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Fowler and Pisanski showed that the Fries number for a fullerene on surface Σ is bounded above by

Several possibilities of generalizations of the fullerene cages have become a research interest soon after establishing fullerene research itself. Fullerenes are closed carbon-cage molecules made up solely of pentagons and hexagons. One generalization is to consider such trivalent structures with faces of other sizes, that is, fulleroid [

The concepts of the Fries number and the Clar number were named after Fries [

Both Clar number and Fries number have their chemical significance. The Fries number was proposed as a stability index for hydrocarbon in the early decades of the nineteenth century, whereas the Clar number was later considered as another stability index for hydrocarbon in the 1960s. For buckminsterfullerene

The chemical significance of leapfrog transformation is that leapfrog cluster with at least one face of size

The paper is organized as follows. In Section

The following lemma is an extension of the result on the sphere in [

Let

Let

Zhang and Ye [

Let

Let

We say that a fullerene on surface

Let

Fowler [

Let

For any edge in a perfect matching

A

Fowler and Pisanski [

A map

If a map

Conversely, if

Illustration for the proof of Theorem

The following theorem is first presented by Fowler for spherical fullerenes [

Let

For a fullerene

The following theorem is a natural generalization of a similar result on planar graphs in [

Given a bipartite map

Since the set of faces in

It is well known that Eulerian triangulation of the plane is 3-colorable. This implies that, for a 3-regular planar graph

Let

The following theorem, presenting an upper bound of Clar number for (4,6)-fullerenes on surface

Let

Let

Since

Let

Let

Let

For any edge in a perfect matching

Now we are going to characterize the (4,6)-fullerenes on surface

A (4,6)-fullerene

Suppose that

Conversely, suppose that

Since a Klein-bottle fullerene (toroidal fullerene) coincides with a Klein-bottle (4,6)-fullerene (toroidal (4,6)-fullerene, resp.), we may use the results of the previous section to obtain more information about fullerenes on the Klein bottle and torus whose Clar numbers attain the upper bound in Theorem

Toroidal fullerenes (resp., Klein-bottle fullerenes) with girth at least 6 are classified into two (resp., five) classes by Thomassen [

(a)

Theorem

For a toroidal or a Klein-bottle fullerene

The following theorem reveals an interesting appearance of leapfrog transformation on Klein-bottle fullerenes.

The leapfrog fullerene of a bipartite Klein-bottle fullerene is nonbipartite, and the leapfrog fullerene of a nonbipartite Klein-bottle fullerene is bipartite.

We prove the first part of this theorem by presenting an odd cycle in the leapfrog fullerene of a bipartite Klein-bottle fullerene. Given a bipartite Klein-bottle fullerene

We prove the second part of this theorem by presenting a bipartition in the leapfrog fullerene of a nonbipartite Klein-bottle fullerene

Illustration for the first part of proof in Theorem

Illustration for the second part of proof in Theorem

Theorem

The following three theorems in [

Combining Theorem

For a toroidal fullerene

For a Klein-bottle fullerene

For a Klein-bottle fullerene

A map on a surface is called

Hitherto, we have already characterized the (4,6)-fullerenes on surface

A map on the projective plane is the antipodal quotient of a centrosymmetric spherical map, that is, its vertices, edges, and faces are obtained by identifying antipodal vertices, edges, and faces of the centrosymmetric spherical map. Maps on the projective plane are usually drawn inside a circular frame where antipodal boundary points are identified [

For an extremal fullerene

Expansion and reverse expansion.

Let

Since

Let

Theorem

For each quadrilateral face of

The following three theorems are the natural generalizations of the similar results on the sphere in [

Suppose

A

Let

The extremal fullerenes

The extremal fullerenes

Perfect diagonalization of a bipartite (4,6)-fullerene

The leapfrog fullerene of a bipartite (4,6)-fullerene on the projective plane is nonbipartite, and the leapfrog fullerene of a nonbipartite (4,6)-fullerene on the projective plane is bipartite.

Suppose

Suppose

For any cycle

Let

The vertices of

Illustration for the proof of Theorem

The following theorem presented by Mohar in [

Let

Theorem

The face chromatic number of a nonbipartite (4,6)-fullerene on the projective plane is 3, the face chromatic number of a bipartite (4,6)-fullerene on the projective plane is 4 except

The theorem holds directly from Theorem

Theorem

The iterated leapfrog fullerenes of (4,6)-fullerenes on the projective plane.

The illustration of constructing extremal projective fullerenes from an extremal projective (4,6)-fullerene.

Since there is a one-to-one correspondence from projective fullerenes to centrosymmetric spherical fullerenes, it is natural to study the relation between the projective fullerenes (resp., (4,6)-fullerenes) attaining maximum Fries number and the spherical fullerenes (resp., (4,6)-fullerenes) attaining maximum Fries number and the relation between extremal projective fullerenes (resp., (4,6)-fullerenes) and extremal spherical fullerenes (resp., (4,6)-fullerenes).

A map on the projective plane is a leapfrog graph if and only if the corresponding centrosymmetric spherical map is a leapfrog graph.

Suppose a map

Conversely, suppose the corresponding centrosymmetric spherical map

As a corollary of Theorems

A projective (4,6)-fullerene is extremal if and only if the corresponding centrosymmetric spherical (4,6)-fullerene is extremal.

Combining Corollary

A projective fullerene is extremal if and only if the corresponding centrosymmetric spherical fullerene is extremal.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by NSFC (Grant no. 11371180) and the Fundamental Research Funds for the Central Universities (no. lzujbky-2014-21).

_{60}. Kekulé counts versus stability

_{60}