Table Of ContentThe distance-3 graph of the Biggs-Smith graph
1
1
0 Italo J. Dejter
2 University of Puerto Rico
t Rio Piedras, PR 00936-8377
c
O [email protected]
0
1
Abstract
O] The distance-3 graph S3 of theBiggs-Smith graph S is shown tobe: (a)
a connected edge-disjoint union of 102 tetrahedra (copies of K4) and as
C
such the K4-ultrahomogeneous Menger graph of a self-dual (1024)-confi-
h. guration; (b) a union of 102 cuboctahedra, (copies of L(Q3)), with no
t twosuchcuboctahedrahavingacommonchordless4-cycle;(c)notaline
a graph. Moreover,S3 isshowntohaveaC-ultrahomogeneouspropertyfor
m
C = {K4}∪{L(Q3)} restricted to preserving a specific edge partition of
[ L(Q3)into2-paths,witheachtriangle(resp. edge)sharedbytwocopiesof
2 L(Q3)plusoneofK4(resp. fourcopiesofL(Q4)). Boththedistance-2and
distance-4 graphs, S2 and S4, of S appearin thecontextassociated with
2
v the above mentioned edge partition. This takes us to ask whether there
8 anynon-line-graphicalconnectedK4-ultrahomogeneousMengergraphsof
8 self-dual (n4)-configurations that are edge-disjoint unions of at least two
5 copies of K4 for positive integers n∈/ {42,102}.
0
.
2
1 Introduction
0
0
1 Given a (di)graph Γ and a positive integer k diameter(Γ), the distance-k
≤
: graph Γk of Γ has V(Γk) = V(Γ) and an arc in Γk from a vertex u to a vertex
v
i v =uwheneverthereisashortestk-arcoflengthk inΓfromutov. Ak-arcin
X 6
a graph is a sequence of vertices (v ,...,v ) such that consecutive vertices are
0 k
r adjacent and v = v when 0 < i < k [9]. A k-arc can also be interpreted
a i−1 6 i+1
as a directed walk of length k in which consecutive edges are distinct [11].
Ultrahomogeneous(or UH) graphs were introducedand treated in [3, 8, 10, 14,
15], but we deal here with the following modified concept of ultrahomogeneity.
Given a collection of (di)graphs closed under isomorphisms, a (di)graph G is
C
said to be -UH if every isomorphism between two induced members of in G
C C
extends to an automorphism of G. If is the isomorphism class of a graph H,
C
thenG is saidto be H -UHorH-UH.If is the isomorphismclass ofagraph
{ } C
H with an edge partition Ω into 2-paths), then G is said to be Ω-preserving
H -UHorH-UHifonlyeveryΩ-preservingisomorphismbetweentwoinduced
{ }
copiesofH extends toanautomorphismofG. In[12], -UHgraphsaredefined
C
1
and studied when is the collection formed either by the complete graphs, or
C
the disjoint unions of complete graphs, or the complements of those disjoint
unions.
Let M be a subgraph of a graph H and let G be both an M-UH and an H-
UH graph. (A particular case has H = L(Q ) with an edge partition Ω into
3
2-paths, as for example the L(Q ) represented in Figure 4, Section 4 below,
3
with M equal to a 2-path in Ω). We say that G is an (Ω-preserving) H -
M
{ }
UH graph if, for each induced copy H of H in G containing an induced copy
0
M of M, there exists exactly one induced copy H = H of H in G with
0 1 0
6
V(H ) V(H ) = V(M ) and E(H ) E(H ) = E(M ). These vertex and
0 1 0 0 1 0
∩ ∩
edge conditions can be condensed as H H = M . We say that such a G is
0 1 0
∩
tightly fastened. This is generalizedby saying that an H -UH graphG is an
M
{ }
ℓ-fastened H -UH graph if given an induced copy H of H in G containing
M 0
{ }
an induced copy M of M, then there exist exactly ℓ induced copies H = H
0 i 0
6
ofH inGsuchthatH H M ,foreachi=1,2,...,ℓ,withH H =M .
i 0 0 1 0 0
∩ ⊇ ∩
Observe that in these definitions we started by assuming that G is M-UH and
H-UH.
Atransformationofdistance-transitivegraphsinto -UHgraphsthatgoesfrom
C
theCoxetergraphon28verticesintotheKleingraphon56vertices[7]isapplied
below to the Biggs-Smith graph [2, 4] seen as a C -UH graph. This
S { 9}P4
allows to show that the graphobtained by the said transformation, namely the
distance-3 graph 3 of , is a connected edge-disjoint union of 102 tetrahedra
S S
(copiesofK )aswellasaunionof102cuboctahedra,(copiesofL(Q ))withno
4 3
twosuchcuboctahedrahavingacommon4-hole(chordless4-cycle). Inaddition,
3 is shown to possess a -UH property for = K L(Q ) restricted to
4 3
S C C { }∪{ }
preserving a specific edge partition of L(Q ) into 2-paths, with each triangle
3
(resp. edge) of 3 shared by two copies of L(Q ) plus one of K (resp. four
3 4
S
copies of L(Q )) exactly. Moreover, 3 is seen to be the Menger graph [5] of a
3
S
self-dual (102 )-configuration,but not a line graph. Both 2 and 4 are shown
4
S S
to be related to the above mentioned edge partition.
Thus, the connected edge-disjoint union of 42 copies of K in [6], forming a
4
X
K -UH graph which is the Menger graph of a (42 )-configuration but is not a
4 4
linegraph,isnottheonlycaseofMengergraphofaself-dual(n )-configuration
4
which is connected edge-disjoint union of copies of K and K -UH but not a
4 4
line graph: 3 provides another such graph. This allows to raise the following
S
question.
Question 1 Are there any non-line-graphical connected K -UH Menger graphs
4
of self-dual (n )-configurations that are edge-disjoint unions of at least two
4
copies of K , for positive integers n / 42,102 ?
4
∈{ }
We recall that the line graph of the d-cube, is K -UH, where 3 d Z [6].
d
≤ ∈
Other considerations related to Question 1 can be found in Section 5.
2
2 Preliminaries
We recall that the Biggs-Smith graph has order n = 102, diameter d = 7,
S
girthg =9andautomorphismgroup =PSL(2,17)whichisalsoshowntobe
A
that of the distance-3 graph 3 of in Theorem7. Let k be the largestinteger
S S
s such that is s-arc transitive. Then k = 4. Finally, we also recall that the
S
number η of 9-cycles of is η =136.
S
Given a finite graph H and a subgraph M of H with V(H) >3, a graph Γ is
| |
strongly zipped (or SZ) H -UH if for each sequence of connected subgraphs
M
{ }
M =M ,M ...,M K such that M is obtained from M by the deletion
1 2 t 2 i+1 i
≡
ofavertex,fori=1,...,t 1,itholds thatG is a(2i 1)-fastened H -UH
− − { }Mi
graph, for i=1,...,t.
Let P be a (k 1)-path. Let C be a cycle of length g. Theorem 1 below as-
k g
−
sertsthat isSZ C -UH,namelySZ C -UH.Also, 3 isSZ H -UH
S { g}Pk { 9}P4 S { }M
(Theorems 6-7), where H = L(Q ) and M is a triangle, but some additional
3
concepts are needed: A graph G is rK -frequent if every edge e of G is inter-
s
section in G of exactly r induced copies of K , and these copies have only e
s
and its ends in common. (For example, K is 2K -frequent, and L(Q ) is 1K -
4 3 3 3
frequent). A graph G is H ,H -UH, where H is iK -frequent, (i = 1,2),
{ 2 1}K3 i 3
if: (a)GisanH -UHgraphandanedge-disjointunionofinducedcopiesofH ;
2 2
(b) there exists an edge partition Ω of H into 2-paths, and G is Ω-preserving
1
SZ H -UH;(c)eachinducedcopyofH inGhaseachofitsinducedcopies
{ 1}K3 2
of K in common with exactly two induced copies of H in G.
3 1
Properties of needed in the rest of the paper are covered in Section 3. In
S
Section 4, consideration of the distance-3 graphs of the 9-cycles of yields a
S
K ,L(Q ) -UH graph via zipping (meaning identification into an edge)
{ 4 3 }K3 Y
of corresponding oppositely oriented arcs of the resulting 3-cycles. Theorem 7
uses Theorem2, that establishes anexplicit 1-1correspondencebetween the 18
disjoint unions of 4 17-cycles each in and the points of the projective line of
S
thefieldGF(17). Theorem8,establishing = 3,alsoyieldstwoothergraphs,
Y S
appearing in the way the 102 copies of L(Q ) intersect, namely 2 and 4.
3
S S
3 Properties of the Biggs-Smith graph
S
Let A = (A , A , ..., A ), D = (D , D , ..., D ), C = (C , C , ..., C ),
0 1 g 0 2 f 0 4 d
F = (F , F , ..., F ) be 4 disjoint 17-cycles. Each y = A,D,C,F has vertices
0 8 9
y with i expressed as an heptadecimal index up to g = 16. We assume that
i
i is advancing in 1,2,4,8 units mod 17, stepwise from left to right, respectively
for y =A,D,C,F. Thenthe Biggs-Smithgraph is obtainedby adding to the
S
disjointunion A D C F, for eachi Z , a 6-vertextree T formedby the
17 i
∪ ∪ ∪ ∈
edge-disjoint union of paths A B C , D E F , and B E , where the vertices A ,
i i i i i i i i i
D , C , F arealreadypresentinthe cyclesA, D, C, F,respectively,andwhere
i i i
3
the verticesB and E are new and introducedwith the purpose ofdefining the
i i
tree T , for 0 i g = 16. A corresponding Frucht diagram [13] of via
i
≤ ≤ S
Z is depictedonthe left ofFigure 2below,withnonzeroFruchtnotationover
17
the loops, and null Frucht notation over the remaining edges. Now, has the
S
collection of 9-cycles formed by:
9
C
S0=(A0A1B1C1C5C9CdC0B0), W0=(A0A1B1E1F1F9F0E0B0),
T0=(C0C4B4A4A3A2A1A0B0), X0=(C0C4B4E4D4D2D0E0B0),
U0=(E0F0F9F1FaF2E2D2D0,) Y0=(E0B0A0A1A2B2E2D2D0),
V0=(E0D0D2D4D6D8E8F8F0), Z0=(F0F8E8B8C8C4C0B0E0),
and those 9-cycles obtained from these, as Sx,...,Zx, by uniformly adding
x Z mod 17 to all subindexes i of vertices y , so that =136.
17 i 9
∈ |C |
An auxiliary table presenting the form in which the 9-cycles above share the
vertexsetsof3-arcs,eitheroppositelyorientedornot,isshownbelow. Thetable
details,foreachoneofthe9-cyclesξ =S0,...,Z0,(expressedasξ =(ξ ,...,ξ )
0 8
in the shownvertexnotation), each9-cycle η in Si,...,Zi;i=0,...,16 ξ
{ }\{ }
intersecting ξ in the succeeding 3-paths ξ ξ ξ ξ , for i = 0,...,8, with
i i+1 i+2 i+3
additionsinvolvingitakenmod9. Eachsuchη inthe auxiliarytable haseither
a preceding minus sign, if the corresponding 3-arcs in ξ and η are oppositely
oriented, or not, otherwise. Each shown η (resp. η ) in the table has the
j j
−
subindex j indicating the equality of initial vertices η =ξ (resp. η =ξ ) of
j i+3 j i
those 3-arcs, for i=0,...,8:
S0:(-Te,T1,-Z1,Sd,S4,-Z9,Td,-T0,U0), W0:(-U0,W8,W9,-U1,X2,-Xb,Y0,-X0,X9),
1 7 4 4 3 3 0 6 8 4 2 1 3 7 4 6 8 5
T0:(S64,-S03,-Y22,T41,T3g,-Y10,-S70,S1g,V80), X0:(-V40,X2f,X12,-V34,-W56,W88,-Z80,W4f,-W70),
U0:(Yg,-U1,Z1,-Wg,-W0,Z9,-Ug,Y0,S0), Y0:(U0,-T0,-Tf,U1,-Y2,Vf,W0,V0,-Yf),
3 6 7 3 0 0 1 0 8 7 5 2 0 8 2 6 5 4
V0:(-Zd,-V4,Y2,-Xd,-X0,Y0,-Vd,-Z0,T0), Z0:(U8,-Z8,-V4,-S8,-Sg,-V0,-Z9,Ug,-X0),
2 6 5 3 0 7 1 5 8 5 6 0 5 2 7 1 2 6
This table is extended to all of by adding x Z uniformly mod 17 to all
9 17
C ∈
superindexes.
Theorem 2 is an SZ C -UH graph, for i = 0,1,...,k 2 = 2. In
S { 9}Pi+2 −
particular, is a C -UH graph and has exactly 2k 23n9 1 =136 9-cycles.
S { 9}P4 − −
Proof. We have to see that is a (2i+1 1)-fastened C -UH graph, for
S − { 9}P4−i
i=0,1,2. Infact,each(3 i)-pathP of issharedexactlyby 2i+1 9-cycles
4 i
− − S
of , for i = 0,1,2. For k = 4, any edge (resp. 2-path, resp. 3-path) of G is
S
shared by 8 (resp. 4, resp. 2) 9-cycles of G. This means that a 9-cycle C of G
9
sharesa P (resp. P , resp. P ) with exactly other 7 (resp. 3, resp. 1)9-cycles.
2 3 4
The statement follows from this and the details previous to it together with a
simple counting argument for the number of 9-cycles.
The reconstruction of from the 17-cycles A,D,C,F will be denoted
S
(A,D,C,F)= (P1,P2,P4,P8),
ℵ ℵ
where P1 = A, P2 = D, P4 = C, P8 = F, in order to see that it yields 8 au-
tomorphisms of obtained by multiplying modulo 17 the successive exponents
S
4
of P, (=P1 =A), by 2:
(P1,P2,P4,P8) (P2,P4,P8,P−1) ... (P−8,P1,P2,P4) (P1,P2,P4,P8).
ℵ →ℵ → →ℵ →ℵ
Observethata9-cyclexof sharingexactlya1-pathx withAsharesexactly
A
S
a2-pathx withF,withdistance3betweenx andx realizedbytwovertex-
F A B
disjoint 3-paths in x. Note that x ,x and the two 3-paths separating them
A F
are edge-disjoint. By using repeatedly this fact, it can be seen , say via a
representationof showingtheverticesofA,D,C,F symmetricallydistributed
S
onfourconcentriccircles,thatthereisanotherreconstruction (A0,D0,C0,F0)
ℵ
of with four disjoint 17-cycles:
S
A0=(A0A1B1C1C5B5E5F5FdF4FcEcBcCcCgBgAg)= (A00,A01,...,A0g),
D0=(F0F8E8D8D6E6B6C6C2CfCbBbEbDbD9E9F9)=(D00,D10,...,Dg0),
C0=(C0C4B4A4A3B3E3D3D1DgDeEeBeAeAdBdCd)=(C00,C10,...,Cg0),
F0=(D0D2E2F2FaEaBaAaA9A8A7B7E7F7FfEfDf)=(F00,F10,...,Fg0),
where A = A0, F =D0, C =C0, D = F0, etc., and where the vertices A0,
0 0 0 0 0 0 0 0 i
D0, C0, F0, for each fixed i Z , are the degree-1 vertices of a 6-vertex tree
i i i ∈ 17
T0, like the tree T above, formed by the disjoint union of the paths A0B0C0,
i i i i i
D0E0F0 and B0E0. An involution for i Z in (A0,D0,C0,F0) is also
i i i i i ∈ 17 ℵ
present whose orbits are y0 , y1,yg , y2,yf , y3,ye , y4,yd , y5,yc ,
{ } { } { } { } { } { }
y6,yb , y7,ya and y8,y9 , where y A,B,C,D,E,F . For example,
{ } { } { } ∈ { }
(A,D,C,F) and (A0,D0,C0,F0) are represented in Figure 1 below, where
ℵ ℵ
A,A0 (resp. D,D0;C,C0;D,D0havethickblack(resp. green;blue;red)edges.
Figure 1: Assemblies (A,D,C,F) and (A0,D0,C0,F0) of
ℵ ℵ S
There is a copy of the dihedral group D as a subgroup of the automorphism
8
group of that has: (a) a cyclic subgroup of order 8 generated by an au-
A S
tomorphism that sends A onto D (given below by a matrix ρ) and (b) an
i 2i
automorphism that sends A onto A0 (given below by a matrix α2), where
i i
i Z . Moreover, contains a subgroup which is a semidirect product of
17 0
∈ A A
this copy of D and a copy of Z generated by a matrix θ, also given below.
8 17
5
There are 18 reconstructions of , of which we just treated (A,D,C, F) and
S ℵ
(A0,D0,C0,F0). The remaining sixteen reconstructions, denoted (Aj,Dj,
ℵ ℵ
Cj, Fj), have 17-cycles Aj,Dj,Cj,Fj, with 0 = j Z , obtained from
17
6 ∈
A0,D0,C0, F0 via uniform addition of indexes mod 17. If we denote by ζ
A|
the order-9 subgroup of generated by an adequate rotation ζ of a 9-cycle of
A
, then the 18 reconstructions above are related by means of the subgroups
S
ζ. Eachsuchsubgroup ζ of : (a)hasasemidirectproductwith equal
0
A| A| A A
to , showing that = 17 32 24 and how is structured by means of
A |A| × × A
semidirect products from D , Z and ζ; (b) partitions V( ) into eleven 9-
8 17
A| S
orbits and one 3-orbit . The stabilizer in ζ of any of the three vertices of
ζ
O A|
V( ) (on which ζ acts) contains a copy of the symmetric group
ζ
O ⊂ S A| ⊂ A
S . Each one of these three vertices have distance 3 from ζ realized by paths
3
whose ends in ζ subdivide it into paths of length 3. (This disposition of ver-
tices will yield in Section 4 three disjoint copies of K in ). For example, if
4
Y
ζ =(F F F F F E D D E ), then =(A D D ).
g 8 0 9 1 1 1 g g ζ 0 7 a
O
1 A D 2 7
✎☞ ✎☞ ✬ 6 ✩
ss s
✍B ✌ ✍E ✌ ✤ 1 ✜
7 ✗4 ✔
s s ☛ ✟ ☛ ✟
4 C F 8 1
✎☞ ✎☞ ✎0☞(9)1(9) 2(9) 3(9) 4(9) 5(9) 6(9) 7(9) 8(9) 9(9) a(9) b(3)
s s s s s s s s s s s s s s
✍✌ ✍✌ ✍✌
Figure 2: Frucht diagrams of via Z and Z
17 9
S
There is an11-arcdepartingfrom eachvertex ofζ, yielding a totalof 9 disjoint
11-arcs, three arriving at each vertex of , with component vertices at each
ζ
O
fixed distance d < 11 from ζ forming a corresponding 9-orbit d, so we can
O
write 0 = ζ and 12 = . A corresponding Frucht diagram [13] of via
ζ
O O O S
Z is depicted on the right of Figure 2, where each number denoting a vertex
9
is followed with the parenthesized size of the Z -orbit it represents, and with
9
nonzero Frucht notation[13] overedges taken as arcs from left to right, so that
the corresponding oppositely oriented arcs have the respective negative Frucht
notation;horizontaledges,nothavinganexplicitFruchtnotationdenoted,must
be taken with null Frucht notation. There is a correspondence between the
originalnotationof andanew onebasedonthisFruchtdiagram,obtainedby
S
re-expressing the 9 11-arcs as follows:
FFFFFEDDg809111gEEEFFBDDa780913eFFBDBCEEf289013eFEACDCBFbf98e2e3FDCABEFC35f96e23FBAAEDBCc5a6e2d7FABAACCE456bd2bdFAABCACBd46b6cfdFAEACBBC53ba7cEfdADBEBAC52b7ca0fDADDEEBA5179ca0gDADDDDAA7077a0a0 →→→→→→→→ 0000000010325476111111111032547622222222103254763333333310325476444444441032547655555555103254766666666610325476777777771032547688888888103254769999999910325476aaaaaaaa10325476bbbbbbbb10022110
EgBgCgCcC8C4B4E4D4D5D8Da → 08182838485868788898a8b2
The 18 reconstructions of cited above correspond in a 1-1 fashion to the 18
S
points of the projective line over the field GF(17); this 1-1 correspondence
P
6
is to be set by the end of this section. Elements of PSL(2,17), represented by
2 2-matricesoverGF(17),actoverthesepointsbyleftmultiplication. Inorder
×
to fitwith the 2 2-matricesprovidedbelow, (A,D,C,F) mustcorrespondto
the point of g×iven by the column vector (cid:0)1(cid:1)ℵ. Then the point at , given by
P 9 ∞
(cid:0)0(cid:1),andtheremainingpoints,givenbyvectorcolumns(cid:0)1(cid:1),for9=j GF(17),
1 j 6 ∈
correspond to the remaining reconstructions (Ai,Di,Ci,Fi), for i GF(17).
ℵ ∈
The stabilizerof the groupPSL(2,17)atanyfixed elementof coincides with
P
theautomorphismgroupofthe3-cubeQ . Toseethatthisisalsothe stabilizer
3
of at the vertex A of , recall that the number of 4-arcs departing
AA0 A 0 S
from a fixed vertex of is 24. Each such 4-arc determines a unique oriented
S
17-cycleandauniqueoriented9-cycleof . Thus,therearetwentyfouroriented
S
17-cycles passing through each vertex of , in particular through A . Two of
0
S
these oriented 17-cycles are A = (A A A ...A ) and its oppositely oriented
0 1 2 g
17-cycle, A. The other oriented 17-cycles through A are obtained from the
0
−
reconstructions of above. In fact, has generators α,β,γ of orders 4, 2
S AA0
and 3 respectively, that map the successive vertices of A, as presented above,
onto the respective successive vertices of D9,D10, C3. Moreover, α,β,γ
− −
form a generating set of the automorphism group of Q . For example, the
3
automorphismthatsendsAonto Aequalsα2βγ2. Likewise,Aissentontothe
−
following17-cycles,bymeansoftheautomorphismsexpressedinthesubsequent
contiguous line:
A0 A1 C3 C4 D7 D8 D9 Da Cd Ce Ag
α2 α3γ α3γ2 αγ αγ2 α3 α3βγ2 β α2βγ α2β βγ
A0 A1 C3 C4 D7 D8 D9 Da Cd Ce Ag
β−γ2 −α2γ2 −γ −γ2 α−2γ α−βγ2 −α α−βγ α−β α−3βγ α−3β
completing the specifications for the 24 elements of , that yield informa-
AA0
tion about 12 of the 18 reconstructions of , with each two oppositely oriented
S
17-cycles displayed belonging to a specific reconstruction. These twelve recon-
structions are respectively (Ai,Di,Ci,Fi), for i=0,1,3,4,7,8,9,a,d,e,g.
ℵ
The generators α,β,γ of can be associated with the following matrices:
AA0
α → (−01 61), β → (−01 01), γ → (−01−11).
These matrices allow to establish a group embedding of onto PSL(2,17).
AA0
ByLagrange’stheorem, partitions into102classes,correspondingtothe
AA0 A
102 vertices of . Also, γ has as cubic roots the following three matrices:
S
(√3γ)1 → (92 181), (√3γ)2 → (62 185), (√3γ)3 → (96 1115).
Any of these cubic roots can be associated to a rotation generator of ζ. Let
A|
(√3γ)1 be such a generator. We can take the copies of the subgroups Z17 and
D in above to be generated in PSL(2,17) by its elements represented by
8
A
θ → ( 08 22), ρ → (−01 81), α2 → (−−16 61),
7
whereθ hasorder17,ρ hasorder8 andρ combineswithα2 (orwithβ) toform
a copyofD in . These copiesofthe two subgroupsZ andD in , related
8 17 8
A A
adequatelyviasemidirectproductstothecopyofZ9 generatedby(√γ)1,yields
an isomorphism PSL(2,17) to be used specifically in Theorem 7.
A→
BecauseAand Aarerepresentedbytheidentityandbyα2βγ2,respectively,it
−
can be seen that the reconstruction (A,D,C,F) of has associatedthe point
(cid:0)19(cid:1) of P with the vertex A0. Similaℵrly, because A0Sand −A0 are represented
by α2 and by βγ2, respectively, the reconstruction (A0,D0,C0,F0) of has
associatedthepoint(cid:0)110(cid:1)ofP withthevertexA0. ThℵeselectionofθismotiSvated
forittakesthispoint,(cid:0)1(cid:1) ,viasuccessivepowersθ,θ2,...ontothefollowing
10 ∈P
points:
(1),(1),(1),(1),(1),(1),(1),(1),(0),(1),(1),(1),(1),(1),(1),(1),(1),
10 15 16 14 11 6 13 0 1 1 5 12 7 4 2 3 8
which correspond to the successive reconstructions (Ai, Di, Ci, Fi) of , for
ℵ S
i = 0,...,8,10,...,16, thus establishing the desired 1-1 correspondence from
the family of 18 reconstructions of onto .
S P
Theorem 3 In , there exist 18 disjoint unions of four 17-cycles each, deter-
S
mining 18 corresponding reconstructions of 3. Those unions in are in 1-1
S S
correspondence with the 18 points of the projective line of GF(17) in such a
P
way that the action of on the family of these reconstructions is equivalent to
A
the natural action of PSL(2,17) on .
P
4 The K ,L(Q ) -UH graph = 3
4 3 K
{ } 3 Y S
We keep using the notation of and its collection of 136 9-cycles in Section
9
S C
3. Let ( )3 be the collection of distance-3 graphs of 9-cycles of . In each
9 9
C C
copy w w of P in a member C3 = (C )3 of ( )3, where C , let us
0 1 2 9 9 C9 9 ∈ C9
denotebyetheedgebetweenw andw . Then,theinitialvertexw ,theinitial
0 1 0
flag w ,e , the terminal flag e,w and the terminal vertex w of w w are
0 1 1 0 1
{ } { }
indicated,respectively,bymeansofthenamesoftheverticesv ,v ,v ,v ofthe
0 1 2 3
copy v v v v of P in C for which w w stands in C3.
0 1 2 3 4 9 0 1 9
For example, if C = S0 = (A A B C C C C C B ), so that C3 = (S0)3 =
9 0 1 1 1 5 9 d 0 0 9
(A C C )(A C C )(B C B ), then the initial flag of the copy A C of P in
0 1 d 1 5 0 1 9 0 0 1 2
C3 = (S0)3 is indicated by A , the terminal flag by B , while A and C
9 1 1 0 1
are indicated by themselves, namely A and C . We get the indications over
0 1
C3 =(S0)3 shown in Figure 3.
9
We zip now corresponding arc pairs of the distance-3 graphs C3 obtained from
9
, in order to get a graph with the desired -UH properties, meaning that
S Y C
8
B0 C0 A0 B0 A1 A0
✎ ☎ ✎ ☎ ✎ ☎
A0 A1 B1 C1 C5 C9 Cd A1 B1 C1 C5 C9 Cd C0 B1 C1 C5 C9 Cd C0 B0
❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜
Figure3: ExampleofindicationsoverC3 =(S0)3 =(A A B C C C C C B )3
9 0 1 1 1 5 9 d 0 0
we identify into an edge corresponding oppositely oriented arcs of the resulting
3-cycles. Thefollowingsequenceofoperationsisperformed;(comparewith[7]):
( )3 . (1)
9 9
S → C → C → Y
Next, we explain how this operation is composed. The distance-3
S → Y
graphsC3 ofthe1369-cyclesC of areformedbythreedisjoint3-cycleseach,
9 9 S
yielding a total of 3 136 = 408 3-cycles. Thus, determines a collection of
9
× C
408 triangles in , with each edge of shared by exactly two such triangles.
Y Y
In fact, we identify or zip in these triangles the pairs of copies of P obtained
2
as distance-3 graphs of copies of P in . This yields 102 copies of K that
4 4
S
can be subdivided into six subcollections Yi of 17 copies of K each, where
4
{ }
Y A,B,C,D,E,F andi 0,1,...,16=g =Z . ThevertexsetsV(Yi),
17
∈{ } ∈{ }
each followed by the set Λ(Y ) of copies of K containing the corresponding
i 4
vertex Y (as in the notation of Section 3), can be taken as follows, showing
i
Z -symmetries produced by the change-of-signinvolution in Z :
2 17
VVVVVV((((((FEDCABiiiiii))))))======{{{{{{CCCADAiiiiii+,,,+,63,, CFDFDDiiiiii,,−−,,63,, ABEBEFiiiiii++++++514782,,,,,, BAEBEFiiiiii−−−−−−514872}}}}}},,,,,, ΛΛΛΛΛΛ((((((EFDCABiiiiii))))))======{{{{{{CACCDAiiiiii,+,,+,12,, CFDFDDiiiiii−,,−,,12,, BABEFEiiiiii++++++486753,,,,,, BABEFEiiiiii−−−−−−486753}}}}}},,,,,,
wherei Z . Thisrevealsaduality mapφ fromthe 102verticesof ontothe
17
∈ S
102 copies of K . In fact, the copies of K are the vertices
4 4
φ(A )=A3i =A , φ(B )=B 7i =B , φ(C )=C3i =C ,
i ∗i i − i∗ i i∗
φ(D )=D5i =D , φ(E )=E6i =E , φ(F )=F5i =F ,
i i∗ i i∗ i i∗
(i Z ), of a graph φ( ) = with a structure similar to that of
17 ∗
∈ S S ≡ S
the vertices A ,B ,C ,D ,E ,F of and whose copies of K can be precisely
i i i i i i 4
S
denotedy =A ,B ,C ,D ,E ,F ,withcorrespondingvertexsetsΛ(y )asspec-
i i i i i i i i
ified above. Moreover, φ: is a graph isomorphism, with the adjacency
∗
S →S
of similar to that of .
∗
S S
The distance-3graphofeach9-cycleof iscomposedby three disjoint3-cycles
S
in . For example, the distance-3 graphs of S0,...,Z0 are:
Y
WTSUV00000→→→→→{{{{{EECAC7b140\\\\\ACACEe541g=====(((((CAEEA00000,,,,,ACDFE41141,,,,,ACEEFd1208))))),,,,,CEEBB1ab53\\\\\EADCF24gf8=====(((((AACFD11040,,,,F,CFAD1a53,6,,,ECD,AF00208)))),),,,,FFBDB99e25\\\\\CFDDF99ab2=====(((((BBDBF11942,,,,,FFCAD92928,,,,B,DBBF00000)))))}}}}};;;;;
YX00→→{{CA10\\FE1d==((EC00,,AE14,,ED20)),,DA42\\EB84==((CB40,,DA42,,ED02)),,DD02\\BAf2==((AB04,,BD22,,DB00))}};;
Z0→{F0\B9=(F0,B8,C0), F8\Bg=(F8,C8,B0), A4\D4=(E8,C4,E0)}.
9
This way, it can be seen that is a K -UH graph. Moreover, the vertices and
4
Y
copies of K in are the points and lines of a self-dual (102 )-configuration,
4 4
Y
which in turn has as its Menger graph. However, in view of Beineke’s char-
Y
acterization of line graphs [1], and observing that contains induced copies of
Y
K , which are forbidden for line graphs of simple graphs, we conclude that
1,3
Y
is non-line-graphical. We obtain the following statement.
Theorem 4 is an edge-disjoint union of 102 copies of K , with four such
4
Y
copies incident to each vertex. Moreover, is a non-line-graphical K -UH
4
Y
graph. Its vertices and copies of K are the points and lines, respectively, of a
4
self-dual(102 )-configurationwhichhas precisely asitsMenger graph. Inpar-
4
Y
ticular, isarc-transitivewithregulardegree12,diameter 3anddistancedistri-
Y
bution(1,12,78,11). ItsassociatedLevigraph[5]isa2-arc-transitivegraphwith
regular degree 4, diameter 6 and distance distribution (1,4,12,36,78,62,11).
Proof. The statement is immediate from the data given by the construction
of via operation (1) and because of the fact that is a distance-transitive
Y S
graph.
Remark 5 A proof yielding the group structure of the automorphism group
B
of is part of the proof of Theorem 6. However, the Magma software allowed
Y
to verify that the automorphism group of is = = PLS(2,17), whose
Y B A
order is 2448, and that the Levi graph associated to has automorphism group
Y
SL(2,17), whose order is 4896.
Theorem 6 is a K -UH graph.
4
Y { }
Proof. Consider an isomorphism Ψ : Θ Θ between two copies Θ ,Θ of
1 2 1 2
K in . By construction, each Θ (i =→1,2) arises from 4 9-cycles θj in
4 Y i i S
(j =1,2,3,4) related so that each one of them have their union as a subgraph
Θ of with 4 vertices vj of degree 3 and 12 vertices of degree 2 which are
i S i
the internal vertices of 6 3-paths whose endvertices are the vertices vj. For
i
example, the vertices v1 = B ,v2 = B ,v3 = F ,v4 = C ,v1 = B ,v2 =
1 0 1 1 1 9 1 9 2 1 2
B ,v 13 =F ,v4 =C in determine such subgraphs Θ ,Θ in and Θ ,Θ
in2 .2Clearlya, Ψ2induaces aSn isomorphism Ψ : Θ Θ t1hat2sendYs say eac1h v2j
S 1 → 2 1
onto a corresponding vj (j = 1,2,3,4). An automorphism Ψ of exists that
2 S
extends Ψ. Moreover,Ψ determines an automorphism of that restricts to Ψ,
Y
which completes our verification that is a K -UH graph.
4
Y { }
Eachofthe102copiesofK in arisesfromthedistance-3graphsoffourofthe
4
Y
1369-cyclesof . Thesubgraphof spannedbythesefour9-cyclescontainsfour
S S
degree-3vertices,(whichareinitialandterminalverticesofcorrespondingcopies
of P , as in the first paragraph of the section), and twelve degree-2 vertices,
4
(internal vertices in those copies of P ). These twelve vertices form a copy of
4
L
L(Q ) in . For the copy A0 of K in , the copy = a0 of L(Q ) in can
3 4 3
Y Y L Y
10
