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Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonal faces, while all other faces are hexagons. Fullerene graphs are mathematical models of fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites and diamonds. We give a survey on fullerene graphs from our perspective, which could be also considered as an introduction to this topic. Different types of fullerene graphs are considered, their symmetries, and construction methods. We give an overview of some graph invariants that can possibly correlate with the fullerene molecule stability, such as: the bipartite edge frustration, the independence number, the saturation number, the number of perfect matchings, etc.

We show that every cubic bridgeless graph G has at least 2^(|V(G)|/3656) perfect matchings. This confirms an old conjecture of Lovasz and Plummer. This version of the paper uses a different definition of a burl from the journal version of the paper and a different proof of Lemma 18 is given. This simplifies the exposition of our arguments throughout the whole paper

A fullerene graph is a planar cubic 3-connected graph with only pentagonal and hexagonal faces. We show that fullerene graphs have exponentially many perfect matchings.Comment: 7 pages, 3 figure

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts in G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an edge-cut of size at least 1.33008n, where n is the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge-cut also applies to random cubic graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut of size 1.33008n.

We show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978 which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid to random cubic graphs as well as it improves existing lower bounds on the maximum cut in cubic graphs with large girth.

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