17 problems

A fullerene graph is a planar, 3-regular graph whose facets are pentagons or hexagons. Barnette–Goodey conjecture. Every fullerene graph is Hamiltonian. The conjecture is a classic…

Character conjecture. For $lpha,\beta\rtf]\in(0,1)$ and feasible $n_1,n_2$, if $T_{n_1}$ and $T_{n_2}$ are dual fullerenes, then

Let $T_n^6$ be the hexagonal subgraph of a dual fullerene graph $T_n$, and consider the degree sequences of the boundary vertices of its facets. Pentagon-recovery conjecture. The p…

Let $T_n$ be a dual fullerene graph and let $T_n^6$ be its hexagonal subgraph, for feasible $n$. A cut-partition of $T_n^6$ is the set of components generated by the stated cut-par…

Let $C_{56,622}$ and $C_{80,31924}$ denote the indicated fullerene graphs. A cut-partition of a hexagonal subgraph is the set of components generated by the stated cut-partition co…

A generalized Stone–Wales operation is an operation on a fullerene associated with a generalized Stone–Wales path, and fullerene isomers are fullerene graphs considered up to isomo…

Wiener complexity and diameter conjecture. The Wiener complexity and the diameter of fullerene graphs of an arbitrary order having the maximal Wiener index are given in Proposition…

Maximal Wiener index conjecture. If a fullerene graph of an arbitrary order has the maximal Wiener index, then it is a nanotubical fullerene graph with caps of types $a$--$d$, and…

Maximal entropy conjecture. The quantity $h_F(m)$ is equal to $\mathbf{h}$, and hence is strictly greater than the entropy $h(m)$ of the family $F(m,k)$.

Andova–Skrekovski conjecture. If $G$ is any fullerene graph on $n$ vertices, then

A fullerene graph is a three-connected planar cubic graph whose faces have only five or six sides. Fullerene Hamiltonian-cycle conjecture. Every fullerene graph has a Hamiltonian c…

Let $G$ be a fullerene graph, and let $\lambda_n(G)$ denote its smallest adjacency-matrix eigenvalue. Let $\phi=(1+\sqrt{5})/2$ be the golden ratio. The truncated icosahedron has s…

Let $G$ be a fullerene graph on $n$ vertices. Let $\alpha(G)$ be its independence number, and let $\alpha^-(G)$ be the maximum size of an independent set $A$ such that exactly half…

Let $G$ be a fullerene graph, and let $n$ be its number of vertices. Let $\alpha(G)$ denote its independence number. The bound under discussion is … Daugherty's equality conjecture…

Let $G$ be a fullerene graph on $n$ vertices, meaning a cubic bridgeless plane graph whose faces have size $5$ or $6$. An independent set is a set of vertices containing no adjacen…

Let $G$ be a fullerene graph on $n$ vertices, meaning a cubic bridgeless plane graph whose faces have size $5$ or $6$. An odd cycle transversal is a set of edges whose removal make…

A fullerene graph is a planar cubic graph in which every face has size five or six. Fullerene Hamiltonicity conjecture. Every fullerene graph is Hamiltonian. This conjecture asks w…