamples for the necessity of Theorem $14$ in \cite{Doslic} and list some forbidden configurations to preclude the existence of a perfect star packing of type $P0$.