Doubly transitive 2–factorizations *

14 Pages·2006·0.23 MB·English

Doubly transitive
2–factorizations *
Arrigo BONISOLI, Marco BURATTI and Giuseppe MAZZUOCCOLO
Dipartimento di Scienze Sociali, Cognitive e Quantitative, Universit`a di Modena e
ReggioEmilia,viaGiglioliValle9,42100ReggioEmilia,[email protected]
Dipartimento di Matematica e Informatica, Universit`a di Perugia, via Vanvitelli
1, 06123 Perugia, Italy. [email protected]
Dipartimento di Matematica, Universit`a di Modena e Reggio Emilia, via Campi
213/B, 41100 Modena, Italy. [email protected]
ABSTRACT
Let F be a 2–factorization of the complete graph K admitting an automor-
v
phismgroupGactingdoublytransitivelyonthesetofvertices. Thevertex–set
V(K ) can then be identified with the point–set of AG(n,p) and each 2–factor
v
of F is the union of p–cycles which are obtained from a parallel class of lines
of AG(n,p) in a suitable manner, the group G being a subgroup of AGL(n,p) in
this case. The proof relies on the classification of 2–(v,k,1) designs admitting
a doubly transitive automorphism group. The same conclusion holds even if
G is only assumed to act doubly homogeneously. (cid:176)c ???JohnWiley&Sons,Inc.
Keywords: graph, 2–factorization, doubly transitive permutation group, design
MSC 2000: 05C70 05C15 05C25
1. INTRODUCTION
InhislectureattheInternationalSymposiumonGraphs,DesignsandApplications,
held in Messina (Italy) from 30 Sept. to 4 Oct. 2003, Alexander Rosa remarked,
among other things, that doubly transitive 1–factorizations of complete graphs are
classified by the work of Cameron and Korchm`aros [5], while no such classification
(cid:176)c ???JohnWiley&Sons,Inc. CCC???
JournalofCombinatorialDesignsVol.???,(???)
1
2 BONISOLI,BURATTI,MAZZUOCCOLO
for doubly transitive 2–factorizations of complete graphs was available yet, see [11,
§6]. Here “doubly transitive” means there exists an automorphism group of the 2–
factorization acting doubly transitively on the vertices of the underlying complete
graph. In particular there was no classification for doubly transitive hamiltonian
decompositionsofcompletegraphs. Thesearenamelythe2–factorizationsinwhich
every 2–factor is a hamiltonian cycle.
One family of such hamiltonian decompositions was known, see [10, Lemma
2.1], [2, Thm 4.2]: if p is an odd prime and V(K ) = Z we can define C =
(cid:161) (cid:162) p p i
0,i,2i,...,(p − 1)i for i = 1,2,...,(p − 1)/2 and obtain the hamiltonian de-
composition H = {C ,C ,...,C } of K . This hamiltonian decomposition
1 2 (p−1)/2 p
is invariant under the group AGL(1,p) in its natural sharply doubly transitive
permutation representation.
Afurtherfamilyofdoublytransitive2–factorizationswasknown[11,§3]. Namely,
if we identify V(K ) with the point–set of the affine space AG(n,3), then each
3n
class of parallel lines yields a 2–factor which is the union of 3–cycles, the vertices
in each 3–cycle being the three points on some line of the class. Since any two
pointslieonauniquelineoftheaffinespace,weseethatthe2–factorsarisingfrom
all parallel classes actually form a 2–factorization of K . The group AGL(n,3)
3n
of affine transformations acts doubly transitively on points and clearly leaves this
2–factorization invariant.
In the present paper we prove that the former examples are the unique doubly
transitivehamiltoniandecompositionsofcompletegraphs,whilethelatterexamples
can be generalized to an arbitrary odd prime p, thus yielding doubly transitive 2–
factorizations of K , n a positive integer. These results were presented by the
pn
first author at the conference “Giornate di Geometria” held at the Universit`a “La
Sapienza” in Rome, 4–6 dec. 2003.
Furthermore, we show that these two families of examples account for ALL
doubly transitive 2–factorizations, up to isomorphisms. Our way of achieving this
classification is by proving that the 2–factorization arises from a 2–(v,k,1) design
andthattheoriginalgroupactingonthe2–factorizationinducesanautomorphism
group of the design. It is therefore possible to refer to the determination of 2–
(v,k,1) designs admitting an automorphism group acting doubly transitively on
points, a result of W.M. Kantor [9]. The fact that the size of blocks must be a
prime rules out many items in this list. Some others are ruled out by properties
of the action of the given group. Only affine spaces over prime fields survive and
the2–factorizationcanbereconstructed. Thisresultwasannouncedbythesecond
authoratthe“InternationalConferenceonIncidenceGeometry”heldinLaRoche–
en–Ardennes (Belgium) 23–29 May 2004.
Meanwhile the paper by T.Q. Sibley [13] has appeared. This paper gives a
classification of edge colorings of the complete graph admitting an automorphism
group acting doubly transitively on vertices. Our case falls within that description
if it is assumed that the color classes are 2–factors. Section 3 of Sibley’s paper [13]
treats the more general case in which the color classes are regular subgraphs. We
showintheAppendix(Section8.below)howthedeterminationofdoublytransitive
2–factorizations can be derived from Theorem 15 in [13].
We can further observe that both Sibley’s and our methods require the classi-
fication of doubly transitive finite permutation groups, therefore both ultimately
DOUBLYTRANSITIVE 2–FACTORIZATIONS 3
rest on the classification of finite simple groups. Our more restricted goal allows
some more direct treatment of the subject matter at various stages.
Weconcludewithaslightgeneralizationandshowthatthefamilyofdoublyho-
mogeneous2–factorizationscoincideswiththatofdoublytransitive2–factorizations,
where“doublyhomogeneous”obviouslymeansthereexistsanautomorphismgroup
of the 2–factorization acting doubly homogeneously on vertices.
2. NOTATION AND PRELIMINARY PROPERTIES
If Γ is a graph then V(Γ) and E(Γ) denote the vertex–set and the edge–set of Γ,
respectively. The edge consisting of the vertices x, y will generally be denoted by
[x,y]. We shall denote by (x ,x ,...,x ) the k–cycle on the vertices x , x , ...,
0 1 k−1 0 1
x consisting of the edges [x ,x ], [x ,x ], ..., [x ,x ]. If x, y are vertices of
k−1 0 1 1 2 k−1 0
the graph Γ, then we denote by d (x,y) the distance from x to y in Γ.
Γ
Let Γ be a regular graph of degree d. A 2–factorization of Γ is a collection
F ={F ,F ,...,F }ofedge–disjoint2–factorsofΓ formingacoveroftheedge–set
1 2 s
E(Γ). Inotherwordsforanytwodistinctindicesiandj wehaveE(F )∩E(F )=∅
i j
and E(F )∪E(F )∪···∪E(F )=E(Γ). A necessary condition for the existence
1 2 s
of such a 2–factorization is d even and s=d/2.
Assume F is a 2–factorization of Γ. The cycles occurring in the 2–factors of F
will be called F–cycles. Clearly for any given edge [x,y] of Γ there exists a unique
F–cycle containing it. If each 2–factor in F consists of a unique F–cycle, which is
thusahamiltoniancycleofΓ, thenweshallspeakofahamiltonian 2–factorization
or a hamiltonian decomposition of Γ and we shall use the letter H to denote it.
If F is a 2–factorization of the regular graph Γ, then an automorphism of F is
simply a permutation of the vertex–set V(Γ) leaving F invariant, that is mapping
2–factors in F to 2–factors in F. Any automorphism of F induces a permutation
on the edge–set E(Γ) and is thus an automorphism of Γ by the very definition.
LetF bea2–factorizationoftheregulargraphΓ. WeshallsaythatF isdoubly
transitive if there exists an automorphism group G of F acting doubly transitively
on the vertices of Γ. If that is the case, then since G is an automorphism group of
Γ acting doubly transitively on V(Γ) we have that Γ is a complete graph K and
v
the existence of a 2–factorization of K forces v to be odd.
v
If C is a k–cycle then the permutations of V(C) fixing C are easily seen to form
a dihedral group of order 2k. It is customary to represent C through a euclidean
k–gon and the dihedral group as a group of euclidean isometries. With that in
mind, if C is a k–cycle in a graph Γ and g is a permutation of V(Γ) leaving C
invariant,thenweshallsaythatg inducesarotation orareflection onC according
to the nature of the corresponding action on the k–gon. If k is odd and g does not
fix V(C) elementwise, then g will induce a reflection if and only if g fixes precisely
one vertex of C and exchanges the remaining ones in pairs, while g will induce a
rotation if and only if it yields a fixed–point–free permutation on V(C) with orbits
of equal length strictly greater than 2.
Proposition 2.1. Let C be a cycle in a given graph Γ. Assume g is a permuta-
tion on V(Γ) with Cg =C. If x, y are adjacent vertices in C with xg =x, yg =y,
then g fixes C elementwise.
4 BONISOLI,BURATTI,MAZZUOCCOLO
Let Γ be a regular graph of degree d with a 2–factorization F which is left
invariant by an automorphism group G of Γ. If [x,y] is an edge in Γ, then there is
aunique2–factorF inF containingtheedge[x,y]andsoitfollowsfromProposition
2.1 that the stabilizer G fixes elementwise the F–cycle of F containing [x,y].
x,y
3. DOUBLY TRANSITIVE HAMILTONIAN DECOMPOSITIONS
Proposition 3.1. LetHbeahamiltoniandecompositionofK ,vodd,admitting
v
an automorphism group G acting doubly transitively on V(K ). Then G acts tran-
v
sitively on the hamiltonian cycles in H and sharply doubly transitively on V(K ).
v
In particular v is an odd prime power.
Proof. Let C and C(cid:48) be H–cycles and choose edges [x,y] in C and [x(cid:48),y(cid:48)] in
C(cid:48) respectively. By the 2–transitivity of G on vertices, there exists g ∈ G with
[x,y]g = [x(cid:48),y(cid:48)]. Since C and C(cid:48) are the unique H–cycles containing [x,y] and
[x(cid:48),y(cid:48)] respectively, we necessarily have Cg =C(cid:48) and so the H–cycles form a single
G–orbit.
Let x, y be distinct vertices and let C be the unique H–cycle containing [x,y].
It follows from Proposition 2.1 that G fixes each vertex of C and so, since the
x,y
hamiltonian cycle C contains all vertices of K , we conclude that G reduces to
v x,y
the identity, in other words G is sharply doubly transitive on V(K ). A sharply
v
doublytransitivefinitepermutationgroupcontainsasharplytransitiveelementary
abelian subgroup, see for instance [7, Lemma 20.7.K1]. In particular v is a prime
power and the assertion follows.
Proposition 3.2. Let p be an odd prime and let a be an integer, a>1. There
cannot exist a doubly transitive hamiltonian decomposition of K .
pa
Proof. Assume H is a hamiltonian decomposition of K and let G be an auto-
pa
morphism group of H acting doubly transitively on V(K ). Let C be an arbi-
pa
trary H–cycle and let S denote the stabilizer of C in G. Proposition 3.1 yields
|G|=pa(pa−1) and since |H|=(pa−1)/2 we also have |S|=2pa. In particular a
Sylowp–subgroupofS hasorderpa andsoitwillcoincidewithaSylowp–subgroup
ofG. NowGpossessesasharplytransitiveelementaryabelianp–subgroupN which
is normal, see [7, Lemma 20.7.K1] and so, in particular, N is the unique Sylow p–
subgroup of G, consequently N is contained in S. We conclude that N fixes each
H–cycle.
Let now g = (x ,x ,...,x )(y ,y ,...,y )...(z ,z ,...,z ) be a non–trivial
1 2 p 1 2 p 1 2 p
(fixed–point–free) permutation in N. Consider the edge [x ,x ] and let C(cid:98) be the
1 2
H–cycle containing [x ,x ]. Since N fixes C(cid:98) we have C(cid:98)g = C(cid:98), consequently g
1 2
maps edges of C(cid:98) to edges of C(cid:98). We have [x ,x ]g = [x ,x ], hence [x ,x ] ∈ C(cid:98).
1 2 2 3 2 3
We iterate the argument and obtain [x ,x ] ∈ C(cid:98), [x ,x ] ∈ C(cid:98), ..., [x ,x ] ∈ C(cid:98),
1 2 2 3 p 1
forcing C(cid:98) to coincide with the cycle (x ,x ,...,x ), a contradiction since a>1.
1 2 p
The argument in the previous proof basically shows that the elementary abelian
p–groupN cannotfixthehamiltoniandecompositionHelementwiseandactsharply
transitively on V(K ), a > 1. It was proved in Theorem 1.1 of [3] that a sharply
pa
DOUBLYTRANSITIVE 2–FACTORIZATIONS 5
transitive cyclic group cannot fix a hamiltonian decomposition of K with p an
pa
odd prime and a>1.
The following stronger property holds.
Proposition 3.3. Ifpisanoddprimeanda>1isaninteger,thentherecannot
exist a hamiltonian decomposition of K admitting an automorphism group acting
pa
sharply transitively on vertices.
Proof. AssumeHisahamiltoniandecompositionofK andletRbeanautomor-
pa
phismgroupofHactingsharplytransitivelyonV(K ). Wehave|H|=(pa−1)/2
pa
and |R| = pa, consequently at least one H–cycle is fixed by R. We have seen in
section 2 that the permutations of V(K ) fixing a pa–cycle form a dihedral group
pa
of order 2pa, consequently R must be the cyclic group of order pa in this dihedral
group. The result now follows from Theorem 1.1 of [3]
The discussion on doubly transitive hamiltonian decompositions can therefore
be limited to the case of a complete graph K in which p is an odd prime and G is
p
a sharply doubly transitive permutation group on V(K ).
p
Proposition 3.4. Let p be an odd prime. There exists up to isomorphisms a
unique doubly transitive hamiltonian decomposition H of the complete graph K .
p
Proof. We have seen in the Introduction that there exists one such hamiltonian
decomposition. Zassenhaus showed in 1936 that a sharply doubly transitive finite
permutationgroupisalwaysisomorphictothegroupofaffinelineartransformations
overafinitenearfield,seeagain[7,§20.7]. ThefieldZ istheuniquefinitenearfield
p
ofprimeorderpandthesharplydoublytransitivegroupGisnecessarilyAGL(1,p)
inthiscase,thatisthegroupofallaffinelineartransformationsZ →Z ,x(cid:55)→ax+b
p p
witha,b∈Z , a(cid:54)=0. Weproveuniquenessbyshowingthatthehamiltoniancycles
p
inthedecompositioncanbeuniquelyreconstructedfromtheactionofGonvertices.
The sharply transitive normal subgroup N of G consists of the mappings x (cid:55)→
x+b, b∈Z . We have seen that N fixes each p–cycle in the decomposition. Take
p
theedge[0,b]andconsiderthemappingx(cid:55)→x+b. Theedge[0,b]mapsto[b,2b]. If
C is the unique H–cycle containing [0,b] we see that since g fixes C the edge [0,b]g
mustbeinC,hence[b,2b]∈C. Astraightforwardrepetitionoftheargumentyields
(cid:161) (cid:162)
C = 0,b,2b,...,(p−1)b . Since b was an arbitrary non–zero element of Z , we
p
conclude that H consists of all such p–cycles as b varies in {1,2,...,(p−1)/2}.
4. 2–FACTORIZATIONS ARISING FROM RESOLVABLE DESIGNS
Itissomewhatstraightforwardtoconstruct2–factorizationsfromresolvabledesigns.
We refer to the monograph [1] for the standard definitions and terminology of
designs. Inwhatfollowsweunderstandthatadesign isa2–(v,k,1)design. Assume
D is a resolvable design in which v is odd, and consequently k is also odd. Setting
r = (v−1)/(k−1) we know that r is the number of resolution classes and v/k is
the number of blocks in each resolution class.
Let L , L , ..., L be the resolution classes. We number the blocks in the
1 2 r
resolution class L as B , B , ..., B .
j j,1 j,2 j,v/k
6 BONISOLI,BURATTI,MAZZUOCCOLO
ForeachblockB wedefinea2–factorizationofthecompletegraphhavingthe
j,s
k pointsofB asverticesanddenoteits2–factorsbyF ,F ,...,F .
j,s j,s,1 j,s,2 j,s,(k−1)/2
Let K be the complete graph with the points of D as vertices. For each j =
v
1,2,...,r and each c = 1,2,...,(k − 1)/2 we form a 2–factor of K by taking
v
the union of F , F , ..., F . In other words, we form a 2–factor from
j,1,c j,2,c j,v/k,c
a resolution class by “glueing” together the 2–factors which exist in each block of
the class. Here glueing occurs by arbitrary labelling of the 2–factors in each block.
The incidence properties of the design immediately yield that the set of all such
2–factors forms a 2–factorization of K .
v
If g is an automorphism of the design D and B is a block, then it may well
happen that the 2–factorization which was chosen on Bg does not coincide with
the g–image of the 2–factorization which was chosen on B, and consequently that
g does not induce an automorphism of the 2–factorization.
The next section shows that, at least in some very special cases, an accurate
choice of the 2–factors in the single blocks can produce a 2–factorization which
inherits a significant automorphism group from the design.
5. A CONSTRUCTION FOR
NON–HAMILTONIAN 2–TRANSITIVE
2–FACTORIZATIONS
Let p be an odd prime and let n be a positive integer. A class of parallel lines in
theaffinespaceAG(n,p)isuniquelydeterminedbyanon–zerovectorw. Eachline
in this class can be written in parametric form as P = P +αw where P is an
0 0
arbitrary point on the line, α∈GF(p).
For i=1,2,...,(p−1)/2 form a p–cycle C on the given line as
i
(P , P +iw , P +2iw , ... , P +(p−1)iw).
0 0 0 0
A different choice of the initial point P on the given line only alters the repre-
0
sentation of C by a cyclic permutation. Replacement of w by −w amounts to
i
moving along C in the reverse order. A different choice of the non–zero vector w
i
in the 1–dimensional vector subspace (cid:104)w(cid:105) still yields the same cycles C , C , ...,
1 2
C , possibly in a different order.
(p−1)/2
Fix a parallel class of lines in AG(n,p) and fix a non–zero vector w which is
parallel to the lines of this class. For each integer between 1 and (p−1)/2 form
a 2–factor F by taking the union of the p–cycles C constructed above, one for
i i
each line of the parallel class. As the parallel class varies or, equivalently, as the
vector w varies in a system of distinct representatives for the 1–dimensional vector
subspaces of GF(p)n, the set of all such 2–factors yields a 2–factorization of the
complete graph whose vertices are the points of AG(n,p). This 2–factorization is
non–hamiltonian for n>1.
Letf ∈AGL(n,p)beanaffinetransformationinducingthelinearautomorphism
ϕ on vectors. Setting P(cid:48) = Pf and wϕ = w(cid:48) the line P = P +αw is mapped
0 0 0
to the line P(cid:48) = P(cid:48) +αw(cid:48). The p–cycle C is mapped to the p–cycle (P(cid:48),P(cid:48) +
0 i 0 0
iw(cid:48),P(cid:48) +2iw(cid:48),...,P(cid:48) +(p−1)iw(cid:48)). The affine transformation f maps thus the
0 0
p–cyclesC obtainedfromtheparallelclassoflinesgivenbythevectorwuniformly
i
to the p–cycles with the same index obtained from the parallel class of lines given
DOUBLYTRANSITIVE 2–FACTORIZATIONS 7
by the vector w(cid:48). Put it another way, the 2–factors F , F , ..., F arising
1 2 (p−1)/2
from the parallel class of lines identified by the vector w are mapped by f to the
2–factors F(cid:48), F(cid:48), ..., F(cid:48) arising from the parallel class of lines identified by
1 2 (p−1)/2
the vector w(cid:48). Since f permutes the classes of parallel lines, we conclude that f
leaves the 2–factorization invariant. In particular, if T is the translation group
of AG(n,p), then the 2–factorization admits T as an automorphism group acting
sharply transitively on vertices.
The above doubly transitive 2–factorization of K admits an alternative alge-
pn
braic description through the notion of a Cayley graph (see [12]).
Let G be an additive group and let Ω ⊂ G−{0} be such that ω ∈ Ω if and
only if −ω ∈ Ω. Then the Cayley graph of G over Ω is the simple graph Cay[G :
Ω] with vertex-set G and edge–set {[g,ω +g] | g ∈ G;ω ∈ Ω}. It is clear that
Cay[G : Ω] is a |Ω|-factor of K(G), the complete graph on G. As observed in [4],
if G has odd order, then all possible 2–regular Cayley graphs of G, namely the
graphs of the form Cay[G : {g,−g}] for some g ∈ G−{0}, form a 2–factorization
of K denoted by N(G) and called the natural 2–factorization of K . Note that
G G
N(G) admits Hol(G), the holomorph of G, as an automorphism group. It is not
difficult to see that the 2-factorization of V(K ) that we described geometrically
pn
aboveisnothingbutN(Zn). Its2-transitivitycanbeestablishedbyobservingthat
p
Hol(Zn)=AGL(n,p).
p
6. SOME FURTHER PROPERTIES
Assume G is an automorphism group of the 2–factorization F of K acting doubly
v
transitively on vertices. The same argument of Proposition 3.1 shows that the
group G acts transitively on F–cycles which have thus equal length k, a divisor of
v. The total number of F–cycles is thus v(v−1)/2k and so the stabilizer G in G
C
of a given F–cycle C has order 2k|G|/v(v−1).
Proposition 6.1. The group induced by G on V(C) is a dihedral group of
C
order 2k.
Proof. The group induced by G on V(C) is the quotient G /G for any two
C C xy
vertices x, y which are adjacent in C. The 2–transitivity of G on vertices implies
|G |=2kv(v−1)|G |/v(v−1), whence |G |/|G |=2k. Since the largest pos-
C x,y C x,y
siblegroupinducedbyG onV(C)isdihedraloforder2k, theassertionfollows.
C
Proposition 6.2. The number of F–cycles through any given pair of distinct
vertices is a constant λ=(k−1)/2.
Proof. If C and C(cid:48) denote the set of F–cycles through the vertices x, y and x(cid:48), y(cid:48)
respectively, then, since G is doubly transitive on vertices, there exists g ∈G with
xg =x(cid:48),yg =y(cid:48). WehavethusCg =C(cid:48) andconsequentlyλisaconstant. Standard
double counting yields
(cid:181)v−1(cid:182) (cid:179)v(cid:180) (cid:161)v(cid:162)
=λ (cid:161)2(cid:162)
2 k k
2
whence λ=(k−1)/2.
8 BONISOLI,BURATTI,MAZZUOCCOLO
Definition 6.1. A Steiner k–cycle system of K is a decomposition of K into
v v
k–cycles with the property that for any pair of distinct vertices x, y and any integer
d ∈ {1,2,...,(k −1)/2} there exists exactly one k–cycle in the decomposition in
which x and y occur at distance d.
Proposition 6.3. The F–cycles yield a Steiner k–cycle system of K .
v
Proof. Let x, y be distinct vertices and take d ∈ {1,2,...,(k − 1)/2}. Since
(k−1)/2 is the number of F–cycles through x, y, all we need to show is that x, y
occuratdistancedinatleastoneF–cycle. LetC beanygivenF–cycleandletx ,
1
y be vertices at distance d on C. Since G is doubly transitive on vertices, there
1
exists g ∈G with xg =x, yg =y. Consequently Cg is a F–cycle in which x and y
1 1
occur at distance d.
Proposition 6.4. ThestabilizerG fixeselementwiseeachF–cyclethroughthe
x,y
distinct vertices x, y.
Proof. Each g ∈G fixes the set of F–cycles through x, y setwise. Since for any
x,y
distance d∈{1,2,...,(k−1)/2} there is a unique F–cycle in which x, y occur at
distancedandsinceg preservesdistancesoncorrespondingF–cycles, wehavethat
g fixes each F–cycle through x, y setwise, whence also elementwise by Proposition
2.1.
Proposition 6.5. Let C be a F–cycle and let x, y ∈ V(C) be distinct vertices.
If g ∈G is such that xg, yg are in V(C) with d (x,y)=d (xg,yg), then we have
C C
Cg =C. InparticularC isfixedbyanyautomorphisminGexchangingtwodistinct
vertices of C.
Proof. Theg–imageoftheF–cycleC istheF–cycleCg andtherelationd (x,y)=
C
d (xg,yg) holds. Our assumption implies d (xg,yg)=d (xg,yg) and if C (cid:54)=Cg
Cg C Cg
then C and Cg would be distinct F–cycles on which xg, yg lie at equal distance.
OurnextaimistoshowthatF–cyclessharingatleasttwoverticesmustshareall
of their vertices. That will be achieved stepwise in the following five propositions.
Proposition 6.6. Let A, B be F–cycles sharing three vertices, two of which are
adjacent in A. Then we have V(A)=V(B).
Proof. If k = 3 then (k−1)/2 = 1 is the number of F–cycles through any two
given vertices and the assertion is obvious in this case. We may therefore assume
k >3.
We write A = (a ,a ,...,a ), B = (b ,b ,...,b ) with a = b , a = b .
0 1 k−1 0 1 k−1 0 0 k−1 h
The indices will be always understood mod k.
By Proposition 6.1 there exists f ∈ G inducing on A the reflection with fixed
vertexa sothatwehaveaf =a foreverys. Inparticular,f exchanges
(k−1)/2 s k−1−s
b with b and hence, by Proposition 6.5, we see that f induces a reflection on B,
0 h
namely bf =b for every t. Thus we have (V(A)∩V(B))f =V(A)∩V(B) and
t h−t
a =b ∈V(A)∩V(B), a =b ∈V(A)∩V(B) (1)
s t k−1−s h−t
Assume that a ∈V(A)∩V(B) with 0≤i<(k−1)/2 and let s be the smallest
i
integer greater than i such that a ∈V(B).
s
DOUBLYTRANSITIVE 2–FACTORIZATIONS 9
If i = 0 we have s (cid:54)= k−1 otherwise we would have V(A)∩V(B)= {a ,a }
0 k−1
contradicting the assumption |V(A)∩V(B)| ≥ 3. If i > 0 we also have s (cid:54)= k−1
since by (1) we have a ∈V(B) and i<k−1−i<k−1.
k−1−i
Take g ∈ G such that ag = a for every j (existing by Proposition 6.1).
A j j+s
Setting a = b and taking account of (1) we obtain bg = b , bg = b and
s t 0 t h−t h
d (b ,b )=d (b ,b ). So, observing that b and b are distinct vertices of B
B 0 h−t B t h 0 h−t
(otherwise we would have s=k−1 that we excluded before), we have g ∈G by
B
Proposition 6.5.
It follows that (V(A)∩V(B))g =V(A)∩V(B) and hence, in particular, a =
s−1
ag ∈ V(B). By the definition of s this implies that a = a . In this way we
k−1 s i+1
haveprovedthatifa ∈V(B)thena isalsoinV(B)for0≤i<(k−1)/2. Thus
i i+1
we have {a ,a ,...,a }⊂V(B) and applying (1) we get the assertion.
0 1 (k−1)/2
Definition 6.2. Let W ⊆ V(K ) be an arbitrary set of at least two vertices.
v
Define L (W) to be the set of all vertices which belong to some F–cycle A with
1 (cid:161) (cid:162)
|V(A) ∩ W| ≥ 2. For m > 1 we inductively define L (W) = L L (W) .
(cid:83) m 1 m−1
Finally, define L(W) = L (W). We have L (W) ⊆ L (W) ⊆ ... and
m≥1 m 1 2
consequentlythereexistsasmallestindexswithL (W)=L (W)or,equivalently,
s s+1
L (W)=L(W).
s
Proposition 6.7. Let U, W be sets of vertices of K with |U|≥2, |W|≥2. If
v
U ⊆L(W) then we have L(U)⊆L(W).
Proof. WriteL(W)=L (W)forsomeindexs. ThenU ⊆L (W)impliesL (U)⊆
(cid:161) (cid:162) s s 1
L L (W) = L (W) = L(W). We conclude that L (U) ⊆ L(W) holds for all
1 s s+1 j
indices j and consequently L(U)⊆L(W) holds as well.
(cid:161) (cid:162)
Proposition 6.8. If A,B are F–cycles with V(A) ⊆ L V(B) then we have
(cid:161) (cid:162) (cid:161) (cid:162)
L V(A) =L V(B) .
Proof. Since G is transitive on F–cycles there exists g ∈ G with B = Ag and we
(cid:161) (cid:162) (cid:161) (cid:162) (cid:161) (cid:162) (cid:161) (cid:162)
have |L V(B) | = |L V(Ag) | = |L V(A) g| = |L V(A) |. Proposition 6.7 yields
(cid:161) (cid:162) (cid:161) (cid:162)
L V(A) ⊆L V(B) , whence the assertion.
Proposition 6.9. IfC isaF–cycleandg ∈Gfixesatleasttwodistinctvertices
(cid:161) (cid:162) (cid:161) (cid:162)
x, y in L V(C) then g fixes L V(C) elementwise.
(cid:161) (cid:162)
Proof. Let A be the F–cycle containing [x,y]. We have V(A) ⊆ L V(C) . Fur-
thermoreg fixesV(A)elementwisebyProposition6.4andconsequentlyg alsofixes
(cid:161) (cid:162) (cid:161) (cid:162)
L V(A) elementwise. Assuming now inductively that g fixes L V(A) ele-
1 m−1
mentwise, then g will also fix elementwise any F–cycle having at least two vertices
(cid:161) (cid:162) (cid:161) (cid:162)
in common with L V(A) , in other words g fixes L V(A) elementwise. The
(cid:161) (cid:162) m−1 m (cid:161) (cid:162) (cid:161) (cid:162)
setL V(A) isthusalsofixedbyg elementwiseand,sinceL V(A) =L V(C) by
Proposition 6.8, the assertion follows.
Proposition 6.10. If A, B are distinct F–cycles with |V(A)∩V(B)|≥2 then
we have V(A)=V(B).
Proof. Assume A, B are distinct F–cycles with |V(A)∩V(B)| ≥ 2. Assume,
further, that there exist two common vertices which are adjacent on one of the
10 BONISOLI,BURATTI,MAZZUOCCOLO
two F–cycles, say A. We write A = (a ,a ,...,a ), B = (b ,b ,...,b ) with
0 1 k−1 0 1 k−1
{a ,a }⊆V(A)∩V(B).
0 k−1 (cid:161) (cid:162)
We have V(A)∪V(B) ⊆ L V(A) . Let h ∈ G induce on A the reflection
A
exchanging a with a . We have Bh = B by Proposition 6.5 and so h also
0 k−1
induces a reflection on B. Let x and y denote the unique vertices which are fixed
(cid:161) (cid:162)
by h on A and B, respectively. If x (cid:54)= y then, since both x and y lie in L V(A) ,
(cid:161) (cid:162)
we see that h fixes L V(A) elementwise by Proposition 6.9, a contradiction since
(cid:161) (cid:162)
h exchanges the vertices a and a in L V(A) . If x=y then A and B share at
0 k−1
least three vertices and Proposition 6.6 yields the assertion.
Assume no two vertices in V(A)∩V(B) are adjacent on either A or B. Choose
two such common vertices z, z(cid:48) and let C be the F–cycle containing the edge
[z,z(cid:48)]. We have C (cid:54)= A, C (cid:54)= B and the previous discussion yields V(C) = V(A),
V(C)=V(B), whence also V(A)=V(B) as desired
Proposition 6.11. The common length k of all F–cycles is a prime.
Proof. Let C be a F–cycle. Define H to be the set of all F–cycles having at
least two, whence all vertices in V(C). If A ∈ H then we have V(A) = V(C)
and |H| = (k −1)/2. It follows from Proposition 6.10 that H is a hamiltonian
decomposition of the complete graph K having V(C) as its vertex–set.
k
Let S be the setwise stabilizer of V(C) in G. Since G preserves F–cycles we
see that S necessarily preserves all F–cycles having V(C) as vertex–set, in other
words S induces an automorphism group on H. Furthermore, the action of S on
V(C) is doubly transitive: as a matter of fact, if x, y and x(cid:48), y(cid:48) are two pairs of
distinct vertices in V(C) and if A and A(cid:48) denote the F–cycles containing [x,y] and
[x(cid:48),y(cid:48)]respectively,thenwehaveV(A)=V(A(cid:48))=V(C)byProposition6.10; there
exists g ∈ G with xg = x(cid:48), yg = y(cid:48) and consequently we have Ag = A(cid:48), g ∈ S. We
conclude that H is a doubly transitive hamiltonian decomposition of K and the
k
discussion in section 4 yields the assertion.
Proposition 6.12. Let F be a 2–factorization of the complete graph K admit-
v
ting an automorphism group G acting doubly transitively on V(K ). Then F arises
v
froma2–(v,k,1)designadmittingGasanautomorphismgroupactingdoublytran-
sitively on points.
Proof. Let V(K ) be the point–set. Declare a k–element subset of the point–set
v
to form a block if there exists a F–cycle having the given k points as vertices. Let
us prove that any two distinct points x, y lie on a uniquely determined common
block. Let A be the unique F–cycle containing the edge [x,y]. The subset V(A) is
a block by definition. Any other block containing x, y must have the form V(C)
for some other F–cycle C having x, y among its vertices. Proposition 6.10 shows
that we necessarily have V(C)=V(A), whence uniqueness. We have thus defined
a 2–(v,k,1) design D.
ThegroupGisadoublytransitivepermutationgrouponthepointsofD bythe
very definition. In order to show that G induces an automorphism group of the
design D, we have to show that each permutation g in G maps blocks to blocks,
but that is clearly a consequence of the fact that g maps F–cycles to F–cycles.