Partitionable graphs arising from near-factorizations of finite groups

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Partitionable graphs arising from near-factorizations of

finite groups

Arnaud Pêcher

To cite this version:

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near-fa torizations of nite groups

Arnaud P^e her

LaBRI, domaine universitaire, 351 ours de la Liberation, 33405 Talen e,Fran e

Abstra t

In1979, two onstru tionsformaking partitionablegraphswereintrodu edin[9 ℄.

The graphs produ ed by the se ond onstru tion are alled CGPW graphs. A

near-fa torization (A;B) of a nitegroup is roughly speaking a non-trivial

fa tor-izationofGminusoneelementintotwosubsetsAandB.EveryCGPWgraphwith

n verti esturns outto be a Cayley graphof the y li group Z n

, with onne tion

set(A A)nf0g, foranear-fa torization(A;B) ofZ n

.Sin ea ounter-exampleto

theStrongPerfe tGraphConje turewouldbeapartitionablegraph[14℄,any'new'

onstru tion formakingpartitionable graphsisof interest.

In this paper, we investigate the near-fa torizations of nite groups in general,

and their asso iated Cayley graphs whi h are all partitionable. In parti ular we

showthatnear-fa torizationsofthedihedralgroupsprodu eeveryCGPWgraphof

evenorder.Wepresentsomeresultsaboutnear-fa torizationsofnitegroupswhi h

implythataniteabeliangroupwithanear-fa torization(A;B) su hthatjAj4

mustbe y li (alreadyprovedin[7 ℄).Oneofthese resultsmaybeusedtospeedup

exhaustive al ulations. At last, we prove that there is no ounter-example to the

StrongPerfe tGraphConje turearisingfromnear-fa torizationsofaniteabelian

groupof even order.

Key words: partitionablegraph,perfe t graph,near-fa torization, group

2000 MSC:MSC 05C17,MSC 05C25,MSC20D60

1 Introdu tion

In 1960, Claude Berge introdu ed the notion of perfe t graphs: a graph is

perfe tif for every indu ed subgraph Hof it,the hromati number ofH does

not ex eed the maximum number of pairwise adja ent verti es in H. A hole

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graphs are exa tly the graphs with no indu ed odd holes and no indu ed

omplementof anoddhole, orequivalentlythat minimalimperfe tgraphsare

odd holes and their omplements. This onje ture is often alled the Strong

Perfe t GraphConje ture and has motivated many works.

Lovasz[12℄andPadberg[14℄gavesomepropertiesofminimalimperfe tgraphs.

Following the paperof Bland,Huang and Trotter[3℄, agraph Gis said to be

partitionableif thereexisttwointegerspand qsu hthat Ghas pq+1verti es

and for every vertex v of G, the indu ed subgraph Gnfvgadmits apartition

in p liques of ardinality q and also admits a partition in q stable sets of

ardinality p. Let ! denote the maximum ardinality of a lique of G and

denote the maximum ardinality of a stable set of G. Then it is lear that

p= and q=!.

Withthis denition, Lovasz [12℄and Padberg [14℄ proved that everyminimal

imperfe tgraphispartitionable.Thusa ounter-exampletotheStrongPerfe t

Graph Conje ture would lie in the lass of partitionable graphs. Hen e an

approa h to Berge's onje ture is to prove that a given lass of partitionable

graphsdoesnot ontainanyminimalimperfe tgraphwhi hisnotanoddodd

hole oranti-hole.

In 1979, Chvatal, Graham, Perold and Whitesides introdu ed two

onstru -tions for making partitionable graphs [9℄. In 1996, Sebo proved that there is

no ounter-example to the Strong Perfe t Graph Conje ture in the rst one

[16℄.In1984,Grinsteadprovedthatthereisno ounter-exampletotheStrong

Perfe t Graph Conje ture inthe se ondone [11℄. A variantof apartitionable

graph is a partitionable graph with the same verti es, the same maximum

liques and the same maximum stable sets. In 1998, Ba so, Boros, Gurvi h,

Maray and Preissmann [1℄ extended Grinstead's result to the wider lass of

the variants of the se ond onstru tion.

A graph with n verti es is ir ular if there exists a y li numbering of its

verti es (modulo n) su h that, for every vertex x, for every maximum lique

C and for every maximum stable set S, the set f( +x) (mod n) j 2 Cg

is a maximum lique and the set f(s+x) (mod n) j s 2 Sg is a maximum

stable set.

A normalized graph is a graph su h that for every edge fi;jg, there exists a

maximum lique ontaining both i and j.

A partitionable graph produ ed by the se ond onstru tion due to Chvatal,

Graham,PeroldandWhitesidesis alledaCGPWgraph,whereCGPWgraph

isthe abbreviationofChvatal-Graham-Perold-Whitesidesgraph.AnyCGPW

graph appears to be a ir ular normalized partitionable graph. The onverse

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onje -Conje ture 1 [1℄ Every ir ular normalized partitionable graph isa CGPW

graph.

We allit the ir ular partitionable graph onje ture.

In 1984, Grinstead laimed, through a omputer he k, that this onje ture

is true for graphs with a number of verti es at most fty, or sixty-one [11℄.

In1998, Ba so,Boros,Gurvi h,Maray and Preissmann proved itfor graphs

with size of maximum liques et most 5[1℄.

LetG be a nitegroup of order n with operation .Two subsetsA and B of

G of ardinality at least 2 are said to form a near-fa torization of G if and

only if n = jAj jBj+1 and there is an element u(A;B) of G su h that

AB = Gnfu(A;B)g. Let S be a symmetri subset of G whi h does not

ontainthe identity element e.The Cayley graphwith onne tion setS isthe

graph with vertex set G and edge set ffi;jg; i 1

j 2 Sg. We denote by

Cay(G;S) this graph. Noti e that the denitions of a Cayley graph given in

the literature may dier. The one we use in this paper isvery lose fromthe

denition given in the book 'Algebrai Graph Theory' of Norman Biggs [2℄.

Sin e S is a symmetri set su h that e 2= S, the graph Cay(G;S) is a simple

graph withoutloops, as are all graphsin this paper.

Let be any ir ular normalized partitionable graph with n verti es. Let C

be a maximum lique of and let S be a maximum stable set of . Then it

is easy to see that (C;S) is a near-fa torization of the group Z n

and that

is the Cayley graphof the nite group Z n

with onne tionset (C C)nf0g.

The onverse is true: if (A;B) is a near-fa torization of Z n

then the Cayley

graphwith onne tionset (A A)nf0g isa ir ularnormalizedpartitionable

graph [1℄.

Due tothis equivalen e, the se ond onstru tion of Chvatal, Graham,Perold

andWhitesideshadbeenrstdes ribedbyN.G.DeBruijnin1956[6℄,though

ina dierent ontext.

If (A;B) is a near-fa torizationof a nite group then the Cayley graph with

onne tion set (A 1

A)nfeg is a normalized partitionable graph (Se tion

2). This observation has motivated this paper: the main aim is to produ e

near-fa torizations of some nite groups, so as givingrise to 'new'

partition-ablegraphs.Wegive'new'near-fa torizations forthe dihedralgroups butthe

asso iatedCayleygraphs turnout alltobeCGPWgraphs(Se tion 3).These

near-fa torizations produ e all CGPW graphs of even order. In Se tion 2,

we give several results about near-fa torizations for nite groups in general,

whi h may be used to speed up exhaustive sear hes by omputer. We give

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abeliangroup ofeven orderisa ounter-example totheStrong Perfe t Graph

Conje ture.

2 Near-fa torizations of nite groups and partitionable graphs

A group is a non-empty set G with a losed asso iative binary operation ,

anidentity element e, and aninverse a 1

for every element a2G.If G has a

nite number of elements, then the ardinality of G is denoted by jGj and is

alledthe order of G.To avoida on i tof notation, we use the symbol to

denotethe standard multipli ationbetween two integers. Anabeliangroup is

a group G su h that is ommutative, that is gg 0

= g 0

g for all elements

g and g 0

of G.

IfX andY aretwosubsetsofG,wedenotebyXY thesetfxy; x2X; y 2

Yg. With a slight abuse of notation, if g is an element of G and X is subset

of G,we denoteby gX the set fggX and Xg the set Xfgg.Furthermore

jXj is the ardinality of X, that is the numberof elements of X. The subset

X issaid tobe symmetri if X =X 1 ,where X 1 isthe set fx 1 ; x2Xg.

Re all that two subsets A and B of ardinality at least 2 of anite group G

of order n forma near-fa torization ofG if and only if n=jAjjBj+1and

there is an element u(A;B) of G su h that AB =Gnfu(A;B)g: u(A;B)

is alledthe un overed elementof the near-fa torization. Sometimes, weshall

write simply u instead of u(A;B). The ondition about the ardinality of A

and B isrequired to avoid the trivial ase A=Gnfug and B =feg. Noti e

that every element x of Gdistin t fromu may be writtenin aunique way as

x = ab with a 2 A and b 2 B. Hen e a near-fa torization (A;B) may be

seen as atilingof Gnfu(A;B)g with prototile A.

The y li groupof ordernisthe groupwhi hisgeneratedby anelementx of

order n. This group is denoted by Z n

. For onvenien e, we use the following

representationof Z n

:the elementsof Z n

are the integers between 0 and n 1

and the operation is dened by x y = (x+y) (mod n). Due to this

denition of the operation of Z n

, we denote this operation by + rather than

.

Example 2 Let Z 13

be the y li group of order 13,

Let A=f0;1;2gand B =f0;3;6;9g.

Then A +0 = f0;1;2g, A+3 = f3;4;5g, A+6 = f6;7;8g and A+9 =

f9;10;11g. Thus A+B =(Z 13

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of Z 13

.

The following gure shows the tiling of Z 13 nf12g givenby (A;B).

0000000000000000000000

1111111111111111111111

000000

111111

000000

111111

000000

111111

00000000

11111111

0 1 2 3 4 5 6 7 8 9 10 11 12 A=f0;1;2g A+0 B =f0;3;6;9g A+3 n =13 A+6 u=12 A+9 Z 13

Figure1.Exampleof a near-fa torizationof Z 13

Note that if A and B are seen as sets of integers and + denotes the usual

addition between integers, then A+B is a tiling of the segment [0;11℄. This

onne tion is somewhat detailed in page 12.

The dihedral group D 2n

of even order 2n (with n 3) is the non-abelian

group generated by two elements r and s su h that:

r isof order n.

s isof order 2.

sr=r 1

s

The problem of hara terizing the near-fa torizations of the dihedral groups

is addressed inSe tion 3.

Letg 1

;:::; g n

betheelementsofthegroupGwithg 1

=e.IfRisanysubsetof

G,we denoteby M(R )the square nn (0;1)-matrixdened by M(R ) i;j =1 if and only if g j 2g i R .

LetI be the nn identity matrixand J bethe nn matrixwith allentries

equal to 1. Then De Caen, Gregory, Hughes and Kreher [7℄ observed that

(A;B) is a near-fa torization of G with un overed element e if and only if

M(A)M(B)=J I.

Sin e M(A)M(B) = J I implies that M(B)M(A) = J I ([5℄), we have

the followingproperty:

Lemma 3 [7℄ Let G be a nite group and A, B be two subsets of G. Then

(A;B) isa near-fa torization of G with u(A;B)=e if and only if(B;A) is a

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D 16

of order 16. Let A =fe;r 5 ;sr 5 g and B = fe;s;r;sr;sr 7 g. A small

al u-lation shows that AB = D 16

nfr 7

g. Thus (A;B) is a near-fa torizationof

D 16

, though (B;A)is not one as sr 5 =esr 5 =sr 5 .

The graph G(A;B) asso iated with a near-fa torization(A;B) is the Cayley

graph with onne tionset (A 1

A)nfeg.

If is a graph, we denote by !( ) the maximum ardinality of a lique of

and ( ) the maximum ardinality of a stable set of . We denote by V( )

the vertex set of and E( )the edge set of .

The graph with vertex set V is isomorphi tothe graph 0

with vertex set

V 0

ifthere exists abije tive mapf fromV ontoV 0

su hthat fi;jgis anedge

of if and only if ff(i);f(j)g is anedge of 0 . If e 0 is an edge of we denote by e 0

the subgraph of with vertex set

V( )and edgeset E( )nfe 0 g.Likewise,if e 0 isa non-edgeof ,wedenoteby +e 0

the graph with vertex set V( ) and with edge set E( )[fe 0

g. If v is

any vertex of , we denoteby nfvg the indu ed subgraph of with vertex

set V( )nfvg and edge set ffx;ygjfx;yg2V( ); x6=v; y6=vg.

Aperfe tmat hinginagraphwith2nverti esisasetofnnode-disjointedges.

Obviously, distin t near-fa torizations of a given group may give rise to the

same graph. In parti ular, we may left-shift A and right-shift B without

al-tering the asso iatedgraph:

Lemma 4 Let x and y be two elements of G. Then (xA;By) is a

near-fa torization of G su h that u(xA;By) = xu(A;B)y and G(xA;By) is

isomorphi to G(A;B).

PROOF. The proof is straightforward. 2

We say that (xA;By) isshift-isomorphi to(A;B).

Thus due to Lemma4, we may always assumethat the un overed element is

e, withoutaltering the asso iatedgraph.

In the ase of abelian groups, De Caen, Gregory, Hughes and Kreher gave a

useful property of near-fa torizations:

Lemma 5 [7℄ Let G be an abelian group and (A;B) be a near-fa torization

of G. Then there existtwo elements x and y of G su h that xA is symmetri

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y) = h(x)h(y) for all x and y of G. An inner-automorphism h of G is an

automorphism of G su h that there exists an element g of G whi h satises

h(x)=gxg 1

for allx of G.

Then we have this obvious Lemma:

Lemma 6 Let Cay(G,S) be a Cayley graph with onne tion set S of a group

G. Let h be any automorphism of G. Then the Cayley graph Cay(G,h(S)) is

isomorphi to Cay(G,S).

Ify isany elementof G,we denoteby hyithe y li subgroupof G generated

by y.The order of y is the smallestinteger k su h thaty k

=e and isdenoted

by o(y). An involution of G is anelement of G of order 2.The enter of G is

the set of all elements inG whi h ommute with every element of G.

Let H be any subgroup of G and (A;B) be a near-fa torization of G with

un overed elementu.

A right oset of H is any subset Hx with x 2 G. A left oset of H is any

subset xH with x2G. The proofof Lagrange's Theorem assertsthat forany

subgroup H of G, there exists a unique partition of G in right osets of H.

Likewise there exists a unique partitionin left osets of H. A subgroup H of

G isnormal if for every g of G, we have gH =Hg.

Aright-tile ofA isthetra e ofA ontoaright- osetof H,thatis thesubset T

is aright-tile of A if and onlyif there exists g in Gsu h that T =A\Hg. A

left-tileof A isthe tra e of A onto a left- osetof H

Theunique partitionofGinright osetsofH indu esaunique partitionof A

in right-tiles: let fHg 1

; :::; Hg d

g be the partitionof G in right- osets, then

the set of right-tilesof A isfA\Hg 1

; :::; A\Hg d

g. IfT is aright-tile of A

whi h is equal to awhole right- oset,then T is alled a H-right- oset.

Let be the partition of A in right-tiles indu ed by a given subgroup H.

Clearly fTb; T 2 ; b 2 Bg is a partition of G n fug. Hen e, given the

subgroup H, a near-fa torization (A;B) may be seen as a tiling of Gnfug

withtheright-tilesofAastiles.LetKbeanysu htileandbbeanyelementof

B.Noti ethatKbliesentirelyinaright- osetofH.ThusthistilingofGnfug

indu esatilingforeveryright- osetofH distin tfromHuandindu esatiling

of (Hu)nfug.Let Hg beany right oset ofH:weshall saythat the right-tile

K is used to over Hg if there exists an element b of B su h that Kb Hg.

Thetri kof many proofs inthis paperisto olle tenoughinformationsabout

the tilingof everyright- osetof H soasbeingable toget informationsabout

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Example 7 Let (A;B) be the near-fa torization of the dihedral group D 16 given byA =fe;r 5 ;sr 5 g and B =fs;r;sr;r 2 ;sr 2 g.

Let H := fe;sg be the y li subgroup of D 16

generated by s. Then fH, Hr,

Hr 2 ;:::;Hr 7 g is the partition of D 16

in right osets of H. Hen e A splits in

exa tly two right-tiles T 1 and T 2 with T 1 =feg=A\H T 2 =fr 5 ;sr 5 g=A\Hr 5 The tile T 2

isa H-right- oset. The set B has 5 elements, this implies that T 2 isused to over5 ofthe8right- osets ofH, namelytheright- osetsHr

3 , Hr 6 , Hr 4 , Hr 7 and Hr 5 be ause Hr 3 =T 2 s, Hr 6 =T 2 r, Hr 4 =T 2 sr, Hr 7 =T 2 r 2 and Hr 5 =T 2 sr 2 . ThetileT 1

isused exa tlytwi e to over theright- oset Hras Hr=fr;srg=

T 1 r[T 1 sr. The tile T 1

is used exa tly twi e to over the right- oset Hr 2 as Hr 2 =fr 2 ;sr 2 g=T 1 r 2 [T 1 sr 2

. Thelast timeT 1

isused, it isto overHnfeg

as Hnfeg=fsg=T 1

s.

The followinggure represents this tiling of the right- osets of H.

H e s T 1 s Hr r sr T 1 r Hr 2 r 2 sr 2 T 1 sr Hr 3 r 3 sr 3 T 1 r 2 Hr 4 r 4 sr 4 T 1 sr 2 Hr 5 r 5 sr 5 T 2 s Hr 6 r 6 sr 6 T 2 sr Hr 7 r 7 sr 7 T 2 sr 2 T 2 r T 2 r 2

The unique partitionof G in left osets of H also indu es a unique partition

of A in left-tiles. If T is a left-tile of A whi h is equal to a whole left- oset,

then T is alled aH-left- oset.

When the un overedelementise,weknowthat (B;A)isa near-fa torization

ofGtoo.Thusweget atilingofGnfeg withthe left-tilesofAastiles. LetK

be any su h tile and b be any element of B. Noti e that bK liesentirely in a

left- oset of H. Hen e we have a tilingfor every left- oset of H distin t from

Heand atilingof(He)nfeg.LetgH beany left osetofH:weshall saythat

the left-tileK is used to overgH if there exists anelement b of B su hthat

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group D 16 given by A = fe;r 5 ;sr 5 g and B = fs;r;sr;r 2 ;sr 2

g and the y li

subgroup H of D 16

generated by s.

As u(A;B)=e, we knowthat (B;A) isa near-fa torization of D 16 too. Noti e that fH;rH;r 2 H;:::;r 7 Hg is the partition of D 1 6 in left osets of H.

Hen e A splits in exa tly three left-tiles T 1 , T 2 and T 3 with T 1 =feg=H\A T 2 =fr 5 g=r 5 H\A T 3 =fsr 5 g=r 3 H\A

Thus no left-tile of A is a left- oset. This means that the tiling indu ed by

(B;A) is a tually dierent of the one indu ed by (A;B).

LetHg 1 ,Hg 2 ,...,Hg d

beapartitionof Ginright- osetsofH.LetX beany

subset of G.We denethe integer disp r H (X)as disp r H (X):=jfi; 1id; ;(Hg i \X( Hg i gj

The ounter disp r H

(X) is the number of right- osetsof H whi h meet X and

are not a subset of X.

Letdisp l H

(X)be the number of left- osets of H whi h meet X and are not a

subset of X. When H is a normal subgroup then we use rather the notation

disp H (X) instead of disp r H (X) ordisp l H

(X). The notationdisp H

isrelated to

the word 'dispersion'.

Let y be any element of G. A subset W of G is a left-y- hain (respe tively

right-y- hain) if jWj 6= jhyij and W an be written wfe; y; :::; y jWj 1

g

(respe tively fe; y; :::; y jWj 1

gw).

If H is a y li subgroup hyi, then it is useful to subdivide any tile of A in

right-y- hains. For onvenien y, these right-y- hains will be onsidered again

as tiles. Let T :=fe; y; :::; y jTj 1 gt and T 0 :=fe; y; :::; y jT 0 j 1 gt 0 be

two maximal right-y- hains of A not ne essarilydistin t. Let b and b 0

betwo

elements of B. The tileT 0

b 0

is said to be used afterthe tile Tb if and only if

t 0 b 0 = y jTj

tb. This implies that t 0 1 y jTj t =b 0 b 1 is an element of B B 1

. When this relation is all we need, we say simply that the tileT 0

is

used afterthe tileT (see gure 2).

The fa t that G(A;B) is a normalized partitionable graph may be dedu ed

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near-T = fe;y;y 2 gt T 0 = fe;ygt 0 > = > ;

are two right y hains of A PSfrag repla ements y 4 g y 3 g y 2 g y 1 g g yg y 2 g y 3 g y 4 g Tb T 0 b 0 hyig

Tb isused to overhyig

T 0 b 0 is used afterTb tb =y 1 g t 0 b 0 = y 2 g 9 > = > ; )b 0 =t 0 1 y 3 tb

Figure2.Fragment ofthe tilingof the oset hyig

fa torization(A;B)andthepartitionablegraphare loselyrelated,by

exhibit-ing the partition in maximum liques and the partition in maximum stable

sets of G(A;B)nfxg forevery x:

Lemma 9 If (A;B) is a near-fa torization of a nite group G su h that A

B =Gnfeg, then thegraph G(A;B) isa normalized partitionable graph with

maximum liques fxA; x2Gg and maximum stable sets fxB 1

; x2Gg.

PROOF.

Claim 10 For every x of G, xA is a lique of G(A;B)

Letx 1

and x 2

betwodistin telementsof xA: thereexista 1 and a 2 of A su h that x 1 =xa 1 and x 2 =xa 2 . Then x 1 1 x 2 =a 1 1 a 2 is an element of (A 1 A)nfeg.Thus fx 1 ;x 2

gis anedge of G(A;B),and so xAis a lique of

G(A;B) 2

Claim 11 For every x of G, xB 1

is a stable set of G(A;B).

Let x 1

and x 2

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su hthat x 1 =xb 1 and x 2 =xb 2 . If fx 1 ;x 2

g is an edge of G(A;B), then x 1 1 x 2 = b 1 b 1 2 is an element of A 1

A. Thusthere exista 1 anda 2 inA su hthat b 1 b 1 2 =a 1 1 a 2 .Hen e a 1 b 1 =a 2 b 2

.Sin e (A;B)isa near-fa torization,this implies thata 1 =a 2 and b 1 =b 2 .Thus x 1 =x 2 ,a ontradi tion. Hen e fx 1 ;x 2

g is not an edge of G(A;B).This implies that xB 1

is a stable

set of G(A;B). 2

Claim 12 For every x of G, G(A;B)nfxg is partitioned by the jBj liques

fxbA; b2Bg andisalsopartitioned bythe jAjstablesetsfxa 1

B 1

; a2Ag.

Hen e G(A;B) is a partitionable graph with ! =jAj and =jBj.

If there exists b in B su h that x 2 xbA then there is an element a in A

su h that x = xba thus e = b a, hen e b = a 1

and so ab = e in

ontradi tionwiththehypothesisAB =Gnfeg.Hen e S b2B xbAGnfxg. If xbA\xb 0 A 6=; with b and b 0

in B, then there are a and a 0 inA su h that xba =xb 0 a 0 thusba=b 0 a 0

. This implieswithLemma 3again that

a = a 0 and b = b 0 . Hen e j S b2B xbAj = P b2B jxbAj = jBjjAj = jGnfxgj. Thus S b2B

xbA=Gnfxgand fxbA; b2Bgis a partitionof Gnfxg.

If there exists a inA su h that x2xa 1

B 1

then there is anelement b in B

su hthat x=xa 1 b 1 thus e=a 1 b 1 andso e=ba: ontradi tion. Hen e S a2A xa 1 B 1 Gnfxg. Ifxa 1 B 1 \xa 0 1 B 1 6=; with a and a 0 in

A, then there are b and b 0 in B su h that xa 1 b 1 =xa 0 1 b 0 1 thus a 1 b 1 = a 0 1 b 0 1 and so b a = b 0 a 0

. This implies that a = a 0 and b=b 0 .Hen e j S a2A xa 1 B 1 j= P a2A jxa 1 B 1 j=jBjjAj=jGnfxgj.Thus S a2A xa 1 B 1 =Gnfxgand fxa 1 B 1 ; a2Ag isapartitionofGnfxg. 2

Claim 13 For every maximum liqueQ of G(A;B), there isan elementx of

Gsu hthatQ=xA, hen e theset ofthe n maximum liquesisfxA; x2Gg.

Likewise the set of the n maximum stable sets of G(A;B) is fxB 1

; x2Gg.

Sin e G(A;B) is a partitionable graph, we know that G(A;B) has exa tly n

maximum liques.Thuswearedoneifweshowthatforeverypair ofelements

x and y of G su h that x 6=y, we have xA 6=yA. This is equivalent to show

that if A=zA then z =e. Suppose A =zA. Then for every element a of A,

wehave that za isan elementof A. Thus A admitsapartitionin

hzi-right- osets. Hen e ! = 0 (modo(z)) where o(z) is the order of z. Thus n = 1

(modo(z)). As o(z) divides the number of elements of G,we alsohave n=0

(modo(z)). Therefore o(z) = 1 and so z = e. This proof also works for the

maximum stable sets. 2

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Letfx;ygbeany edge ofG(A;B). Then x y2A A, thus there exists a 2 A su h that y 2 xa 1 A. Obviously x 2 xa 1 A. Hen e G(A;B) is a normalized graph. 2

Sin ethe ardinalityofamaximum liqueofG(A;B)isequaltojAj,wedenote

by ! the value of jAj. Likewise, wedenote by the value of jBj.

A graph = (V;E) on ! + 1 verti es is alled a web, if the maximum

ardinality of a lique of is !, the maximum ardinality of a stable set of

is , and there is a y li al order of V so that every set of ! onse utive

verti es in this y li al order is an !- lique. Equivalently, normalized webs

with n verti es are graphs indu ed by any near-fa torization (A;B) of Z n su hthat A isan interval.

In 1979, V. Chvatal, R.L. Graham, A.F. Perold and S.H. Whitesides [9℄

in-trodu edamethodtoprodu ealarge lass ofnear-fa torizationsof the y li

groups Z n . Twosubsets A 1 and B 1

of Nare said to formanear-fa torizationin integers

if and only if A 1 +B 1 =[0::(jA 1 jjB 1 j 1)℄. Obviously, a near-fa torization

inintegersindu es anear-fa torization of Z

jA1jjB1j+1 . Let(A 1 ;B 1

)beanear-fa torizationinintegerssu hthatA 1 +B 1 =[0::n 1 2℄. Letk;k 0

be any positive integers.

One mayobtaina near-fa torizationinintegers(A 2 ;B 2 )su h thatA 2 +B 2 = [0::n 2 2℄with n 2 :=(jA 1 jk)(jB 1 jk 0 )+1 by dening: A 2 :=A 1 +(n 1 1)[0::k 1℄ and B 2 :=B 1 +(n 1 1)k[0::k 0 1℄

A CGPW graph is a graph G(A;B) where (A;B) is obtained with a nite

number of appli ations of this method starting from a basi fa torization,

that is a near-fa torization (A 1 ;B 1 ) su h that A 1 = [0::jA 1 j 1℄ and B 1 = jA 1 j[0::jB 1 j 1℄.

Expli itly, the CGPW graph G given by 2p positive integers k 1

;:::;k 2p

is

onstru ted inthis way:

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takek =k 5 and k =k 6 then al ulateA 3 and B 3 startingfromA 2 andB 2 . Set n 3 =k 1 k 2 k 3 k 4 k 5 k 6 +1. ... until k =k 2p 1 and k 0 =k 2p . Gis G(A p ;B p )and isdenoted by C[k 1 ;:::;k 2p ℄. By onstru tion,jA p j=k 1 k 3 ::: k 2p 1 =!,jB p j=k 2 k 4 ::: k 2p =andn p =k 1 k 2 ::: k 2p + 1= !+1.

Noti e that normalized webs are CGPWgraphs su h that p=1.

Following [1℄, a near-fa torization produ ed by this method is alled a De

Bruijn near-fa torization.

LetX be any subset of the group G. We set

INT (X)= max x2G;y2G; x6=y

fjxX\yXjg

Noti e that INT(A) denotes the maximum ardinality of the interse tion

be-tween twodistin t!- liquesof G(A;B)andthat INT (B 1

)denotes the

max-imum ardinality of the interse tion between twodistin t -stable sets.

An edge e of a graph is said to be an - riti al edge if and only if (

e) > ( ). Similarly, a non-edge e 0

is said to be o- riti al if and only if

!( +e 0

)>!( ).Itiseasyto he kthatagraphG(A;B)hasa o- riti al

non-edge(respe tively - riti aledge)if and onlyifINT (A)=! 1(respe tively

INT (B 1 )= 1). Lemma 15 INT (X)= max g2Gnfeg fjX\gXjg

PROOF. The proof is straightforward. 2

Next lemmawill beused inthe proofs of this arti le:

Lemma 16 Let Gbe anite group havinganear-fa torization (A;B).Let H

be any normal subgroup of G. If there is a H- oset (Ha) in A, then in every

oset of H, a tile T of A may be used at most on e.

PROOF. LetT beany tile of A: thereexists y of G su h that T =A\Hy.

Letg beany elementof Gand letB g

bethe set fb 2B; Tb Hgg.We want

toshow that jB g

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If jB g

j 2 then there exist two distin t elements b and b of B su h that

Tb Hg and Tb 0

Hg. From T Hy, we get Hg = Hyb and Hg =

Hyb 0

. Then Hab = ay 1

Hyb be ause H is a normal subgroup. Thus Hab =

ay 1 Hg =ay 1 Hyb 0 =Hab 0

.Sin e(A;B)isanearfa torizationandHaA,

fb;b 0

gB, this implies that b=b 0

: a ontradi tion. Hen e jB g

j1. 2

Noti e that Example 7 shows that the hypothesis that H must be normal is

a tually needed.

We are now ready to state the main result of this paper.

Theorem 17 Let G be a nite group admitting a near-fa torization (A;B).

Let H be a non-trivial proper subgroup of G. Then

(1) disp r H

(A)>0 and disp l H (A)>0. (2) if disp r H (A)=1 or disp l H (A)=1 thenjHj=2.

(3) if H is a normal subgroup, disp H

(A)=2 and jAj6=2, then jHj= n 2 .

PROOF. Sin e no spe ial property is required for B, we may assume that

u(A;B)=e sin eotherwise allwehavetodoistoright-shiftB by u(A;B) 1

.

Hen e we haveAB =Gnfeg=BA (Lemma 3).

(1) Ifdisp r H

(A)=0,then everyright-tile of A isa H-right- oset.Let T be a

right-tile of A whi h is used to over the right- oset He. There exists b

of B su h that Tb He. Sin e T is a H-right- oset, we have Tb = He.

Hen e e2AB, a ontradi tion. Thus disp r H

(A)>0.

Likewise, we have disp l H

(A)>0.

(2) Suppose that disp r H (A)=1. LetHg 1 , Hg 2 , ..., Hg d be apartition of G

in right- osets of H. Sin e disp r H

(A) = 1 there exists a unique integer p

between 1and dsu hthat ;(A\Hg p (Hg p .LetA 0 :=A\Hg p .Thus

the set of right-tiles of A is A 0

and some H-right- osets.

Letb be anelement of B su h that A 0

b He. Then we have Hg p

b =

He, whi himplies that (g p

b) 2He. Thus, if for every b in B, we have

A 0

b He, then g p

B He. We know that (B;A) is a near-fa torization

with u(B;A)=e. Hen e (g p

B;A) isa near-fa torizationwith un overed

elementg p .Asg p B He,g p

B hasonlyoneright-tile.Sin eH isaproper

subgroup of G, there exists a right oset Hx distin t from He. Thus

jHxj = 0 (modjg p

Bj) = 0 (mod ), whi h implies n =0 (mod ),

ontradi ting the relationn =!+1.

Hen ethereexists binB su hthat A 0

b liesina osetHxdistin tfrom

He. Obviously A 0

is the only tile of A whi h an be used to over Hx

be ause the other tiles are H-right- osets thus jHxj = 0 (mod jA 0

j).

The tile A 0

is again the only tile whi h an be used to over He, thus

jHej=1 (mod jA 0

j). Hen e jA 0

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Let H be the onjugate subgroup g p Hg p of H. Let Hg 1 , Hg 2 , ..., H 0 g 0 d be a partition of G in right- osets of H 0 . For every i between 1 and d, let B i := B \H 0 g 0 i

. Then for every i between 1 and d, we have

(A 0 B i )(Hg p g 1 p Hg p g 0 i )=Hg p g 0 i .

Leti beany integerbetween 1and d.IfB i 6=;then A 0 isused atleast on e to over Hg p g 0 i . Thus Hg p g 0 i

is overed with the right-tile A 0 only. Hen ewehave(Hg p g 0 i )nfeg=[ b2B; A 0 bHgpg 0 i A 0

b.Letbbeanyelementof

B and letj bethe integersu hthatb 2B j . ThusA 0 bHg p g 0 j =g p H 0 g 0 j . Hen e, if b is not in B i then A 0 b is not a subset of Hg p g 0 i . Thus we have A 0 B i =(Hg p g 0 i )nfeg.Sin ejA 0 j=1,wemusthavejB i j=j(Hg p g 0 i )nfegj.

Hen e we have for all i between 1 and d, jB i j = 0 or jB i j = jHg p g 0 i n

fegj. Thus disp r

H 0

(B) 1. We know that disp r

H

0(B) = 0 is impossible

a ording tothe rst se tion of the proof of this Theorem. Therefore we

have disp r

H 0

(B) = 1. There exists a unique integer p 0 between 1 and d su h that B p 0 6= ; and B p 0 6= H 0 g 0 p 0. We set B 0 := B p 0 . Then we get jB 0

j=1 aswe have seen forA 0 . We have A 0 B 0 = (Hg p g 0 p 0) n feg. If Hg p g 0 p 0

6= He, then we have

jHj = jA 0

B 0

j = 1, hen e H is the trivial subgroup: a ontradi tion.

ThusHg p g 0 p 0

=He, whi h implies jHj=2 as required.

If disp l H

(A) = 1 then the same proof may be applied to the

quasi-fa torization (B;A) by working with the left- osetsof H.

(3) Noti e that H is assumed tobenormal.

Sin e disp H

(A)=2,there exist two distin t osets Hg 1

and Hg 2

of G

su hthat;(A\Hg 1 (Hg 1 and;(A\Hg 2 (Hg 2 .LetA 1 :=A\Hg 1 and A 2 :=A\Hg 2 .

If there is a H- oset in A then by Lemma 16, A 1

(and A 2

) annot

be used twi e on the same oset. Thus A 1

is used at least on e on a

oset distin t from He otherwise we would have 1. LetHv be su h

a oset. Obviously Hv is not overed with only A 1 be ause A 1 is not a H- oset. Hen e A 1 and A 2

are used exa tly on e to over Hv. Thus

jHvj=jA 1 j+jA 2 j.Hen e n =0 (modjA 1 j+jA 2 j). IfC is any H- oset of A, we have jCj= jHj= jA 1 j+jA 2 j. Thus ! =0 (modjA 1 j+jA 2 j).

Fromn=!+1,wegetn=1 (modjA 1 j+jA 2 j) ontradi tingn=0 (modjA 1 j+jA 2

j). Therefore there isnoH- oset in A.

ThusA=A 1

[A 2

.AsH isapropersubgroup ofG,thereexists xsu h

that He\Hx=;.

IfjA 1

j=jA 2

j,thendue tothe overof Hx,weget n=0 (mod jA 1

j).

From n =!+1, we have n = 1 (mod jA 1

j). Thus jA 1

j = 1. This

means that jAj = 2, whi h is ontradi tory to the hypothesis of the

Theorem. Hen e jA 1

j6=jA 2

j and we may assume that jA 1

j>jA 2

j.

IfzisanyelementofG,letn z (A 1 )(respe tivelyn z (A 2 ))bethenumber

of times the tileA 1

(respe tively A 2

) isused to over the oset Hz, that

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max 2 z2G z 2 min 2 z2G z 2 Claim 18 n max (A 1 )=n max (A 2 ) n min (A 1 )=n min (A 2 )

PROOF. Letb beany elementof B and z beany elementof G.

IfA 1 bHzthenb2Hg 1 1 zasA 1 Hg 1

andHisanormalsubgroup

of G. From A 2 Hg 2 , we get A 2 b Hg 2 Hg 1 1 z =Hg 2 g 1 1 z. Likewise, if A 2 b Hg 2 g 1 1 z then A 1 b Hz. Hen e A 1 Hz if and only if A 2 b Hg 2 g 1 1

z. And so for any z in G, there exists z 0 and z 00 su h that n z (A 1 )=n z 0(A 2 )and n z (A 2 )=n z 00(A 1 ). Thus n min (A 1 ) = n min (A 2 ) and n max (A 1 ) = n max (A 2 ). Let n max := n max (A 1 ) and n min :=n min (A 1 ). 2 Claim 19 n max >n min PROOF. Ifn max =n min then jHxj=n min (jA 1 j+jA 2 j)and son=0 (mod!), ontradi ting n =!+1. 2

Tosimplify the notation, leta 1 =jA 1 j and leta 2 =jA 2 j. Claim 20 n max =n min +1, a 1 =a 2 +1 and jHj=n max a 1 +n min a 2 .

PROOF. If g is any element of G, we set (g) = 1 if Hg = H and we

set (g)=0otherwise.

Letz bean element of Gsu h that n z

(A 2

)=n max

(by denition su h

anelement exists), we rst show that n z (A 1 )=n min .

Bydenitionthereexistsg inGsu hthatn g (A 1 )=n min .Letkn min and ln max

beintegers su h thatjHzj=ka 1 +n max a 2 +(z)=jHgj= n min a 1 +la 2 +(g).Weget that(k n min )a 1 =(l n max )a 2 +(g) (z). Sin e k n min 0, a 1 > a 2 1, 1 n max 0, (g) (z) 1, we get that k =n z (A 1 )=n min .

Now leth be anelement of G su h that n h (A 1 )=n max . WehavejHzj=n min a 1 +n max a 2 +(z)=jHhjn max a 1 +n min a 2 +(h) and so (z) (h) (n max n min )(a 1 a 2 ). Sin e n max > n min 0, a 1 > a 2 0 and (z) (h) 1, we get n max = n min +1, a 1 =a 2 +1, (z)=1, (h) =0 and n h (A 2 )=n min

. Noti e that from these equalities

jHj=n max a 1 +n min a 2 =n min a 1 +n max a 2 +1. 2 Claim 21 H is of ardinality n 2

PROOF. Letz beany element of G.From what pre edes itis not

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z 2 max then jBj = P i=1;:::;d n g i (A 1 ) = P i=1;:::;d n g i (A 2 ) = (d 1)n max +n min = (d 1)n min +n max . Sin e n max 6=n min

, this implies that d=2.

2

Example 22 Let (A;B)be thenear-fa torization ofD 16 introdu edin Exam-ple 7: A=fe;r 5 ;sr 5 g and B =fr;r 2 ;s;sr;sr 2 g. Let H 1 :=fe;sr 5 g. Sin e disp H 1 (A)=1, H 1 must be of ardinality 2. Let H 2 := fe;r;r 2 ;r 3 ;r 4 ;r 5 ;r 6 ;r 7 g. Sin e disp H 2

(A) = 2, jAj 6= 2 and H 2 is normal, H 2 must be of ardinality 16 2 =8.

Theorem 17 may be used to de rease the number of ases to be investigated

when looking for a near-fa torization for a given group with the help of a

omputer. From the list of all subsets A of G of ardinality !, we may keep

onlythose satisfyingTheorem 17and then for every of these A he k if there

exists asubsetB of ardinalitysu h that(A;B)isa near-fa torization.For

every group of smallorder (that is less than 1000),it is quite easyto get the

listofallsubgroups ofGandthe listofallnormalsubgroups ofGusingGAP

[10℄forinstan e. Theorem17isaninterestinglterbe auseitmaybeapplied

to any group. Our implementation [15℄ revealed that it performs quite well

when! or issmallasonemightexpe t. In somegroups,there isnosubsets

at all satisfying Theorem 17 with the required ardinality. For instan e, the

only groups of order 16 with a subset A of ardinality 3 satisfying Theorem

17are the dihedralgroups and y li groups.

We willuse Theorem 17to deriveLemma 24and Lemma28.

Lemma 23 l If ! =3, A issymmetri and n isodd then G(A;B) is a web.

PROOF. Sin en isodd, there is noinvolution inG. This implies with A=

A 1

that there is a in G su h that A = fa 1

;e;ag. Let H be the y li

subgroupgenerated bya. Noti ethatA H,thusdisp r H (A)=disp l H (A)=1.

If H is distin t fromG then by Theorem 17, we must have jHj=2,whi h is

impossible as n is odd. Thus G is a y li group. Sin e ! = 3, G(A;B) is a

web [1℄. 2

Andras Sebo proved in [16℄ that the minimal imperfe t graphs ontaining

ertain ongurations oftwo - riti aledges and one o- riti alnon-edge are

exa tlythe odd holes or anti-holes.

S.Markossian, G.Gasparian, I.Karapetianand A. Markosian alsostudiedin

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Re allthatagraphG(A;B)hasa o- riti alnon-edgeifandonlyifINT (A)=

! 1. Next Lemma partially hara terizes graphs G(A;B) with a o- riti al

non-edge.

Lemma 24 Let G be a nite group su h that every involution z ommutes

with every element of G. If (A;B) is a near-fa torization of G su h that

INT (A)=! 1 then G is a y li group and G(A;B) is a web.

PROOF. Sin eINT (A) =! 1, by Lemma 15there exists anelement y of

Gsu hthat jA\yAj=! 1.LetH bethe y li subgroupofGgeneratedby

y. Noti e that Aadmits aunique partitionin maximalright-y- hains and

H-right- osets. Letk be the number of maximalright-y- hains inthis partition.

Then we have jA\yAj= ! k. Thus there is exa tly one maximal

right-y- hain inA. LetT :=fe; y; y 2

; :::; y jTj 1

gt be thismaximal right-y- hain.

Noti ethat T isasubsetof aH-right oset.Therefore wehavedisp r H

(A)=1,

asthe right-tilesof A are T and H-right- osets,

Obviouslyy6=e, hen eH isnot the trivialsubgroup ofG. Thusby Theorem

17,we have H=G orjHj=2.

IfjHj=2thenyisaninvolutionofGdistin tfrome,andwemusthavejTj=

1. Hen e there must be some H-right- osets in A. The element y ommutes

with every element of G, hen e H is a normal subgroup of G. If T is used

only on the oset Hu(A;B), then 1, whi h is impossible. Therefore T is

used inthe over ofanother oset Hx.As onlyT isused on Hx,it isused at

least twi e, whi h is in ontradi tion with Lemma 16 be ause H is a normal

subgroup of G.

Therefore H =G, that isG is a y li group.

Hen e A=T and G(t 1

A;B) isa web. Thus G(A;B)whi his isomorphi to

G(t 1

A;B) is aweb. 2

Lemma 24 is not true if the hypothesis that every involution is in the enter

of G is not assumed. Indeed the dihedral groups are examples of non- y li

groups having near-fa torizations (A;B) and INT (A) = ! 1 (see Se tion

3). Besides we give in Se tion 4, a graph G(A;B) with 50 verti es su h that

INT (A)=! 1, whi h isnot aweb.

Corollary 25 IfGisanon- y li niteabeliangroupthenitadmitsno

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near-fa torization (A;B) su h that INT (A)=! 1.

PROOF. Indeed thereis no involution ina group of odd order. 2

Example 27 Let G be any group of order 3p+1 (p a prime) su h thatits

enter ontainsallitsinvolutions,withasymmetri near-fa torization(A;B).

WemayassumethatjAj=3. Sin ejAj isoddandA issymmetri ,there must

be an element w in A su hthat w 2

=e. Let a be another element in A. Thus

fa; wgA\awAand soINT (A)2. ThenbyLemma24, Gmustbe y li .

This impliesfor instan e that7 groups,out of the 14 groups of order 16, have

no symmetri near-fa torizations.

There are manynon-abeliangroups ontainingin their enteralltheir

involu-tions:a ordingtoGAP [10℄ thereare58 su hgroupsout ofthe 267groupsof

order 64, and52su hgroupsoutof the231groupsoforder 96.Noti ethatfor

n=64 or 96, ! or must be prime, hen e any CGPW graph of these orders

isa web. Thus if any of these groups has a near-fa torization (A;B) then the

graph G(A;B) is nota CGPW graph. Noti e thatfor n=64, these groups do

not have any symmetri near-fa torization (A;B) su hthat jAj=3.

Lemma 28 Let G be a nite group su hthat allits y li subgroups are

nor-mal and admitting a near-fa torization (A;B) su h that INT (A) = ! 2.

Then

If G is abelian then G is y li .

If G isnot abelian thenthe order of G isa multiple of 4, G has an element

y of order n 2 and y n 4

is the only involution of G .

PROOF. Sin eINT (A)=! 2, wehave !3 andthere exists an element

y of Gsu h that jA\yAj=! 2.Let T 1 :=fe; y; y 2 ; :::; y jT 1 j 1 gt 1 and T 2 :=fe; y; y 2 ; :::; y jT 2 j 1 gt 2

bethe twomaximal right-y- hainsof A.Let

u be the un overed element. Let H be the y li subgroup generated by the

element y.Hen e by assumption onG,H isa non-trivialnormal subgroupof

G:

IfG=HthenGisabelianand y li ,thuswearedone.Hen ewemayassume

that H (G.

Sin e A is made of T 1

, T 2

and some H- osets, we have disp r H

(A) 2. By

Theorem 17, we have disp r H

(A) > 0. If disp r H

(A) = 1 then by Theorem 17

again, we get jHj = 2. Sin e disp r H (A) = 1, T 1 and T 2

must lie in the same

right- osetofH.ThusT 1

[T 2

isaH- oset,andthisimpliesthatdisp r H

(A)=0,

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Hen edisp H

(A)=2andbyTheorem17again,H has ardinality n 2 .Therefore y is anelement of order n 2

and there is noH- oset inA.

Claim 29 We have jT 1

j6=jT 2

j.

PROOF. Suppose that jT 1

j = jT 2

j. As there is no H- oset in A, we have

jHj = 1 (mod jT 1

j) due to the over of the oset Hu(A;B). Then we also

havejHj=0 (mod jT 1

j) dueto the overof the other oset.Hen e jT 1

j=1.

This implies that jAj=2. This is impossibleas !3. 2

ThusjT 1

j6=jT 2

j and we mayassume that jT 2

j<jT 1

j.

Claim 30 The pair fHt 1

; Ht 2

g is a partition of G in right osets.

PROOF. Ift 1

and t 2

liein the same right oset then disp r H (A) 1, ontra-di tingdisp r H (A)=2. ThusHt 1 \Ht 2 =;. As jHj= n 2 , we are done. 2 Claim 31 We have (Ht 1 ) 1 =Ht 1 and (Ht 2 ) 1 =Ht 2 .

PROOF. Suppose that H = Ht 1

then we obviously have (Ht 1 ) 1 = Ht 1 .

Sin e the inversion map is a bije tive map, this implies that (Ht 2 ) 1 =Ht 2 .

The proof for the ase H =Ht 2

issimilar. 2

Claim 32 If G is abelian then G isa y li group.

PROOF. If G is abelian then let b be any element of B distin t from t 1 2 y jT2j u, that is, T 2

b is not followed by the un overed element u. Hen e

T 2 b is followed by a tile T 2 b 0 or by a tile T 1 b 0 , that is t 2 b 0 = y jT 2 j t 2 b or t 1 b 0 =y jT 2 j t 2 b.Thusb 0 =y jT 2 j borb 0 =y jT 2 j t 1 1 t 2 b.Ifb 0 =y jT 2 j b then t 1 b 0 =t 1 y jT2j b.Sin ejT 2 j<jT 1 j,y jT2j t 1 isanelementofT 1 .Thusy jT2j t 1 isanelementofAandwehavea ontradi tion.Therefore b

0 =y jT 2 j t 1 1 bt 2 . Let y 0 := y jT 2 j t 1 1 t 2

. We have seen that for every element b of B ex ept

maybe one, y 0

b is anelementof B. Thus INT (B)= 1.Sin e Gis abelian,

(B;A)isobviouslyanear-fa torizationofG.Hen e by Lemma24,G must be

y li . 2

Claim 33 If G is not abelian then n is a multiple of 4 and y n 4

is the only

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Letq be anelement of Gsu h that Hq 6=H.

Ifnisnotamultipleof4thenjHjisodd.Hen edue toFa t 31thereexists at

least one elementz inHq su h that z 2

=e. Sin ehzi isanormal subgroupof

G, z must ommute with every element of G and in parti ular with y. Sin e

z is an element of Hq, there exists an integer i su h that z = y i q. From zy =yz, we get y i qy =y i+1

q. Thus qy =yq. Due to Fa t 30,

G must be abelian, whi h is impossible. Thus n is a multiple of 4 and so y n 4

is aninvolutionof G.

Obviouslyinthe osetH thereare exa tlytwoinvolutions:the elementseand

y n 4

. Thusif there isanother involution inGthen there must be aninvolution

z in Hq, and we have seen that in this ase G must be abelian, whi h is

impossible.Hen e weare done.

2

Corollary 34 If (A;B) is a near-fa torization of a nite abelian group G

su h that jAj4 then G is y li [7℄ and G(A;B) is a CGPW graph.

PROOF. Let(A;B)be anear-fa torizationof Gsu hthat jAj4.Sin e G

is abelian,we use the additivenotation+ todenote the operation of G.

IfjAj3thenobviouslyINT (A)! 2.ThusGis y li byLemma 28and

Corollary25. Then itis proved in[1℄ that G(A;B)must be aCGPW graph.

If jAj =4 then n is odd and there is noinvolution in G. By Lemma 5, there

existx andy inGsu h that(x+A;B+y)isa symmetri near-fa torization.

LetA 0 :=x+A.Sin eA 0 = A 0

andthere isnoinvolution,there area anda 0 inG su hthat A 0 =fa;a 0 ; a; a 0 g. Thenfa;a 0 gA 0 \A 0 +(a+a 0 ).Hen e INT (A 0

)! 2.ByLemma28and Corollary25,Gmust bethe y li group.

ThusG(A;B)G(A 0

;B 0

) is aCGPW graph [1℄. 2

Example 35 The Quaternion group Q 8

of order 8 is an example of a

non-abelian nite group su hthat all its y li subgroups are normal.

There does not seem to be many non-abelian groups su h that all their y li

subgroups are normal.A ording to GAP, there isonly one (out of 267) su h

group of order 64: the262 th

group.As it has no element of order 32, weknow

that is has no near-fa torization (A;B) su h that jAj = 7 and INT (A) 5.

There is also only one (out of 231) su h group of order 96: the 222 th

group.

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the minimal imperfe t graphs in the lass of the graphs produ ed by

near-fa torizationsof nitegroups. We rstneed tore all someresults about

min-imalimperfe t graphs.

A small transversal is a subset of verti es T su h that T is of ardinality at

most!+ 1andT meetseverymaximum lique andeverymaximumstable

set.

In1976,V. Chvatalfound avery useful propertyof minimalimperfe t graphs

whi hstates thataminimalimperfe tgraph ontainsnosmalltransversal[8℄.

In 1998, G. Ba so, E. Boros, V. Gurvi h, F. Maray and M. Preissmann

[1℄ introdu ed a suÆ ient ondition for partitionable graphs to have a small

transversal alledthe'ParentsLemma'.Amaximum liqueKofGisamother

ofavertex x2K ifeverymaximum liqueK 0

ontainingxsatisesjK\K 0

j

2.Similarly,amaximumstablesetS ofGisafatherofavertexx2S ifevery

maximum stable set S 0

ontainingx satises jS\S 0

j2.

Lemma 36 'The Parents Lemma' [1℄ If a vertex of a partitionable graph

has a fatherand a mother then the graph has a small transversal.

Then we have the followingresult:

Lemma 37 Let G be a nite group of even order su h that every

involu-tion y ommutes with every element of G. If (A;B) is any symmetri

near-fa torization of Gthen G(A;B)has asmalltransversal, hen eis notminimal

imperfe t.

PROOF. Sin en iseven, ! and are ne essarily odd.

As!isodd,thereisanelementyofAsu hthaty 2

=e.Weare goingtoshow

that Ais amotherof y.LetpA be any!- lique ontaining ydistin tfrom A.

Hen ethere isainAsu hthat y=pa.Ifa 1 =ythen p=ya 1 =y 2 =e

and so pA =A, a ontradi tion. Thus a 1

is not equal to y. We have a 1

=

yp = py be ause y ommutes with p. Thus a 1

is an element of pA.

Hen e fa 1

; ygA\pA. This means that A isa mother of y.

Likewise there exists an element x of B su h that x 2 = e and B = B 1 is a fatherof x.Hen e yx 1 B =yx 1 B 1

isafatherof y.Byapplyingthe Parents

Lemma,we see that the graph G(A;B)has asmall transversal. 2

Corollary 38 Let G be a nite abelian group of even order. If (A;B) is any

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In this se tion,we showhow to arry any near-fa torizationof a y li group

of even order tothe dihedral groupof the same order.

We begin by introdu ing amap from Z 2n intoD 2n . Aneven elementof Z 2n isanelementof 2Z 2n

. Theodd elementsare the other

elements of Z 2n

. Noti e that if x is an even element of Z 2n

, then there exists

a unique integer y between 0 and (n 1)su h that x=2y. Wedenote by x

2

this integer.

If xand y are two even elements of Z 2n then we have x+y 2 = x 2 + y 2 (modn)

and if x is any element of Z 2n

then we have 2x

2

=x (mod n).

Let be the bije tivemap of Z 2n onto D 2n dened by: : Z 2n ! D 2n x is even 7! r x 2 x is odd 7! sr x 1 2

We now state some properties of whi h are useful for the proofs:

Lemma 39 For every x and y of Z 2n , we have if y is even, (x)(y) 1 =(x y) and (x+y)=(x)(y). if y is odd, (x)(y) 1 =(y x).

PROOF. If x and y are even then we have (x +y) = r x+y 2 = r x 2 + y 2 = r x 2 r y 2 =(x)(y)and(x y)=r x y 2 =r x 2 + y 2 =r x 2 r y 2 =(x)(y) 1 .

If x is odd and y is even then we have (x+ y) = sr x+y 1 2 = sr x 1 2 + y 2 = sr x 1 2 r y 2 = (x)(y) and (x y) =sr x y 1 2 =sr x 1 2 y 2 =sr x 1 2 r y 2 = (x)(y) 1 .

Hen e, if y is even then we have (x+y)=(x)(y) and (x)(y) 1

=

(x y).

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If x is odd and y is odd then we have (x)(y) 1 = sr x 1 2 (sr y 1 2 ) = r y x 2 =(y x).

Hen e, if y is odd then we have (x)(y) 1

=(y x). 2

From a near-fa torization (A;B) of Z 2n

, we get a near-fa torization of D 2 n this way:

Algorithm 1 Carrying anear-fa torization of Z 2n

into D 2n

Input: a near-fa torization(A;B) of Z 2n Output: anear-fa torization(A 0 ;B 0 )of D 2n Step 1: nd an element x of Z 2n

su h that A +x is symmetri and let

A 1

:=A+x (exists by Lemma 5).

Step 2:take anelement a 1 of A 1 and letA 2 :=A 1 +a 1 . Step3:letB 0

be theset of the even elementsofB and B 1

bethe set ofthe

odd elements of B. Then take A 0 :=(A 2 ) and B 0 :=(B 0 )[(B 1 )r a1 . We say that (A 0 ;B 0

) is a dihedralnear-fa torization asso iated to (A;B). We

allDe Bruijn dihedralnear-fa torization anydihedral near-fa torizations

as-so iated toa De Bruijn near-fa torization.

Obviouslyonemayget several distin tnear-fa torizations ofD 2n

throughthis

algorithm from one near-fa torization of Z 2n

as x is not uniquely dened in

Step 1 and neither isa 1

in Step 2.

We rst prove that any ouple (A 0

;B 0

) produ ed by this algorithm is indeed

anear-fa torizationof D 2 n

.

Theorem 40 Let (A;B) be a near-fa torization of Z 2n . Let (A 0 ;B 0 ) be an

output of algorithm 1 with input (A;B). Then(A 0 ;B 0 ) is a near-fa torization of D 2n .

PROOF. Re all that due to the algorithm, we have A 0 = (A 2 ) and A 2 = A 1 +a 1 where A 1 issymmetri and a 1 is anelement of A 1 .

Claim 41 For every b of B, there exists b 0

in B 0

su h that (A 2 +b)=A 0 b 0 .

PROOF. Ifb iseventhenletabeanyelementofA 2

.ByLemma39,wehave

(a+b) =(a)(b). Hen e (A 2

+b)(A 2

)(b). Sin e is a bije tive

map, we get (A 2

+b)=(A 2

)(b) with (b)2B 0

. Thus we are done.

If b is odd then let a be any element of A 2 . By denition of A 2 , a a 1 is an elementofA 1

,whi hisasymmetri set.Hen ea 1

aisanelementofA 1

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2a 1 a is an element of A 2 . Noti e that 2a 1

+b is odd. Let b :=(2a 1 +b). As(2a 1 +b)=sr a 1 + b 1 2 =sr b 1 2 r a 1 ,b 0 isanelementof B 0 .Ifa iseven then (2a 1 a)b 0 =r a 1 a 2 sr a 1 + b 1 2 =sr a+b 1 2

=(a+b). Hen e (a+b)2A 0

b 0

.

If a is odd then (2a 1 a) b 0 = sr 2a 1 a 1 2 sr 2a 1 +b 1 2 = r a+b 2 = (a +b). Thus (a+b) 2A 0 b 0

. Therefore we have (A 2

+b) A 0

b 0

. This implies that

(A 2 +b)=A 0 b 0

be ause is abije tive map. 2

Claim 42 The ouple (A 0 ;B 0 ) is a near-fa torization of D 2n .

PROOF. Wehaveseenthatf(A 2 +b); b 2BgfA 0 b 0 ; b 0 2B 0 g.Sin e is

abije tivemap,thereexistsuinD 2 n

su hthatf(A 2 +b); b2Bgisapartition ofD 2 n nfug.As B and B 0

are ofequal ardinality,weget thatfA 0 b 0 ; b 0 2B 0 g is apartitionof D 2n nfug. Therefore (A 0 ;B 0 )is a near-fa torizationof D 2n . 2 Example 43 A 2 =f0;1;2;9;10;11;18;19;20g B=f0;3;6;27;30;33;54;57;60g A 0 =fe;s;r;sr 4 ;r 5 ;sr 5 ;r 9 ;sr 9 ;r 10 g B 0 =fe;r 3 ;sr 11 ;r 15 ;sr 23 ;sr 26 ;r 27 ;r 30 ;sr 38 g The ouple (A 0 ;B 0 ) is a near-fa torization of D 8 2

indu ed by the

near-fa tori-zation (A 2

;B) of Z 82

We now prove that the graph G(A 0

;B 0

) is not altered by the hoi e of x in

Step 2 orby the hoi e of a 1

inStep 3.

Lemma 44 Let (A;B) be a near-fa torization of Z 2n . Let (A 0 ,B 0 ) and (A 00 , B 00

) be two dihedral near-fa torizations asso iated to (A, B). Then the graph

G(A 0

, B 0

) is isomorphi to the graph G(A 00

, B 00

).

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A 00 =(A+y) =fr i j0in 1; 2i (mod2n) 2A+yg [fsr i j0in 1; 2i+1 (mod2n)2A+yg

Ify x iseven then by taking the unique integerj between 0and n 1 su h

that 2j =2i+x y (mod 2n),we get

A 00 = n r j+ y x 2 j0j n 1; 2j (mod 2n)2A+x o [ n sr j+ y x 2 j0j n 1; 2j+1 (mod2n)2A+x o Hen e, A 00 = A 0 r y x 2 . Thus we have A 00 1 A 00 = r y x 2 A 0 1 A 0 r y x 2 . This means

thatthe onne tingset(A 00

1 A

00

)nfegistheimageof(A 0

1 A

0

)nfeg underthe

innerautomorphismg 7!r y x 2 gr y x 2

.ThenLemma6impliesthatthe Cayley

graph G(A 00

;B 00

) isisomorphi to the Cayley graphG(A 0

;B 0

).

The ase y x is odd is slightly tri kier.

Letk bean element ofZ 2n

su hthat A+k issymmetri . LetA sym :=A+k. We have A 0 =(A sym +(x k)) and A 00 =(A sym +(y k)).Thus A 0 =(A sym +(x k)) =fr i j0in 1; 2i (mod2n)2A sym +(x k)g [fsr i j0in 1; 2i+1 (mod 2n)2A sym +(x k)g and A 00 =(A sym +(y k)) =fr i j0in 1; 2i (mod2n) 2A sym +(y k)g [fsr i j0in 1; 2i+1 (mod2n)2A sym +(y k)g

Forevery integer pbetween 0 and n 1, we have:

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[ r p+i j0in 1; 2i 1 (mod2n)2A sym +(k x) = n sr p+i j0in 1; 2i+x 2k+y (mod 2n)2A sym +(y k)g [ n r p+i j0in 1; 2i 1+x 2k+y (mod2n)2A sym +(y k)g Thus by taking p = k + (y+x) 1 2 (modn), we have A 0 sr p = A 00 . Hen e A 00 1 A 00 = sr p A 0 1 A 0 sr p

. Therefore the onne ting set (A 00 1 A 00 )nfeg is the image of (A 0 1 A 0

) nfeg under the inner automorphism g 7! sr p

gsr p

. This

implies that the Cayley graph G(A 00

;B 00

) is isomorphi to the Cayley graph

G(A 0 ;B 0 ). 2

Thus from a near-fa torization (A;B) of Z 2n

, we get a unique partitionable

graphG(A 0 ;B 0 )where(A 0 ;B 0

)isanydihedralnear-fa torizationasso iatedto

(A;B).Itremainstoknowif wemayget some'new'partitionablegraphsthis

way. We have not su eeded in proving that in general the graph G(A 0

;B 0

)

is isomorphi to G(A;B) when (A;B) is any near-fa torization of the y li

group.

Nevertheless, in Theorem 45 we prove that this is true for all the graphs

G(A;B)on y li groups known sofar.

Theorem 45 If(A;B)isaDeBruijnnear-fa torizationofZ 2n

thenthegraph

G(A;B)isisomorphi tothegraphG(A 0 ;B 0 )where(A 0 ;B 0

)isa dihedral

near-fa torization asso iated to (A;B).

PROOF. We rst al ulatea dihedralnear-fa torization(A 0

;B 0

) asso iated

to (A;B). Noti e that due to Lemma 44,we may pro eed without having to

fearany lossof generality.

Let k 1

;:::;k 2p

be the parameters of the graph G(A;B), that is G(A;B) =

C[k 1

;:::;k 2p

℄. As 2n is even, jAj and jBj must be odd. This implies that

the 2p parameters k i

are all odd. Thus for every j between 1 and p, n j = k 1 k 2 k 3 k 2j +1 is even. We set n 0

:= 2 in order to avoid a spe ial

ase inthe proof.

Leta + :=(k 1 1)+ P p 1 j=1 2j i=1 k i (k 2j+1 1).Noti e that a + isthe greatest

elementof Aseen asaset of integersand that itisaneven elementofA su h

that A a

+ 2

is symmetri .Thus inStep 1,we may take x= a

+ 2

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0000

1111

000

111

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0000

1111

0000000000000000000000000000000000000000000000000000000000000

1111111111111111111111111111111111111111111111111111111111111

0000

1111

0000

1111

000

111

000

111

111 0000

0000

1111

0000

1111

0000

1111

000

111

111 0000

0000

1111

0000

1111

000

111

111 0000

0000

1111

0000

1111

000

111

111 00000

00000

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0000

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111 0000

0000

1111

PSfrag repla ements 1 1 n 1 =10 1 1 A 2 =f0;1;2g+9f0;1g n 2 =19 ! 2 =! 1 2 B 2 =f0;3;6g+18f0;1g 2 = 1 2 n =37 9 18 36

Figure3.The DeBruijnnear-fa torizationgivenbya 1 =3,a 2 =3,a 3 =2,a 4 =1, a 5 =1and a 6 =2 Sin e x is an element of A a + 2 , we may take A 2 := A in Step 2. Hen e by taking A 0 := (A) and B 0

as dened in Step 3, we get a dihedral

near-fa torization asso iatedto (A;B).

Claim 46 We have A 0 A 0 1 =(A A).

PROOF. We have to prove that (A)(A) 1

=(A A).

We rst provethe in lusion(A)(A) 1

(A A). Letwbeany element

of(A)(A) 1

:thereexistaand a 0

inAsu hthatw=(a)(a 0

) 1

.Hen e

by Lemma 39, we have w = (a a 0

) or (a 0

a). In both ases, w is an

element of (A A). Thus(A)(A) 1

(A A).

Wenowprovethe onverse in lusion.Letwbeanyelementof(A A);there

exista and a 0

inA su h that w=(a 0

a).

If a 0

iseven then w=(a)(a 0

) 1

hen eit isan element of (A)(A) 1

.

Ifa 0

isodd,thendue tothedenitionofA,thereexistintegersÆ 0 ; Æ 1 ; :::;Æ p 1 and Æ 0 0 ; Æ 0 1 ; :::;Æ 0 p 1 su h that a = Æ 0 +(n 1 1)Æ 1 +:::+(n p 1)Æ p 1 and a 0 =Æ 0 0 +(n 1 1)Æ 0 1 + :::+(n p 1)Æ 0 p 1 with0Æ i ; Æ 0 i (k 2i+1 1)forevery

i between 0 and p 1. Sin e a 0

is odd, there must be an integer j between 0

and p 1 su h that 0 < Æ 0 j

< (k 2j+1

1) be ause all the k 2i+1

1 are even.

Thusk 2j+1

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IfÆ j =0thena+(n j 1)isanelementofAanda+(n j 1)isanelementofA. Thenw=(a a 0 )=((a + n j 1) (a 0 + n j 1))=(a + n j 1) (a 0 + n j 1) 1 be ause a 0 +n j 1 iseven asn j =a 1 a 2 a 3 :::a 2j +1iseven. Therefore

w isan element of (A)(A) 1 . IfÆ j >0thena (n j 1)isanelementofAanda 0 (n j 1)isanelementofA. Thenw=(a a 0 )=((a n j + 1) (a 0 n j + 1))=(a n j + 1) (a 0 n j + 1) 1 be ause a 0 n j

+1is even. Hen e w is anelement of (A)(A) 1

.

Thus(A A)(A)(A) 1

.

Therefore (A A)=(A)(A) 1

. 2

Claim 47 Let bethegraphwithvertexsetD 2 n

andwithedgesetffx;yg; x

y 1 2(A 0 A 0 1

)nfegg. Then G(A;B) isisomorphi to .

PROOF. Letfi;jgbeanyedgeofG(A;B).Theni j 2(A A)nf0g.Thus

j i 2(A A)nf0g.Hen e(i j)2((A A)nf0g)and(j i)2((A A)n

f0g).Thus(i)(j) 1

2((A A)nf0g).So(i)(j) 1

2((A)(A) 1

)nfeg.

Therefore f(i);(j)gis an edgeof .

Letf(i);(j)gbeanyedgeof .Then(i)(j) 1 2((A)(A) 1 )nfeg.Sin e (i)(j) 1

isequal to(i j)or (j i),we get (i j)2((A A)nf0g)

or(j i)2((A A)nf0g), by Fa t 46.Hen e i j 2(A A)nf0g, that

is fi;jgis an edge ofG(A;B). 2

Claim 48 There exists an element g su h that gA 0

is a symmetri subset of

D 2n

PROOF. Letk bean element of Z 2n

su h that A+k is asymmetri subset

of Z 2n

.

Let A 0

be the set of the even elements of A and letA 1

be the set of the odd

elements of A. LetH be the subgroup of D 2n generated by r. Ifk iseven thenr k 2 A 0 =r k 2 (A)=r k 2 (A 0 )[r k 2 (A 1 )=(A 0 +k)[r k 2 (A 1 ). Thenr k 2 (A 1

)isasubset ofsH,thus itisasymmetri subsetof D 2n

asevery

of its elements is an involution. The set (A 0 +k) is a symmetri subset of D 2n be ause A 0 +k is asymmetri subsetof Z 2n . Hen e r k 2 A 0 is symmetri . Ifk isodd thensr k +1 2 A 0 =sr k +1 2 (A 0 )[sr k +1 2 (A 1 ).Theset sr k +1 2 (A 0 ) is a symmetri subset of D 2n

as it is a subset of sH. We have (A+k) =

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1 n 1 metri ,thussr k +1 2 (A 1 )is symmetri .Therefore sr k +1 2 A 0 issymmetri . 2

Claim 49 The graph G(A 0

;B 0

) is isomorphi to the graph G(A;B).

PROOF. Allwe have toshow isthat G(A 0 ;B 0 )is isomorphi to . Letg bean elementof D 2 n su h that gA 0

is symmetri and let A 00 :=gA 0 . Obviously, G(A 0 ;B 0 ) is isomorphi to G(A 00 ;B 0 ). Let 0

be the graph with

vertex set D 2n

and with edge set ffx;yg; xy 1 2(A 00 A 00 1 )nfegg.

Letinvbethebije tivemapofD 2 n

ontoitselfwhi hmapsanelementontoits

inverse.fx;ygisanedgeofG(A 00 ;B 0 )ifandonlyifx 1 y2(A 00 1 A 00 )nfeg,

that is if and only if inv (x)inv(y) 1 2 (A 00 A 00 1 )nfeg as A 00 =A 00 1 , hen e

if andonlyif finv (x); inv(y)gisanedge of 0 . Hen eG(A 00 ;B 0 )is isomorphi to 0 .

Let h denote the inner automorphism of D 2 n

whi h maps an element x onto

g 1

xg.Then fx;ygis anedge of 0

if and onlyif fh(x);h(y)gisanedge of .

Thus 0 is isomorphi to . Therefore G(A 0 ;B 0 )is isomorphi to . 2

In1990,D.DeCaen,D.A.Gregory,I.G.HughesandD.L.Kreher[7℄des ribed

a lassofnear-fa torizationsofthedihedralgroups:if!isanydivisorof2n 1,

then let := 2n 1 ! and dene A:= r i ; 1i ! 1 2 [ sr i ; 0i ! 1 2 B := r i! ; 0i 1 2 [ sr i! ; 1i 1 2

The graphs asso iated to these near-fa torizations are a stri t subset of the

CGPW graphsof even order:

Lemma 50 The graphs G(A;B) produ ed bythis method are webs.

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2n 0 1gandby B 0 :=!f0;:::; 1g.LetA 0 :=(A 0

).Weknowthatthereexists

B 0 su h that (A 0 ;B 0 ) is anear-fa torizationof D 2n with G(A 0 ;B 0 ) isomorphi to G(A 0 ;B 0 ). We have A 0 = n e;s;r;:::;r ! 1 2 o . Thus A 0 = Asr ! 1 2 . Hen e A 0 1 A 0 = sr ! 1 2 A 1 Asr ! 1 2

. This means that the onne tion set of G(A;B)

is the image under an inner automorphism of D 2 n

of the onne tion set of

G(A 0

;B 0

).ThusG(A;B)isisomorphi toG(A 0 ;B 0 ).AsG(A 0 ;B 0 )isisomorphi toG(A 0 ;B 0

)whi h isa web, we are done. 2

4 Some open questions

This paper gives rise to several questions. We rst re all the ir ular

parti-tionablegraph onje ture:

Conje ture 51 If (A;B) is a near-fa torization of the y li group Z n

then

there exists a De Bruijn near-fa torization (A 0

;B 0

) su h that G(A;B) is

iso-morphi to G(A 0

;B 0

).

Grinstead has veried by omputer this onje ture for groups of order less

than 50, and Ba so, Boros, Gurvi h,Maray and Preissmann have proved it

when A isof ardinalityat most 5.

We do not know any near-fa torization (A;B) of the dihedral groups whose

asso iated graph G(A;B) is not a CGPW graph. Thus we ask this question,

whi h may be seen as the ir ularpartitionable graph onje ture in dihedral

groups:

Problem 52 If (A;B) is a near-fa torization of the dihedral group D 2n

, is

G(A;B) always isomorphi to a graph G(A 0 ;B 0 ) with (A 0 ;B 0 ) a De Bruijn

dihedral near-fa torization ?

We believe that this is not true be ause in a dihedral group, a tile may be

used 'ba kwards', whi h is not possible in the y li group. Hen e a tiling of

D 2n

nfugdoesnot behave inthesame way thana tilingofZ 2n

nfug,whereas

apositiveanswer toProblem 52would suggest the opposite.

Withthehelp ofTheorem 17,anexhaustivesear hby omputer [15℄revealed

that the only groups of order stri tly less than 64 having a symmetri

near-fa torization are the y li groups and the dihedral groups. Hen e this leads

tothis naturalquestion:

Problem 53 Are the y li groups and the dihedral groups the only groups

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tionablegraphsgeneralizingtherst onstru tionofChvatal,Graham,Perold

andWhitesides.Letus allBGH-graphsthepartitionablegraphsprodu edby

this new method. All the BGH-graphs ontain a riti al !- lique, that is an

!- liqueQ su h thatthe riti al edgesof Qindu ea tree overing allverti es

of Q.

Our omputer experiments revealed that the group D 10

Z 5

has a

near-fa torization (A;B) below, su h that the graph G(A;B) does not have any

riti al!- lique. We denote this graph by 50 . A=f(e;0);(s;0);(e;3);(s;3);(r;4);(sr;4);(r 2 ;4)g B=f(s;1);(r;1);(sr 2 ;1);(sr 3 ;3);(r 4 ;3);(sr 3 ;4);(r 4 ;4)g

Lemma 54 Thegraph 50

doesnothaveany riti aledge,whereasthe riti al

edges of 50

form a perfe t mat hing of 50

.

PROOF. If 50

has a riti aledge then there exists an element y su h that

jB 1

\yB 1

j=6.LetH bethe y li subgroup generatedby y.ByTheorem

17appliedtothenear-fa torization(B 1

;A 1

),wehavejHj=2,thus ymust

be aninvolution.

Thesetofinvolutionsisf(s;0);(sr;0);(sr 2 ;0);(sr 3 ;0);(sr 4 ;0)g.Aqui k

om-putation shows that y an not be any of these 5 values, thus we have a

on-tradi tion: 50

does not haveany riti aledge.

fi;jg is a riti al edge of 50

if and only if there exist k and k 0 su h that fig=kAnk 0 A and fjg=k 0 AnkA. Thus jA\k 1 k 0 Aj =6and by Theorem 17 we get that k 1 k 0

must be an involution. Then it is lear that k 1

k 0

must

be equal to (s;0). Thus if fi;jg is a riti al edge then there exists k su h

that fig = kA nk(s;0)A and fjg = k(s;0)AnkA, that is i = k(r 2

;4) and

j =k(sr 2

;4).This implies that j =i(sr 4

;0).

Hen e any riti al edge of 50

is aleft oset of the subgroup H 0

generated by

the involution (sr 4

;0).As any left oset of H 0

form a riti al edge of 50

, we

have that the riti al edges of 50

form the perfe t mat hing of 50

given by

the left osets of H 0

. 2

Thusthisgraph,aswellasits omplement,doesnothaveany riti al!- lique.

ThereforeitisnotaBGH-graph,andneitherisitaCGPW-graph.Hen e

near-fa torizations of nite groupsdo produ e 'new'partitionable graphs.

Problem 55 Is it possible to des ribe a lass of near-fa torizations of a

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I am indebted to an unknown referee for simplifyingthe end of the proof of

Theorem 17.

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