Open AccessArticle

The Structural Properties of (2, 6)-Fullerenes

Rui Yang

^* and

Mingzhu Yuan

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China

Author to whom correspondence should be addressed.

Symmetry 2023, 15(11), 2078; https://doi.org/10.3390/sym15112078

Submission received: 19 September 2023 / Revised: 10 November 2023 / Accepted: 13 November 2023 / Published: 17 November 2023

(This article belongs to the Special Issue Symmetry in Algorithmic Graph Theory and Interconnection Networks)

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Abstract

(2, 6)

-fullerene F is a 2-connected cubic planar graph whose faces are only

2

-length and

6

-length. Furthermore, it consists of exactly three

2

-length faces by Euler’s formula. The

(2, 6)

-fullerene comes from Došlić’s

(k, 6)

-fullerene, a 2-connected 3-regular plane graph with only

k

-length faces and hexagons. Došlić showed that the

(k, 6)

-fullerenes only exist for

k = 2

, 3, 4, or 5, and some of the structural properties of

(k, 6)

-fullerene for

k = 3

, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of

(2, 6)

-fullerene. We discover that the edge-connectivity of

(2, 6)

-fullerenes is 2. Every

(2, 6)

-fullerene is 1-extendable, but not 2-extendable (F is called

n -

e x t e n d a b l e

(

| V (F) | \geq 2 n + 2

) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant (

k \geq 1

) if the deleting of any i (

0 \leq i \leq k

) disjoint even faces of F results in a graph with at least one perfect matching. We have that every

(2, 6)

-fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if

F - S

is connected and has no perfect matchings, where

F - S

denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by

a k (F)

, is the cardinality of a smallest anti−Kekulé set of F. We have that every

(2, 6)

-fullerene F with

| V (F) | > 6

has anti−Kekulé number 4. Further we mainly prove that there exists a

(2, 6)

-fullerene F having

f_{F}

hexagonal faces, where

f_{F}

is related to the two parameters n and m.

Keywords:

(2, 6)-fullerene; edge-connectivity; anti-Kekulé number; resonance

1. Introduction

(2, 6)

-fullerene F is a 2-connected cubic planar graph whose faces are only

2

-length and

6

-length. The

(2, 6)

-fullerene comes from Došlić’s

(k, 6)

-fullerene. A

(k, 6)

-fullerene is a 2-connected cubic planar graph whose faces are only

k

-length and 6-length. Došlić showed that all

(k, 6)

-fullerenes only exist for

k = 2

, 3, 4, or 5, and are 1-extendable [1]. A

(5, 6)

-fullerene is the usual fullerene as the molecular graph of a sphere carbon fullerene. A

(4, 6)

-fullerene is the molecular graph of a boron–nitrogen fullerene. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph [2,3]. And the number of perfect matchings is closely related to the stability of molecular graphs [4,5,6,7,8]. Therefore, many articles have studied the structural properties of graphs in both mathematics and chemistry [9,10,11]. In 2002, Došlić showed that fullerene graphs are bicritical and 2-extendable [9,12]. At the same time, he showed that every fullerene graph is cyclically 5-edge-connected [1]. In 2008, Kardoš characterized the cyclic edge-cuts of fullerene graphs by means of three operations [13]. For the resonance of fullerene graphs, in 2009, Ye et al. proved that every fullerene graph is 1-resonant and all leapfrog fullerene graphs are 2-resonant [14]. Later, Kaiser showed that the IPR fullerene graphs are also 2-resonant [15]. In 2010, Zhang et al. showed that boron–nitrogen fullerene graphs are bipartite, 3-connected, and 1-extendable [16]. They also proved that every boron–nitrogen fullerene is 2-resonant [16]. In 2011, Jiang et al. proved that boron–nitrogen fullerene graphs have the forcing number at least two [17]. A

(3, 6)

-fullerene is 1-extendable, and has the connectivity 2 or 3 [18]. In 2012, Yang et al. showed that each hexagon of a

(3, 6)

-fullerene with connectivity 2 is not resonant, and each hexagon of a

(3, 6)

-fullerene with connectivity 3 is resonant except for one graph [19]. This paper is mainly concerned with the structural properties of

(2, 6)

-fullerenes.

(2, 6)

-fullerene F is a cubic planar graph such that every face is either

2

-length or

6

-length. A graph with two vertices and n parallel edges joining them is denoted by

n \times K_{2}

. The smallest

(2, 6)

-fullerene is

3 \times K_{2}

. A

p l a n e

g r a p h

is a graph that can be embedded in the plane such that its edges intersect only at their ends. Any such embedding divides the plane into connected regions called

f a c e s

. Two different faces,

f_{1}, f_{2}

, are

a d j a c e n t

if their boundaries have an edge in common. A face is said to be

i n c i d e n t

w i t h

the vertices and edges in its boundary, and vice versa. An edge is said to be

i n c i d e n t

w i t h

the ends of the edge, and vice versa. Two vertices that are incident with a common edge are

a d j a c e n t

, and two distinct adjacent vertices are

n e i g h b o r s

. If S is a set of vertices in a graph, F, the set of all neighbors of the vertices in S is denoted by

N (S)

, and

| N (S) |

denotes the number of neighbors of S.

Let F be a

(2, 6)

-fullerene graph with vertex-set

V (F)

and edge-set

E (F)

. We denote the number of vertices and edges in F by

| V (F) |

and

| E (F) |

. For

H \subseteq H

, we let

F - H

be the subgraph of F obtained from F by removing the elements in H. A

m a t c h i n g

of F is a set of disjoint edges, M, of F. A perfect matching of F is a matching M that covers all vertices of F. A perfect matching of a graph coincides with a Kekulé structure of some molecular graph in organic chemistry. A set,

H

, of disjoint even faces of a graph, F, is a

r e s o n a n t

p a t t e r n

if F has a perfect matching M such that the boundary of each face in

H

is an

M

-alternating cycle. F is said to be k-resonant (

k \geq 1

) if any i (

0 \leq i \leq k

) disjoint even faces of F form a resonant pattern. Moreover, F is called

n -

e x t e n d a b l e

(

| V (F) | \geq 2 n + 2

) if any matching of n edges is contained in a perfect matching of F. F is

b i c r i t i c a l

if F contains an edge and

F - u - v

contains a perfect matching, for every pair of distinct vertices

u, v \in V (F)

. In this paper, we show that every

(2, 6)

-fullerene is 1-extendable, 1-resonant but not 2-extendable, bicritical.

The anti−Kekulé set of a

(2, 6)

-fullerene F with perfect matchings is an edge set,

S \subseteq E (F)

, such that

F - S

is connected and has no perfect matchings. The anti−Kekulé number of F, denoted by

a k (F)

, is the cardinality of a smallest anti−Kekulé set of F. It is NP-complete to find the smallest anti−Kekulé set of a graph. Moreover, it has been shown that the anti−Kekulé set of a graph significantly affects the whole molecule structure by the valence bond theory. We know the

(5, 6)

(4, 6)

, and

(3, 6)

-fullerenes have the anti−Kekulé numbers 4, 4, and 3, respectively. In this paper, We show that every

(2, 6)

-fullerene F has the anti−Kekulé number 4, with

| V (F) | > 6

2. Main Results

An edge-cut of F is a subset of edges

E^{^{'}} \subseteq E (F)

such that

F - E^{^{'}}

is disconnected. An

k -

e d g e - c u t

is an edge-cut with k edges. The

e d g e - c o n n e c t i v i t y

of F, denoted by

κ^{'} (F)

, is equal to the minimum cardinality of edge-cuts. F is

k -

e d g e - c o n n e c t e d

if F cannot be separated into at least two components by removing fewer than k edges.

Lemma 1.

The

(2, 6)

-fullerene F has edge-connectivity 2, where

| V (F) | > 2

Proof.

Since every edge of F is incident with a 2-length face or a 6-length face, there is no cut edge in F. Therefore, F is 2-edge-connected. For one 2-length face, C, in F, we denote it by

C = x y x

. Then either

F ≅ 3 \times K_{2}

or the two edges incident with x and y, respectively, other than

x y

form an 2-edge-cut of F. Therefore,

κ^{'} (F) = 2

, where

| V (F) | > 2

. □

We say an edge, e, is

i n c i d e n t

t o

a subgraph, H, if

| V (e) ⋂ V (H) | = 1

Lemma 2.

Every 2-edge-cut of a

(2, 6)

-fullerene isolates a 2-length face.

Proof.

Let F be a

(2, 6)

-fullerene. If

| V (F) | = 2

, then

F ≅ 3 \times K_{2}

, and the conclusion holds as F has no 2-edge-cut. So, next we suppose

| V (F) | > 2

. By Lemma 1, F has an 2-edge-cut. Let

E = {e_{1}, e_{2}}

be an 2-edge-cut whose deletion separates F into two components,

F^{'}

and

F^{″}

. Then E is a matching of F, as F is 3-regular and has edge-connectivity 2. Suppose every edge,

e_{i}

, has one endpoint, say

x_{i}

, on

F^{'}

, and the other endpoint, say

y_{i}

, on

F^{″}

i = 1, 2

. Suppose the outer face of

F^{″}

is exactly the outer face of F, thus

F^{'}

lies in some inner face of

F^{″}

. Then, there are two hexagons, denoted by

f_{1}

and

f_{2}

, such that both

f_{1}

and

f_{2}

are incident with

x_{1}

x_{2}

y_{1}

, and

y_{2}

. If one of

F^{'}

and

F^{″}

contains a cut edge, without losing generality, assume that

F^{'}

contains a cut edge,

e = u v

, then

F^{'} - e

has two connected components, say

F_{1}

and

F_{2}

. Then, both

e_{1}

and

e_{2}

cannot be incident to the same component

F_{i}

(

i = 1, 2

), otherwise there exists a cut edge, e, in F, which is a contradiction. Then,

V (f_{1}) = {u, v, x_{1}, y_{1}, x_{2}, y_{2}}

V (f_{2}) = {u, v, x_{1}, y_{1}, x_{2}, y_{2}}

. That is, all of

F_{1}

F_{2}

, and

F^{″}

are 2-length faces and we obtain a

(2, 6)

-fullerene with six vertices, and thus the conclusion holds. If both

F^{'}

and

F^{″}

contain cut edges, then there is a face with length more than 6, which is a contradiction. If neither

F^{'}

nor

F^{″}

has a cut edge, then

F^{'}

and

F^{″}

are 2-edge-connected, and in each of them there is only one face that is not 2-length or 6-length, and we denote these two boundaries of the exceptional faces by

C^{'}

and

C^{″}

, respectively. Let

v^{'}

e^{'}

, and

f^{'}

be the number of vertices, edges, and faces in

F^{'}

, respectively. Let

l^{'}

be the length of

C^{'}

, and

f_{2}^{'}

and

f_{6}^{'}

be the number of 2-length faces and 6-length faces in

F^{'}

, respectively. By Euler’s formula and the structure of

F^{'}

, it follows that

\{\begin{matrix} 3 v^{'} = 2 e^{'} + 2 \\ v^{'} - e^{'} + f_{2}^{'} + f_{6}^{'} = 1 \\ 2 f_{2}^{'} + 6 f_{6}^{'} + l^{'} = 2 e^{'} . \end{matrix}

(1)

By (1), we obtain that

l^{'} = 4 f_{2}^{'} - 2 .

(2)

Since F has no face with length more than 6, each of the two faces,

f_{1}

and

f_{2}

, has at most two additional vertices on

C^{'}

. Hence,

2 \leq l^{'} \leq 6

. By (2), we can obtain

1 \leq f_{2}^{'} \leq 2

. If

f_{2}^{'} = 1

, we have

l^{'} = 2

, which means that

F^{'}

is a 2-length face, and thus the conclusion holds. If

f_{2}^{'} = 2

, then

l^{'} = 6

and there are no additional vertices on

C^{″}

, which implies that

F^{″}

is a 2-length face, and thus the conclusion holds. Therefore, every 2-edge-cut of a

(2, 6)

-fullerene isolates a 2-length face. □

In [1], Došlić proved that the

(k, 6)

-fullerene is 1-extendable for

k = 3, 4

, and 5. In fact, we may observe that the conclusions remain valid for

k = 2

Lemma 3

([1]). Let F be a

(2, 6)

-fullerene graph. Then F is 1-extendable.

The resonance of faces of a plane bipartite graph is closely related to a 1-extendable property. It was revealed that every face (including the infinite one) of a plane bipartite graph G is resonant if and only if G is 1-extendable [20]. Combing with Lemma 3, we know that every

(2, 6)

-fullerene is 1-resonant.

Corollary 1.

Every

(2, 6)

-fullerene is 1-resonant.

Moreover, we know no

(2, 6)

-fullerene is 2-extendable.

Theorem 1.

(2, 6)

-fullerene is 2-extendable.

Proof.

Let F be a

(2, 6)

-fullerene graph. Let f be a 2-length face of F with the boundary

v_{1} v_{2} v_{1}

. By the definition of extendability, we know that

| V (F) | \geq 4

. Then, there exist two vertices,

u_{1}

and

u_{2}

, of F, which are different from

v_{1}

and

v_{2}

such that

u_{1} v_{1} \in E (F)

and

u_{2} v_{2} \in E (F)

. Since the four vertices,

u_{1}, u_{2}, v_{1}

, and

v_{2}

, must be contained in the same hexagon of F, there is a vertex,

u_{3} \neq u_{1}

u_{3} \neq v_{2}

, of F such that

u_{2} u_{3} \in E (F)

. Obviously,

u_{1} v_{1}, u_{2} u_{3}

is a matching and cannot be contained in a perfect matching of F. Thus, no

(2, 6)

-fullerene is 2-extendable. □

Similarly, we can show no

(2, 6)

-fullerene is bicritical.

Theorem 2.

(2, 6)

-fullerene is bicritical.

Proof.

Let F be a

(2, 6)

-fullerene graph, and f be a 2-length face of F with the boundary

v_{1} v_{2} v_{1}

. If

F ≅ 3 \times K_{2}

, then

F - v_{1} - v_{2}

has no perfect matchings. If

F ≇ 3 \times K_{2}

, then there exists a vertex, u, of F, which is different from

v_{2}

such that

u v_{1} \in E (F)

. Then,

F - u - v_{2}

has a single vertex,

v_{1}

, as a component. So,

F - u - v_{2}

has no perfect matchings. That is, F is not bicritical. □

Theorem 3

(Tutte’s Theorem [21]). A graph G has a perfect matching if and only if

c_{o} (G - U) \leq | U |

for any

U \subseteq V (G)

, where

c_{o} (G - U)

is the number of odd components of

G - U

Theorem 4

(Hall’s Theorem [21]). Let F be a bipartite graph with bipartition W and B. Then F has a perfect matching if and only if

| W | = | B |

and for any

U \subseteq W

| N (U) | \geq | U |

holds.

For the connected cubic simple bipartite graph, we know its anti−Kekulé number is 4 [22].

Theorem 5

( [22]). If G is a connected cubic simple bipartite graph, then

a k (F) = 4

The above result can be used to determine the anti−Kekulé numbers of some interesting graphs, such as

(4, 6)

-fullerenes [22], toroidal fullerenes [22], etc. Theorem 5 is also valid for

(2, 6)

-fullerene when

| V (F) | > 6

Theorem 6.

Let F be a

(2, 6)

-fullerene graph with

| V (F) | > 6

. Then

a k (F) = 4

Proof.

Let F be a

(2, 6)

-fullerene. For any vertex, u, in F, if

| N (u) | = 1

, then

F ≅ 3 \times K_{2}

(see Figure 1a the graph

3 \times K_{2}

). For any vertex, u, in F, if

| N (u) | = 2

, then

F ≅ F_{6}

(see Figure 1b the graph

F_{6}

). We can easily see that both

3 \times K_{2}

and

F_{6}

cannot exist the anti−Kekulé set. On the other hand, if we let n and

f_{6}

be the number of vertices and the hexagons of F, respectively, then, by Euler’s formula and the formula of degree sum, we can obtain

n = 2 f_{6} + 2

. Thus, if

f_{6} = 0

, then

n = 2

and

F ≅ 3 \times K_{2}

. If

f_{6} = 1

, then

n = 4

, which is impossible as every hexagonal face must contain six vertices. If

f_{6} = 2

, then

n = 6

and

F ≅ F_{6}

(see Figure 1b the graph

F_{6}

). Therefore, when

| V (F) | \leq 6

, there is no anti−Kekulé set in F.

Next, we discuss the anti−Kekulé number of F with

| V (F) | > 6

. Then, there is a vertex, u, in F and

| N (u) | = 3

. Let x, y, and z be the three neighbors of u. Let

e_{1}

and

e_{2}

be two edges incident with x other than

u x

, and let

e_{3}

and

e_{4}

be two edges incident with y other than

u y

. Since every face of F is 2-length or 6-length and F is 2-edge-connected, the four edges

e_{1}, e_{2}, e_{3}

, and

e_{4}

are pairwise different. We claim that

{e_{1}, e_{2}, e_{3}, e_{4}}

is an anti−Kekulé set. It is obvious that

F - {e_{1}, e_{2}, e_{3}, e_{4}}

has no perfect matchings as the two vertices, x, y, cannot be contained in the same perfect matching. If

F - {e_{1}, e_{2}, e_{3}, e_{4}}

is no connected, then we obtain a cut edge,

u z

, in F, contradicting Lemma 1. Then, we find an anti−Kekulé set of size 4, and so

a k (F) \leq 4

In the following, we show

a k (F) \geq 3

. Let A be an anti-Kekulé set of size

a k (F)

. Then

F^{'} = F - A

is connected and has no perfect matchings. According to Theorem 3, there exists

S \subseteq V (F^{'})

such that

c_{0} (F^{'} - S) > | S |

. If we choose such an S with the maximum size, then

F^{'} - S

has no even components. On the contrary, suppose that

F^{'} - S

has an even component, H. For any vertex

v \in V (H)

c_{0} (H - v) \geq 1

. Let

S^{'} = S \cup {v}

, then

c_{0} (F^{'} - S^{'}) \geq c_{0} (F^{'} - S) + 1 > | S | + 1 = | S^{'} |

, which is a contradiction to the choice of S.

Since

| V (F^{'}) |

is even, then

c_{0} (F^{'} - S) \geq | S | + 2

by parity. For any edge

e \in A

, adding e to

F^{'} - S

will connect at most two odd components, then

c_{o} (F^{'} + e - S) \geq c_{0} (F^{'} - S) - 2

. Since A is the smallest anti-Kekulé set of F, then

F^{'} + e

has a perfect matching for any edge

e \in A

. Hence, by Theorem 3, for the above subset S,

c_{o} (F^{'} + e - S) \leq | S |

. Therefore,

| S | \geq c_{o} (F^{'} + e - S) \geq c_{0} (F^{'} - S) - 2 \geq | S |

. We obtain

c_{0} (F^{'} - S) = | S | + 2

, and the edge, e, connects exactly two components of

F^{'} - S

Let

F_{i}

be the odd components of

F^{'} - S

, where

1 \leq i \leq | S | + 2

. For

F_{i} \subseteq F

, let

d (F_{i})

denote the number of the set of edges with one end in

F_{i}

and the other end in

F - F_{i}

. Denote the number of edges between S and the odd components by N. Since F is cubic, S sends out at most

3 | S |

to N. In addition,

⋃_{i = 1}^{| S | + 2} F_{i}

sends out exactly

\sum_{i = 1}^{| S | + 2} d (F_{i}) - 2 a k (F)

edges to N. Hence

N = \sum_{i = 1}^{| S | + 2} d (F_{i}) - 2 a k (F) \leq 3 | S | .

(3)

Because F is 2-edge-connected,

d (F_{i}) \geq 2

for every i. On the other hand,

d (F_{i}) = 3 | V (F_{i}) | - 2 | E (F_{i}) |

, which implies that

d (F_{i})

and

| V (F_{i}) |

are of the same parity. Every

F_{i}

sends odd number edges, hence

d (F_{i}) \geq 3

. Substituting it into (3), we have

3 (| S | + 2) - 2 a k (F) \leq \sum_{i = 1}^{| S | + 2} d (F_{i}) - 2 a k (F) \leq 3 | S |,

the above inequality gives

a k (F) \geq 3

We find that the anti-Kekulé number of F is either 3 or 4. Suppose, on the contrary, that

a k (F) = 3

. Then there exists an anti-Kekulé set,

A = {e_{1}, e_{2}, e_{3}}

, of cardinality three in F, such that

F - A

is connected and has no perfect matchings. Assume W and B are the bipartition of F. By Hall’s theorem, there exists

U \subseteq W

such that

| N_{F - A} (U) | \leq | U | - 1

(4)

where

N_{F - A} (U)

means

N (U)

F - A

. Moreover, for

e_{i} \in A

, since A is the smallest anti-Kekulé set,

F - A + e_{i}

has a perfect matching. Immediately, by Theorem 4, for the above subset U,

| U | \leq | N_{F - A + e_{i}} (U) |

(5)

for

i = 1, 2

, and 3, where

N_{F - A + e_{i}} (U)

means

N (U)

F - A + e_{i}

. In addition, the neighbors of U will be increased by at most one if we add an edge,

e_{i}

, to

F - A

. Hence

| N_{F - A + e_{i}} (U) | \leq | N_{F - A} (U) | + 1 .

(6)

Combining inequalities (4)–(6), we have

| U | = | N_{F - A} (U) | + 1

, and

e_{i}

is incident with the vertices of U and

B - N_{F - A + e_{i}} (U)

F - A + e_{i}

. Thus, the edges going out from

U \subseteq V (F)

either go into A or go into the edges going out from

N_{F - A} (U)

. Then, the number of edges between U and

N_{F - A} (U)

3 | U | - 3

. Since

| N_{F - A} (U) | = | U | - 1

3 | N_{F - A} (U) | = 3 (| N_{F - A} (U) | + 1) - 3 = 3 | U | - 3

, that is, there is no edge between

N_{F - A} (U)

and

W - U

F - A

. As a result, A is an edge-cut, which is a contradiction to the definition of an anti-Kekulé set. □

In [23], Grünbaum and Motzkin showed that

(5, 6)

-fullerene and

(4, 6)

-fullerene having n hexagonal faces exist for every non-negative integer, n, satisfying

n \neq 1

, and gave a similar result for

(3, 6)

-fullerene. Therefore, we consider whether

(2, 6)

-fullerene having n hexagonal faces also exists for any n. We tried to give a positive answer to this question, but we found that the conclusion seems quite elusive. Therefore, in this part, we mainly prove that there exists a

(2, 6)

-fullerene F having

f_{F}

hexagonal faces, where

f_{F}

is related to the two parameters n and m.

Let F be a

(2, 6)

-fullerene. A

f r a g m e n t

, H, of F is a subgraph of F consisting of a cycle together with its interior and every inner face of H is also a face of F. We define

\partial (H)

as the

b o u n d a r y

of the exterior face of H. A face, f, of F is a

n e i g h b o r i n g

f a c e

of H if f is not a face of H and f has at least one edge in common with H. A path of length k (the number of edges) is called a k-

p a t h

. Denote by

f_{H}

the number of hexagons of H.

Proposition 1.

In all the

(2, 6)

-fullerenes, there exists a fragment, say

G_{n}

, such that

f_{G_{n}} = n^{2} + n

n \in Z

Proof.

Let

G_{0}

be a 2-length face and

f_{11}

and

f_{12}

be two neighboring faces of

G_{0}

(see Figure 2a). Then

f_{G_{0}} = 0

. Suppose that

f_{11}

and

f_{12}

are hexagons. Set

G_{1} = G_{0} ⋃ {f_{11}, f_{12}}

, suppose both

f_{11}

and

f_{12}

are inner faces of

G_{1}

, and let

f_{21}, f_{22}, f_{23}

, and

f_{24}

be four neighboring faces of

G_{1}

along the clockwise direction, such that

f_{21}

is incident with the two consecutive 2-degree vertices on

\partial (G_{1})

(see Figure 2b). Then

f_{G_{1}} = 2

. Suppose that

f_{21}, f_{22}, f_{23}

, and

f_{24}

are hexagons, pairwise different, and intersecting if and only if

f_{2 i}

and

f_{2, i + 1}

are intersecting at only one edge for

i = 1, 2, 3, 4

f_{25} = f_{21}

. Set

G_{2} = G_{1} ⋃ {f_{21}, f_{22}, f_{23}, f_{24}}

. Suppose

f_{21}, f_{22}, f_{23}

, and

f_{24}

are the inner faces of

G_{2}

, and let

f_{31}, f_{32}, f_{33}, f_{34}, f_{35}

, and

f_{36}

be six neighboring faces of

G_{2}

along the clockwise direction, such that

f_{31}

is incident with the two consecutive 2-degree vertices on

\partial (G_{2})

(see Figure 2c). Then

f_{G_{2}} = 2 + 4 = 6

Suppose that the proposition holds for any integer less than n, where

n > 2

. According to the induction hypothesis,

f_{G_{n - 1}} = n^{2} - n

and

f_{n 1}, f_{n 2}, \dots, f_{n, 2 n}

are

2 n

neighboring faces of

G_{n - 1}

along the clockwise direction, such that

f_{n 1}

is incident with the two consecutive 2-degree vertices on

\partial (G_{n - 1})

. Suppose that

f_{n 1}, \dots, f_{n, 2 n}

are hexagons, pairwise different, and intersecting if and only if

f_{n i}

and

f_{n, i + 1}

are intersecting at only one edge for

i = 1, 2, \dots, 2 n

f_{n, 2 n + 1} = f_{n 1}

. Set

G_{n} = G_{n - 1} ⋃ {f_{n 1}, \dots, f_{n, 2 n}}

. Suppose

f_{n 1}, \dots, f_{n, 2 n}

are all inner faces of

G_{n}

(see Figure 3). Then

f_{G_{n}} = n^{2} - n + 2 n = n^{2} + n

n \in Z

. □

Proposition 2.

In all the

(2, 6)

-fullerenes, there exists a fragment, say

C_{n}

, such that

f_{C_{n}} = n

n \in Z

Proof.

Let

C_{0}

3 \times K_{2}

, then

f_{C_{0}} = 0

. Let

d_{1}

and

d_{2}

be two 2-length faces. Its boundary,

\partial (d_{i})

, is labelled

v_{i 1}

v_{i 2}

(

i = 1, 2

). Let

P_{i}

be a path that connects two vertices,

v_{1 i}

and

v_{2 i}

(

i = 1, 2

), and

V (P_{1}) ⋂ V (P_{2}) = \emptyset

. If both

P_{1}

and

P_{2}

are 2-paths, then, as F is 2-connected, there is a hexagon, say

f_{1}

, such that

f_{1}

contains the paths

P_{1}

and

P_{2}

and the edges

v_{11} v_{12}

and

v_{21} v_{22}

. Set

C_{1} = d_{1} ⋃ d_{2} ⋃ f_{1}

, without loss of generality, suppose

f_{1}

is the inner face of

C_{1}

(see Figure 4a). Thus,

f_{C_{1}} = 1

. If both

P_{1}

and

P_{2}

are 4-paths, then all whose internal vertices are denoted by

x_{1}, x_{2}, x_{3}

and

y_{1}, y_{2}, y_{3}

, respectively, such that

P_{1} = v_{11} x_{1} x_{2} x_{3} v_{21}

and

P_{2} = v_{12} y_{1} y_{2} y_{3} v_{22}

. Let

x_{2} y_{2} \in E (F)

, then there are 2 hexagons, denoted by

f_{1}

and

f_{2}

, such that

\partial (f_{1}) = v_{11} v_{12} y_{1} y_{2} x_{2} x_{1} v_{11}

and

\partial (f_{2}) = v_{21} x_{3} x_{2} y_{2} y_{3} v_{22} v_{21}

. Set

C_{2} = d_{1} ⋃ d_{2} ⋃ f_{1} ⋃ f_{2}

, also suppose

f_{1}

and

f_{2}

are two inner faces of

C_{2}

(see Figure 4b), then

f_{C_{2}} = 2

. Suppose

P_{1}

and

P_{2}

are

2 n

-paths,

n \in N^{+}

. Let

P_{1} = v_{11} x_{1} \dots x_{2 n - 1} v_{21}

and

P_{2} = v_{12} y_{1} \dots y_{2 n - 1} v_{22}

. Suppose that

x_{i} y_{i} \in E (F)

(

i = 2, 4, \dots, 2 n - 2

), then there are n hexagons between

P_{1}

and

P_{2}

, denoted by

f_{1}, f_{2} \dots f_{n}

. Set

C_{n} = d_{1} ⋃ d_{2} ⋃_{i = 1}^{n} f_{i}

, such that

f_{1}, f_{2} \dots f_{n}

are the inner faces of

C_{n}

(see Figure 4c). Therefore,

C_{n}

is a fragment and

f_{C_{n}} = n

n \in N^{+}

. Thus, there exists a fragment,

C_{n}

, such that

f_{C_{n}} = n

n \in Z

. □

Proposition 3.

In all the

(2, 6)

-fullerenes, there exists a fragment, say

L_{n}^{m}

, such that

f_{L_{n}^{m}} = 2 n^{2} + (m + 3) n

n \in N^{+}

m \in Z

Proof.

Let

G_{n}^{^{'}}

and

G_{n}^{^{''}}

be two fragments, as indicated in Figure 3. By Proposition 1, we know that

G_{n}^{^{'}}

and

G_{n}^{^{''}}

both have

n^{2} + n

hexagons. Suppose n is a positive integer. Since there are

2 n + 2

2-degree vertices on

\partial (G_{n}^{^{'}})

, we can record them clockwise as

u_{1}, u_{2}, \dots, u_{2 n + 2}

, such that

u_{1}

and

u_{2 n + 2}

are adjacent. Similarly,

2 n + 2

2-degree vertices on

\partial (G_{n}^{^{''}})

are denoted by

v_{1}, v_{2}, \dots, v_{2 n + 2}

along the anticlockwise direction of

G_{n}^{^{''}}

, such that

v_{1}

and

v_{2 n + 2}

are adjacent. For

G_{1}^{^{'}}

and

G_{1}^{^{''}}

. Let

e_{1} = u_{1} v_{1}, e_{2} = u_{2} v_{2}

, then

e_{1}

and

e_{2}

are contained in the hexagon, say

f_{1}

. Set

K_{1} = G_{1}^{^{'}} ⋃ G_{1}^{^{''}} ⋃ f_{1}

(see Figure 5a), then

f_{K_{1}} = 5

. For

G_{n}^{^{'}}

and

G_{n}^{^{''}}

. Let

e_{i} = u_{i} v_{i}

i = 1, 2 \dots n + 1

, then

e_{i}

and

e_{i + 1}

are contained in the hexagon, say

f_{i}

i = 1, 2 \dots n

. Set

K_{n} = G_{n}^{^{'}} ⋃ G_{n}^{^{''}} ⋃_{i = 1}^{n} f_{i}

, suppose all of

f_{1} \dots f_{n}

are the inner faces of

K_{n}

(see Figure 5b, the embedding of

K_{n}

), then

f_{K_{n}} = 2 (n^{2} + n) + n = 2 n^{2} + 3 n

Next, we construct the fragment

L_{n}^{m}

from

K_{n}

as follows. We replace each edge,

e_{i} = u_{i} v_{i}

, by a path,

P_{i}

, such that

P_{i} = u_{i} x_{i 1} x_{i 2} \dots x_{i, 2 m} v_{i}

i = 1, 2 \dots n + 1

m \in Z

. Suppose that

x_{i 2} x_{i + 1, 1}, x_{i 4} x_{i + 1, 3} \dots x_{i, 2 m} x_{i + 1, 2 m - 1}

be the edges of F,

i = 1, 2 \dots n

. Therefore, there are

m + 1

hexagons between

P_{i}

and

P_{i + 1}

, denoted by

f_{i 1}, f_{i 2} \dots f_{i, m + 1}

i = 1, 2 \dots n

. Set

L_{n}^{m} = G_{n}^{^{'}} ⋃ G_{n}^{^{''}} ⋃_{i = 1}^{n} {f_{i 1}, f_{i 2} \dots f_{i, m + 1}}

m \in Z

(see Figure 5c, the embedding of

L_{n}^{m}

). Therefore,

L_{n}^{m}

is a fragment and

f_{L_{n}^{m}} = 2 (n^{2} + n) + (m + 1) n = 2 n^{2} + (m + 3) n

n \in N^{+}

m \in Z

. □

Proposition 4.

In all the

(2, 6)

-fullerenes, there exists a fragment, say

H_{n}

, such that

f_{H_{n}} = 2 n + 2

n \in Z

Proof.

Let

H_{0} ≅ F_{6}

be the

(2, 6)

-fullerene with six vertices. Without loss of generality, suppose the exterior face of

H_{0}

is a 2-length face with the boundary

u_{1} v_{1} u_{1}

, and the remaining two 2-length faces are connected by an edge,

u_{2} v_{2}

(see Figure 6a the embedding of

H_{0}

and the labelling of

u_{1}, v_{1}, u_{2}, v_{2}

). Next, we construct the fragment

H_{n}

from

H_{0}

as follows: we replace the two parallel edges,

u_{1} v_{1}

, and one edge,

u_{2} v_{2}

, by two paths,

P_{1}

and

P_{3}

, and one path,

P_{2}

, such that

P_{i} = u_{i} x_{i 1} x_{i 2} \dots x_{i, 2 n} v_{i}

i = 1, 2

, and

P_{3} = u_{1} x_{31} x_{32} \dots x_{3, 2 n} v_{1}

n \in N^{+}

. Suppose that

x_{i 2} x_{i + 1, 1}, x_{i 4} x_{i + 1, 3} \dots x_{i, 2 n} x_{i + 1, 2 n - 1}

be the edges of F,

i = 1, 2

. We construct

n + 1

hexagons between

P_{i}

and

P_{i + 1}

, denoted by

f_{i 1}, f_{i 2} \dots f_{i, n + 1}

i = 1, 2

. Set

H_{n} = H_{0} - {u_{1} v_{1}, u_{2} v_{2}} ⋃_{i = 1}^{2} {f_{i 1}, f_{i 2} \dots f_{i, n + 1}}

, such that

f_{i 1}, f_{i 2} \dots f_{i, n + 1}

are the inner faces of

H_{n}

n \in N^{+}

(see Figure 6b). Therefore,

H_{n}

is a fragment and

f_{H_{n}} = 2 (n + 1) = 2 n + 2

n \in N^{+}

. Thus, there exists a fragment,

H_{n}

, such that

f_{H_{n}} = 2 n + 2

n \in Z

. □

By Propositions 1–4, we can find a

(2, 6)

-fullerene F having

f_{F}

hexagonal faces that relates to the parameters n and m.

Theorem 7.

There exists a

(2, 6)

-fullerene F such that

f_{F} = n^{2} + 2 n

n \in Z

Proof.

Let

G_{n}

be a fragment of F, as shown in Figure 3. Its boundary,

\partial (G_{n})

, is labelled

u_{1}, u_{2}, \dots, u_{4 n + 2}

along the clockwise direction, where

u_{1}

and

u_{2}

are two consecutive 2-degree vertices. Let

C_{n}

be a fragment of F, as shown in Figure 4c. Its boundary,

\partial (C_{n})

, is labelled

v_{1}, v_{2}, \dots, v_{4 n + 2}

along the clockwise direction, where

v_{1}

and

v_{2}

are two consecutive 3-degree vertices. Next, we assume each of the graphs

G_{n}

and

C_{n}

drawn on a hemisphere, with the boundary as equator. If

\partial (G_{n}) = \partial (C_{n})

, then set

F = G_{n} ⋃ C_{n}

. By Propositions 1 and 2, then

f_{F} = f_{G_{n}} + f_{C_{n}} = n^{2} + 2 n

n \in Z

. □

Theorem 8.

There exists a

(2, 6)

-fullerene F such that

f_{F} = 3 n^{2} + m^{2} + 3 m n + 6 n + 3 m + 2

n \in N^{+}

m \in Z

Proof.

Let

G_{m + n + 1}

be a fragment of F, as shown in Figure 3. Its boundary,

\partial (G_{m + n + 1})

, is labelled

u_{1}, u_{2}, \dots, u_{4 m + 4 n + 6}

along the clockwise direction, where

u_{1}

and

u_{2}

are two consecutive 2-degree vertices. Let

L_{n}^{m}

be a fragment of F, as shown in Figure 5c. Its boundary,

\partial (L_{n}^{m})

, is labelled

v_{1}, v_{2}, \dots, v_{4 m + 4 n + 6}

along the clockwise direction, where

v_{1}

and

v_{2}

are two consecutive 3-degree vertices. Next, we assume each of the graphs

G_{m + n + 1}

and

L_{n}^{m}

drawn on a hemisphere, with the boundary as equator. If

\partial (G_{m + n + 1}) = \partial (L_{n}^{m})

, then set

F = G_{m + n + 1} ⋃ L_{n}^{m}

. By Propositions 1 and 3, then

f_{F} = f_{G_{m + n + 1}} + f_{L_{n}^{m}} = 3 n^{2} + m^{2} + 3 m n + 6 n + 3 m + 2

n \in N^{+}

m \in Z

. □

Theorem 9.

There exists a

(2, 6)

-fullerene F such that

f_{F} = n^{2} + 3 n + 2

n \in Z

Proof.

Let

G_{n}

be a fragment of F, as shown in Figure 3. Its boundary,

\partial (G_{n})

, is labelled

u_{1}, u_{2}, \dots, u_{4 n + 2}

along the clockwise direction, where

u_{1}

and

u_{2}

are two consecutive 2-degree vertices. Let

H_{n}

be a fragment of F, as shown in Figure 6b. Its boundary,

\partial (H_{n})

, is labelled

v_{1}, v_{2}, \dots, v_{4 n + 2}

along the clockwise direction, where

v_{1}

and

v_{2}

are two consecutive 3-degree vertices. Next, we assume each of the graphs

G_{n}

and

H_{n}

drawn on a hemisphere, with the boundary as equator. If

\partial (G_{n}) = \partial (H_{n})

, then set

F = G_{n} ⋃ H_{n}

. By Propositions 1 and 4, then

f_{F} = f_{G_{n}} + f_{H_{n}} = n^{2} + 3 n + 2

n \in Z

. □

3. Conclusions

In this paper, we characterized the structural properties of

(2, 6)

-fullerene F. By the conclusion of Došlić on

(k, 6)

-fullerenes, we know that every

(2, 6)

-fullerene is 1-extendable, and then, by means of the relationship between the extendable and resonance, we know that every

(2, 6)

-fullerene is 1-resonant. But, by the definition of n-extendable, we find that every

(2, 6)

-fullerene is not 2-extendable, and then not k-extendable (

k \geq 3

). Moreover, we find that no

(2, 6)

-fullerene is bicritical. On the other hand, for the anti−Kekulé number of

(2, 6)

-fullerene F, we firstly obtain the upper bound 4 of the anti−Kekulé number of F by the definition of an anti−Kekulé set. Secondly, with the help of Tutte’s Theorem, and combing with the structure of

(2, 6)

-fullerene, we obtain the lower bound 3 of the anti−Kekulé number of F. Finally, by analyzing that the anti−Kekulé number of F cannot be 3, we find that the anti−Kekulé number of F is 4.

Grünbaum and Motzkin showed that

(5, 6)

-fullerene and

(4, 6)

-fullerene having n hexagonal faces exist for every non-negative integer, n, satisfying

n \neq 1

, and showed a similar result for

(3, 6)

-fullerene. Therefore, at the end of the paper, we consider whether

(2, 6)

-fullerene having n hexagonal faces also exists for any n. We try to give a positive answer to this question, but we find that the conclusion seems quite elusive. So, we mainly prove that there exists a

(2, 6)

-fullerene F having

f_{F}

hexagonal faces, where

f_{F}

is related to the two parameters n and m. There are, however, still several important open questions.

Problem 1.

Whether every

(2, 6)

-fullerene is k-resonant

(k \geq 2)

Problem 2.

Whether for every

n \neq 1

there exists a

(2, 6)

-fullerene F such that

f_{F} = n

Author Contributions

Methodology, R.Y.; Writing & original draft, M.Y.; Review & editing, R.Y. and M.Y.; Supervision, R.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported by NSFC (grant no.11801148 and 11626089) and the Foundation for the Doctor of Henan Polytechnic University (grant no. B2014-060).

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and useful comments, which helped improve the presentation of the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

References

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Figure 1. The

(2, 6)

-fullerenes

3 \times K_{2}

(a),

F_{6}

(b).

Figure 1. The

(2, 6)

-fullerenes

3 \times K_{2}

(a),

F_{6}

(b).

Figure 2. The fragments

G_{0}

(a),

G_{1}

(b), and

G_{2}

(c).

Figure 2. The fragments

G_{0}

(a),

G_{1}

(b), and

G_{2}

(c).

Figure 3. The fragment

G_{n}

Figure 3. The fragment

G_{n}

Figure 4. The fragments

C_{1}

(a),

C_{2}

(b), and

C_{n}

(c).

Figure 4. The fragments

C_{1}

(a),

C_{2}

(b), and

C_{n}

(c).

Figure 5. The fragments

K_{1}

(a),

K_{n}

(b), and

L_{n}^{m}

(c).

Figure 5. The fragments

K_{1}

(a),

K_{n}

(b), and

L_{n}^{m}

(c).

Figure 6. The fragments

H_{0}

(a) and

H_{n}

(b).

Figure 6. The fragments

H_{0}

(a) and

H_{n}

(b).

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Yang, R.; Yuan, M. The Structural Properties of (2, 6)-Fullerenes. Symmetry 2023, 15, 2078. https://doi.org/10.3390/sym15112078

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Yang R, Yuan M. The Structural Properties of (2, 6)-Fullerenes. Symmetry. 2023; 15(11):2078. https://doi.org/10.3390/sym15112078

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Yang, Rui, and Mingzhu Yuan. 2023. "The Structural Properties of (2, 6)-Fullerenes" Symmetry 15, no. 11: 2078. https://doi.org/10.3390/sym15112078

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Yang, R., & Yuan, M. (2023). The Structural Properties of (2, 6)-Fullerenes. Symmetry, 15(11), 2078. https://doi.org/10.3390/sym15112078

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The Structural Properties of (2, 6)-Fullerenes

Abstract

1. Introduction

2. Main Results

3. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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