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a _ fullerene _  @xcite is a cubic plane graph where all faces are pentagons or hexagons .
euler s formula implies that a fullerene with @xmath4 vertices contains exactly 12 pentagons and @xmath5 hexagons .
the _ dual _ of a fullerene is the plane graph obtained by exchanging the roles of vertices and faces : the vertex set of the dual graph is the set of faces of the original graph and two vertices in the dual graph are adjacent if and only if the two faces share an edge in the original graph .
the dual of a fullerene with @xmath4 vertices is a _ triangulation _
( i.e.  a plane graph where every face is a triangle ) which contains 12 vertices with degree 5 and @xmath5 vertices with degree 6
. the _ face - distance _ between two pentagons is the distance between the corresponding vertices of degree 5 in the dual graph .
the first fullerene molecule ( i.e.  the @xmath6 `` buckyball '' ) was discovered in 1985 by kroto et al .
@xcite . among the fullerenes , the _ isolated pentagon rule _ ( ipr )
fullerenes are of special interest as they tend to be more stable  @xcite .
ipr fullerenes are fullerenes where no two pentagons share an edge , i.e.  they have minimum face - distance at least  2 .
raghavachari  @xcite argued that steric strain will be minimized when the pentagons are distributed as uniformly as possible and therefore proposed the _ uniform curvature rule _ as an extension of the ipr rule .
also , more recently rodrguez - fortea et al .
@xcite proposed the maximum pentagon separation rule where they argue that the most suitable carbon cages are those with the largest separations among the 12 pentagons .
these observations lead us to investigate the maximum separation between pentagons that can be achieved for a given number of atoms , or conversely how many atoms are needed to achieve a given separation .
we will refer to the least face - distance between pentagons of a fullerene as the _ pentagon separation _ of the fullerene .
in the next section we determine the smallest fullerenes with a given pentagon separation .
we also show that the minimum fullerenes for each @xmath1 are unique up to mirror image and that for each @xmath1 there is an @xmath2 such that fullerenes with pentagon separation at least @xmath1 and any number of hexagons greater than or equal to @xmath2 exist .
the latter was already proven for @xmath7 ( i.e. , for all fullerenes ) by grnbaum and motzkin in  @xcite and for @xmath8 ( i.e. , for ipr fullerenes ) by klein and liu in  @xcite .
finally , we also determine the number of fullerenes of pentagon separation  @xmath1 , for @xmath3 , up to 400 vertices .
in this section we determine the smallest fullerenes with a given pentagon separation .
we remind the reader of the icosahedral fullerenes  @xcite .
these fullerenes are uniquely determined by their coxeter coordinates @xmath9 and are obtained by cutting an equilateral goldberg triangle with coordinates @xmath9 from the hexagon lattice and gluing it to the faces of the icosahedron . as a goldberg triangle with coordinates @xmath9 has @xmath10 vertices , an icosahedral fullerene with coxeter coordinates @xmath9 has @xmath11 vertices . also note that an icosahedral fullerene with coxeter coordinates @xmath9 has pentagon separation  @xmath12 .
[ theorem : min_face_distance_nv ] for odd @xmath15 , the smallest fullerenes with pentagon separation at least @xmath1 are the icosahedral fullerenes with coxeter coordinates @xmath16 and @xmath17 .
these are mirror images and have @xmath18 vertices . for even @xmath1 , the unique smallest fullerene with pentagon separation
at least @xmath1 is the the icosahedral fullerene with coxeter coordinates @xmath19 , which has @xmath20 vertices .
* proof in the case that @xmath15 is odd : * + the _ penta - hexagonal net _ is the regular tiling of the plane where a central pentagon is surrounded by an infinite number of hexagons .
the number of faces at face - distance @xmath21 from the pentagon in the penta - hexagonal net is @xmath22 .
so the number of faces at face - distance at most @xmath21 from the pentagon in the penta - hexagonal net is @xmath23 .
figure  [ fig : d=7patch ] shows this situation for @xmath24 .      in a fullerene with pentagon separation at least  @xmath1 , for odd  @xmath1
, the sets of faces at face - distance at most @xmath25 from each pentagon are pairwise disjoint .
consequently the smallest such fullerenes we can hope to find consist of 12 copies of the above patch for @xmath26 , which comes to @xmath18 vertices .
since the patch boundary has no more than two consecutive vertices of degree  2 , it is impossible to join any number of them into a larger patch with a boundary having more than two consecutive vertices of degree  2 .
therefore , considering the complement , no union of these patches which is completable to a fullerene has more than two consecutive vertices of degree  3 .
now , every way to overlap the boundaries of two patches produces three consecutive vertices of degree  3 , such as indicated in the left side of figure  [ fig : d=7patchjoin ] , except for the way shown in the right side of figure  [ fig : d=7patchjoin ] or its mirror image . for each of these two starting points , there is only one way to attach a third patch to those two patches , and so on , leading to a unique completion in each case .
it is easy to see that these two fullerenes are the icosahedral fullerenes mentioned in the theorem .
* proof in the case that @xmath1 is even : * + the proof in this case is similar except that we use a different type of patch . in
@xcite it was proven that the number of vertices at distance @xmath21 from the pentagon in the penta - hexagonal net is @xmath27 .
so the total number of vertices at distance at most @xmath21 from the pentagon in the penta - hexagonal net is @xmath28 . if @xmath21 is even , @xmath29 is equal to @xmath30 .
so the total number of vertices at distance at most @xmath21 from the pentagon in the penta - hexagonal net for even @xmath21 is @xmath31 .    in a fullerene with pentagon separation at least @xmath1 , for even @xmath1 ,
the sets of vertices at distance at most @xmath32 from every pentagon are pairwise disjoint .
the case of @xmath33 is shown in figure  [ fig : d=6patch ] , excluding the ends of the dangling edges .
therefore , the smallest such fullerene of pentagon separation @xmath1 we can hope to construct consists of 12 of these patches for @xmath34 , joined together by identifying dangling edges .
this would give us @xmath20 vertices altogether .
since we are only permitted to create hexagons incident with the dangling edges , dangling edges distance two apart in one patch can only be identified with dangling edges distance two apart in another patch .
otherwise , a face of the wrong size is created , such as the pentagon indicated in the left side of figure  [ fig : d=6patchjoin ] .
this allows us to join two adjacent patches in only one way , as shown by the right side of figure  [ fig : d=6patchjoin ] .
extra patches can then be attached in unique fashion , leading to a single fullerene that is easily seen to be the icosahedral fullerene with coxeter coordinates @xmath19 .
next we will prove that for each @xmath1 there is an @xmath2 such that fullerenes with pentagon separation at least @xmath1 and any number of hexagons greater than or equal to @xmath2 exist . to prove this , we need lemmas  [ lemma : boundarylength_plus1 ] and  [ lemma : add_hexagons_same_boundary ] .    a _
fullerene patch _ is a connected subgraph of a fullerene where all faces except one exterior face are also faces in the fullerene and all boundary vertices have degree 2 or 3 and all non - boundary vertices have degree 3 .
the _ boundary sequence _ of a patch is the cyclic sequence of the degrees of the vertices in the boundary of a patch in clockwise or counterclockwise order .
a _ cap _ is a fullerene patch which contains 6 pentagons and has a boundary sequence of the form @xmath35 .
such a boundary is represented by the parameters @xmath36 . in the literature , the vector @xmath36
is also called the _ chiral vector _ ( see  @xcite ) .        in figure
[ fig : change_cap_bound ] we show how the @xmath37 cap which contains a boundary pentagon ( see figure  [ fig : change_cap_bound_step1 ] ) can be transformed into a cap with parameters @xmath38 without decreasing the minimum face - distance between the pentagons .
this is done by changing the boundary pentagon into a hexagon @xmath39 , adding a ring of hexagons ( see figure  [ fig : change_cap_bound_step2 ] ) and changing a hexagon in the boundary which is adjacent to @xmath39 into a pentagon ( see figure  [ fig : change_cap_bound_step3 ] ) .
[ lemma : add_hexagons_same_boundary ] given a cap @xmath40 with parameters @xmath36 with @xmath41 and @xmath42 and which consists of @xmath43 faces . a cap @xmath44 with the same parameters @xmath36 which contains @xmath40 as a subgraph and has @xmath45 , respectively @xmath46 faces can be constructed from @xmath40 by adding @xmath47 or @xmath48 hexagons to c , respectively .    given a cap @xmath40 with parameters @xmath36 with @xmath41 and @xmath42 . in figure
[ fig : add_hexagons_same_cap_step ] we show how a cap @xmath44 with the same parameters @xmath36 which contains @xmath40 as a subgraph and has @xmath45 faces can be constructed from @xmath40 by adding @xmath47 hexagons to c.        [ theorem : min_face_distance_existence ] for each @xmath1 there is an @xmath2 such that fullerenes with pentagon separation at least @xmath1 and any number of hexagons greater than or equal to @xmath2 exist .
brinkmann and schein  @xcite have proven that every icosahedral fullerene with coxeter coordinates @xmath9 contains a fullerene patch with 6 pentagons which is a subgraph of a cap with parameters @xmath53 .
so @xmath50 contains a fullerene patch with 6 pentagons which is a subgraph of a cap with parameters @xmath54 .
it follows from  @xcite that such a fullerene patch can be completed to a cap with parameters @xmath54 by adding hexagons .
it follows from lemma  [ lemma : boundarylength_plus1 ] that this cap can then be transformed to a cap with parameters @xmath55 without decreasing the minimum face - distance between the pentagons of the cap .
we form a fullerene @xmath56 with pentagon separation at least @xmath1 by gluing together two copies of the @xmath55 cap and adding @xmath55 rings of hexagons if necessary .
let @xmath57 denote the number of hexagons of @xmath56 .
now a fullerene with pentagon separation at least @xmath1 and any number of hexagons greater than @xmath57 can be obtained by recursively applying lemma  [ lemma : add_hexagons_same_boundary ] to @xmath56 .
the counts of the number of fullerenes up to 400 vertices with pentagon separation at least @xmath1 , for @xmath3 , can be found in tables  [ table : fuller_counts_1]-[table : fuller_counts_4 ] .
( note that @xmath13 gives the set of all fullerenes and @xmath58 gives the set of all ipr fullerenes ) .
these counts were obtained by using the program _ buckygen _
@xcite ( which can be downloaded from http://caagt.ugent.be/buckygen/ ) to generate all non - isomorphic ipr fullerenes and then applying a separate program to compute their pentagon separation .
note that fullerenes which are mirror images of each other are considered to be in the same isomorphism class and are thus only counted once .
some of the fullerenes from tables  [ table : fuller_counts_1]-[table : fuller_counts_4 ] can also be downloaded from the _ house of graphs _
@xcite at http://hog.grinvin.org/fullerenes  .
figures [ fig : smallest_d=3]-[fig : smallest_d=5 ] show the smallest fullerenes with pentagon separation @xmath1 , for @xmath59 .          _
acknowledgements : _ jan goedgebeur is supported by a postdoctoral fellowship of the research foundation flanders ( fwo ) .
brendan mckay is supported by the australian research council .
most computations for this work were carried out using the stevin supercomputer infrastructure at ghent university .
we also would like to thank gunnar brinkmann , patrick fowler and jack graver for useful suggestions .
for each @xmath0 , we find all the smallest fullerenes for which the least distance between two pentagons is  @xmath1 . we also show that for each @xmath1 there is an @xmath2 such that fullerenes with pentagons at least distance @xmath1 apart and any number of hexagons greater than or equal to @xmath2 exist . 
 we also determine the number of fullerenes where the minimum distance between any two pentagons is at least @xmath1 , for @xmath3 , up to 400 vertices .    
 ( received )    = 0.30 in