Minimal normal graph covers

0.18 MB·

by  David Gajser

Minimal normal graph covers
DavidGajser Bojan Mohar
∗ †‡
FMF, Department ofMathematics Department ofMathematics
UniversityofLjubljana SimonFraser University
6
Ljubljana,Slovenia Burnaby,BC V5A1S6
1
0 [email protected] [email protected]
2

n
a
J
Abstract
6
A graphis normalif it admits a clique cover and a stable set cover such that each clique in
] andeachstablesetin haveavertexincommonC. Thepair( , )isanorSmalcoverofthegraph. WCe
O S C S
presentthe followingextremalpropertyof normalcovers. For positive integersc,s, if a graphwith n
C
vertices admits a normal cover with cliques of sizes at most c and stable sets of sizes at most s, then
.
h c+s log2(n). Forinfinitelymanyn,wealsogiveaconstructionofagraphwithnverticesthatadmits
≥
t anormalcoverwithcliquesandstablesetsofsizeslessthan0.87log (n). Furthermore,weshowthatfor
a 2
m alln,thereexistsanormalgraphwithnvertices,cliquenumberΘ(log2(n))andindependencenumber
[ Θ(log2(n)).
When c or s are very small, we can describe all normal graphs with the largest possible number

1 of vertices thatallow a normalcover with cliques of sizes at mostc and stable sets of sizes at mosts.
v
However,suchextremalgraphsremainelusiveevenformoderatelysmallvaluesofcands.
9
2
1
1 Introduction
1
0
1. A graph G is said to be normal if it admits a set of cliques and a set of stable sets with the following
C S
0 properties:
6
1 (1) isacoverofG,i.e.,everyvertexinGbelongs tooneofthecliquesin ;
: C C
v
i (2) isacoverofG,i.e.,everyvertexinGbelongs tooneofthestablesetsin ;
X S S
r (3) Everycliquein andeverystablesetin haveavertexincommon.
a C S
Wesaythatthepair( , )isanormalcover ofG.
C S
Clearly, agraph is normal if and only if its complement is normal. This property is reminiscent on the
notion of perfect graphs, see [4, 6, 8]. Namely, normality is one of the basic properties that every perfect
graph satisfies. Of course, normality is much weaker condition since every odd cycle of length at least 9
is normal. Actually, De Simone and Körner conjectured [3] that every graph without induced C , C , and
5 7
thecomplement ofC isnormal. Theirconjecture wasrecently disproved [5]. Nevertheless, normalgraphs
7
make aninteresting class of graphs. Their importance lies intheir close relationship to thenotion ofgraph
entropy, oneofcentralconcepts ininformation theory;seeCsiszárandKörner[1]or[2,6,7].
∗AlargepartoftheresearchwasdonewhenemployedatIMFM,DepartmentofMathematics,Ljubljana,Sloveniaandduringa
visitonSimonFraserUniversitywhichwassupportedbyJavniskladRSzarazvojkadrovinštipendije,163. JR.
†SupportedinpartbyanNSERCDiscoveryGrant(Canada),bytheCanadaResearchChairprogram,andbytheResearchGrant
P1–0297ofARRS(Slovenia).
‡Onleavefrom: IMFM&FMF,DepartmentofMathematics,UniversityofLjubljana,Ljubljana,Slovenia.
1
Forc,s N,wesaythatanormalcoverisa(c,s)-normalcover,ifthesizesofcliquesinthecoverareat
∈
mostcandthesizesofstablesetsinthecoverareatmosts. Furthermore,ifagraphGadmitsa(c,s)-normal
cover,wesayitis(c,s)-normal. Inthispaperweconsiderthefollowing extremalproblem:
Problem1. Howlargecana(c,s)-normal graphbe?
We define N(c,s) as the maximum number of vertices of a (c,s)-normal graph. The obtained results
thatarereminiscent onthebasicsofRamseytheoryarefarfromconclusive.
Notefirstthatitisnotcompletely obvious thatN(c,s)isalwaysfinite. However,inSection3weprove
anupperboundgivenbythefollowingtheorem.
Theorem1.1. If Gisa(c,s)-normal graph,then G 2c+s,i.e.,N(c,s) 2c+s.
| | ≤ ≤
Tostatethetheoremindifferentwords,ifGhasnverticesandis(c,s)-normal,thenc+s lognwhere
≥
the logarithm has base 2, as it is assumed for all logarithms in this paper. Hence, each normal cover of G
musthaveatleastoneelement (i.e.,astablesetoraclique) ofsizeΩ(logn).
In analogy with perfect graphs, one might think that N(c,s) should be closer to c s (see [9, 10]).
·
However, the bound in Theorem 1.1 mustbe exponential, because in Section 2.3 wepresent aconstruction
thatprovesthefollowinglowerbounds.
c
Theorem1.2. Forallintegersc 6,N(c,c) √5 . Additionally, N(5,5) 55andN(4,4) 25.
≥ ≥ ≥ ≥
To compare this bound with the bound in Theorem 1.1, note that 0.86 < 1 < 0.87. Hence, every
log√5
normalcoverofagraphonnverticesmusthaveastablesetoracliqueofsizeatleast 1 logn,butforinfinitely
2
manynwecanconstructanormalgraphwithanormalcoverwhereallcliquesandstablesetsfromthecover
areofsizelessthan0.87logn.
c
ToproveTheorem 1.2weconstruct, foreachpositive integer c,a(c,c)-normal graphwithroughly √5
vertices. We also present a different, simpler construction of a (c,c)-normal graph in Section 2.1 which
gives a bit weaker lower bound for N(c,c). The construction determines only some pairs of vertices to be
connected by an edge or a non-edge. Any other pair of vertices can be adjacent or non-adjacent without
affectingthe(c,c)-normality. Ifforthosepairswechooseindependentlywithprobability 1 whethertheyare
2
connected byanedge ornot, then weargue thatfor largec, wegetsmallindependence andclique numbers
withpositive probability. Thisway,weobtaininSection2.2thefollowing theorem.
Theorem1.3. Foralln,thereexistsanormalgraphGwithnverticesthathascliquenumberandindepen-
dencenumberoforderΘ(logn).
FromRamseytheoryweknowthateverygraphonnverticesmustcontainastablesetoracliqueofsize
1logn. Moreover, random graphs have independence number andclique number oforderΘ(logn). Thus,
2
Theorem 1.3 shows that normal graphs can have clique number and independence number comparable to
thoseofrandomgraphs. Inrelationtothis,itshouldbementionedthatpartialmotivationforthisstudycame
fromthequestionwhetherrandomgraphsG(n,p)arenormalornotwithhighprobability. Asoftoday, this
question isstillunresolved.
Finally, inSection 4wedescribe, for smallvalues ofcand s(c 2ors 2), all(c,s)-normal graphs
≤ ≤
withN(c,s)vertices. Forc = 3(ors = 3)wederiveapolynomialboundbyshowingthatN(3,s) = Θ(s3).
Itisworthobserving thattheresults ofthispaperarereminiscent tothebasic factsabout Ramseynum-
bers: the upper bound for N(c,s) is exponential as is the lower bound for N(c,c); the bases of exponents
inthe lower and upper bound are different and there isno clear evidence whether any of them isasymptot-
ically tight. Small values of N(c,s) are easy to obtain while the exact values for even a reasonably small
values of c and s seem out of reach. Although our techniques are different from those in Ramsey theory,
theyexhibitthenon-constructiveness familiarfromRamseytheory. Whilethelowerboundisobtained bya
direct, nonprobabilistic construction, theupper boundisprovedbyanon-combinatorial technique basedon
analgebraicargumentusingvectorspacedimension.
2
2 Constructions of normal graphs with small cliques and stable sets
In this section we prove Theorem 1.2 and Theorem 1.3. For each integer c > 1, we will first present a
simple construction of a family of (c,c)-normal graphs with roughly 2c vertices. Then we will continue in
two directions. For the first direction, we will show with a probabilistic argument that there is some graph
in the constructed family of (c,c)-normal graphs that has independence number and clique number of size
Θ(c), which will prove Theorem 1.3. For the second direction, we will enhance the construction to get a
c
(c,c)-normal graphwithatleast√5 vertices, whichwillproveTheorem1.2.
2.1 The construction ofG
c
A red-blue graph is a graph with each edge of color red or blue. When drawing such a graph, we will
additionally makeblueedgessolidandrededgesdashed. Foranedge-colored graphH,wesaythatagraph
GisoftypeH if
V(G) = V(H),
•
eachblueedgeofH isanedgeofG,
•
norededgeofH isanedgeofG.
•
For a red-blue graph H and for positive integers c,s, let = C ,C ,...,C be a cover of H with blue
1 2 k
C { }
cliques ofsizeatmostcandlet = S ,S ,...,S beacoverofH withredcliques ofsizeatmosts. If
1 2 l
S { }
thecliqueC andthestablesetS intersectforanyiandj,thenwesaythatthepair( , )isa(c,s)-normal
i j
C S
coverofH. IfH admitsa(c,s)-normalcover, wesaythatitis(c,s)-normal. Fromthedefinitionofgraphs
oftype H itisclear that if( , ) isa(c,s)-normal cover ofH, then ( , ) isalso a(c,s)normal cover of
C S C S
anygraphoftypeH.
0 1
v v
1 2
Figure1: Thered-bluegraphG . Bluecliques 0,v , 1,v andredcliques 0,v , 1,v forma(2,2)-
2 1 2 2 1
{ } { } { } { }
normalcover.
Foranintegerc 1,defineared-blue graphG recursively asG = K and, forc 2,G isobtained
c 1 1 c
≥ ≥
fromtwocopiesofthegraphG ,denoted byG (0)andG (1),byaddingtwonewvertices 0and1,
c 1 c 1 c 1
− − −
suchthateachvertex ofG (0)isconnected to0byablueedgeandto1byarededgeandeachvertexof
c 1
−
G (1)isconnected to1byablueedgeandto0byarededge(seeFigure2).
c 1
−
ThegraphsG ,G andG arepresented inFigures1,3and4,respectively.
2 3 4
3
0 1
G (0) G (1)
c 1 c 1
− −
Figure2: Therecursiveconstruction ofG .
c
0 1
0 1
00 01 10 11
v1 = 000 v2 = 010 v3 = 100 v4 = 110 vv11 vv22 vv33 vv44 vv55 vv66 vv77 vv88
Figure 3: Thegraph G3 withvertices labeled as Figure 4: The graph G4. To make the picture
binarysequences. moreclear,theedgesfrom0and1arenotdrawn
completely.
Next,wegiveanalternative,non-recursivedescriptionofG . LetV(G )bethesetofallbinarysequences
c c
oflengthatleast1andatmostcundertheconditionthatallbinarysequences oflengthexaclychavetoend
with 0. Let a = a a ...a 0,1 k and b = b b ...b 0,1 l be two vertices of G . We may
1 2 k 1 2 l c
∈ { } ∈ { }
assume k l. If k = l, then a and b are not adjacent. Otherwise, ab is an edge if b = a , b = a ,...,
1 1 2 2
≤
b = a . Theedgeisblueifa = b andredifa =b . ThegraphG inFigure3islabeledaccording
k 1 k 1 k k k k 3
− − 6
tothisdescription. Theproofthatbothdescriptions ofG givethesamegraphislefttothereader.
c
Proposition 2.1. Foreachpositive integerc,thegraphG is(c,c)-normal andhas3 2c 1 2vertices.
c −
· −
Proof. Fromthedescription ofG bybinarysequences, itisclearthat
c
V(G ) = 2 2c 1+2c 2+2c 3+ +2 = 3 2c 1 2.
c − − − −
| | · ··· · −
Let m = 2c 1 and let v ,v ,...,v be the vertices that correspond to all binary sequences of length
− 1 2 m
c in V(G ). We will use these vertices as “generators” for a (c,c)-normal cover of G . For each ver-
c c
tex v = a a ...a , define the set C as the set of all beginnings of the sequence v , i.e., C =
i i,1 i,2 i,c i i i
a a ...a 1 k c . It is clear from the definition of G that C is a blue clique of size c.
i,1 i,2 i,k c i
{ | ≤ ≤ }
Similarly,foreachvertexv ,definethesetS = a a ...a a¯ 1 k < c v ,whereweputa
i i i,1 i,2 i,k 1 i,k i
{ − | ≤ }∪{ }
bar over a to indicate a = a¯ . In other words, S is the set of all sequences that are either v or have
i,k i,k i,k i i
6
thesamebeginning asv ,butthendifferfromthebeginning ofv bythelastbit. Fromthedefinition ofG ,
i i c
weseethatS isaredcliqueofsizec.
i
It is clear that each vertex of G is in some blue clique C and in some red clique S . Moreover, each
c i j
bluecliqueandeachredcliqueintersect. Toseethis,takeabluecliqueC andaredcliqueS . Ifi = j,then
i j
4
C and S bot contain v . If i = j, then let k be the first index where the elements of sequences v and v
i j i i j
6
aredifferent. Itfollows that thesequence ofthefirstk elements ofv isinboth C andS . Hence, theblue
i i j
cliquesC andtheredcliquesS forma(c,c)-normalcoverofthegraphG ,whichisthus(c,c)-normal.
i j c
Forthelateruse,the(c,c)-normalcoverofthegraphG asdescribedintheproofofProposition2.1will
c
bereferredtoasthestandard normalcoverofG .
c
2.2 Normalgraphs withsmallcliquenumber andindependence number
IntheproofofTheorem1.3wewillusethefollowingtwolemmas.
Lemma2.2. Foranylargeenoughintegerc,everysubsetU ofverticesofG ofsize4chasatleast4.1c2pairs
c
of vertices that are not connected by ablue edge and at least 4.1c2 pairs of vertices that are not connected
byarededge.
Proof. Wewillgivetheproofonlyforblueedgessincethered-bluegraphobtainedfromG byinterchanging
c
red and blue edges is isomorphic to G (this follows easily by inspection of the recursive construction). In
c
theproof,wewillidentify vericesofG withthecorresponding binarysequences andwewillregularly use
c
thetermheightofavertex torefertothelengthofthesequence corresponding tothisvertex. Wesaythata
vertex v V(G )isapredecessor ofavertex u V(G ), ifuandv areconnected by ablue edgeand the
c c
∈ ∈
height ofv issmallerthan theheight ofu. Wealsosaythat uisanancestor ofv. Ifv isapredecessor ofu
andhasheightonelessthanu,wesaythatv isadirectpredecessor ofuanduisadirectancestor ofv.
LetU beasubsetofverticesofG thatinducesthemostblueedgesamongallsubsetsofvertices ofG
c c
ofsize4c. WecanassumethefollowingthreepropertiesforthesetU (wemayneedtochangeU togetthese
properties, butU willremainasetwith4cvertices anditwillinduce thesamenumberofblueedges asthe
currentU).
a) Foreachu U,allpredecessors ofuarealsoinU. Ifthereissomepredecessor v ofuthatisnotinU,
∈
thenbecause theneighbors (weonlyconsider blueedges)ofuarealsoneighbors ofv,wecanreplace u
byvinU withoutdecreasingthenumberofblueedgesinit. Afterwedosuchatransformation once,the
sum ofheights ofvertices inU strictlydecreases soafter finitelymanysuchtransformations allvertices
ofU willhavealltheirpredecessors inU.
b) Ifavertexv U ofheightkhasnodirectancestorsinU,thenallotherverticesofU ofheightatleastk
∈
have alltheir ancestors inU. Ifthiswasnot thecase, then there existsavertex u U ofheight atleast
6∈
k+1that isadirect ancestor ofsomevertex from U. Bya),uhasstrictly more neighbors inU than v,
thus ifwesubstitute v foruinU wegetstrictly moreblueedges inU,whichisacontradiction withthe
choiceofU.
c) Foreachk,1 k < c 1,U hasatmostonevertexofheightk thatdoesnothavetwodirectancestors.
≤ −
If, for some k, there are twosuch vertices in U, say uand v, then by b)itmust be the case that uand v
eachhasexactly onedirectancestor inU. Hence, wecanreplace thedirect ancestor ofv inU andallof
its ancestors in U with the direct ancestor of u which is not in U and with its corresponding ancestors.
With this trasformation of U, the number of vertices in U as well as the number of blue edges induced
by U ispreserved. Furthermore, the assumption a)ispreserved and, foreach l = k, 1 l < c 1, the
6 ≤ −
numberofverticesofheightlinU thatdonothavetwodirectancestorsisalsopreserved. Becausevnow
hasno direct ancestors inU,itmustbethecase thatallother vertices ofheight k inU have exactly two
direct ancestors inU by b). Because wecan dothe sametransformation for allk, weproved that c)can
indeedbeassumed.
5
Fori = 1,2,...,c,letx bethenumberofverticesofheightiinU. Withassumptionsb)andc)inhand,
i
wecanarguethatx islarge. Letavertex v U bealowestvertexfromU thathasnodirectancestors and
c
∈
let k be the height of v. If k c 1, then using b)and c) we can prove by induction on c i that, for all
≥ − −
1 i< c,itholds
≤
x +1
c
x .
i ≤ (cid:24)2c i 1(cid:25)
−−
Ifk < c 1,thenusinginduction onc iweget
− −
x
c
x = ifk < i < c (usea)andb)toprovethat)
i 2c i 1
−−
and
x /2+1
x k+1 ifi k (useb)andc)toprovethat).
i ≤ (cid:24) 2k i (cid:25) ≤
−
Inthesecondcase,itfurthermore holds
x /2+1 x
k+1 k+1
x +1. (1)
i ≤(cid:24) 2k i (cid:25)≤ 2k+1 i
− j − k
The second inequality in (1) is easy to see using the fact that x is even, which follows from b). In any
k+1
case,wehave
x
c
x +1
i ≤ 2c i 1
−−
foralli,whichgives
x x x
c c c
4c x + x +1 + +1 + +1 + + +1 < 3x +c
≤ c c 2 4 ··· 2c 2 c
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) − (cid:17)
andhencex > c.
c
SoweknowthatthereareatleastcverticesinU ofheightc,whichimpliesthatthereareatleastcvertices
inU ofheightc 1and c verticesofheightc 2etc. LetX U bethesetofallverticesinU ofheight
− ⌈2⌉ − ⊆
atleastc 4. BecauseeachvertexinX hasatmost22possibleancestors,thesetX inducesonlyO(c)blue
−
edges. Hence,thereareatleast
2c+ c + c + c
⌈2⌉ ⌈4⌉ ⌈8⌉ O(c) 4.13c2 O(c)
(cid:18) 2 (cid:19)− ≥ −
pairs of vertices from X not connected by a blue edge. Hence, for large enough c, we have at least 4.1c2
pairsofverticesinU thatarenotconnected byablueedge. Thiscompletestheproof.
Lemma2.3. Forevery large enough integer c,there exists agraph Goftype G such thatα(G) < 4cand
c
ω(G) < 4c.
Proof. Foragivenintegerc,letGbearandomgraphoftypeG whichisobtainedfromG byaddingeach
c c
missingedgeofG withprobability 1 independently foreachedge,removingrededgesanduncoloringblue
c 2
edges. Letnbethenumberofvertices ofG. ByLemma2.2,wegetthat,forlargeenough c,
Pr[ω(G) 4c] Pr[U inducesaclique]
≥ ≤
X
U V(G)
⊆
U =4c
| |
n 2 4.1c2.
−
≤ (cid:18)4c(cid:19)
6
ByProposition 2.1wehavec logn 1. Therefore,
≥ −
Pr[ω(G) 4c] n4c2 4.1clogn+4.1c = 24.1cn 0.1c,
− −
≥ ≤
whichconvergesto0whenctendstoinfinity. Hence,forlargeenough c,
1
Pr[ω(G) 4c] < .
≥ 2
Fromthesymmetrybetweenblueandrededgesintherecursive definitionofG itfollowsthat
c
1
Pr[α(G) 4c] < .
≥ 2
Hence,forlargeenoughc,
Pr[Ghasastablesetoracliqueofsize4c] < 1.
ThismeansthatthereexistsagraphGoftypeG thathasnocliqueorstablesetofsize4c.
c
ProofofTheorem1.3. Let n be a positive integer and let c be such that V(G ) < n V(G ). By
c 1 c
| − | ≤ | |
Lemma2.3, there exists a graph Gof type G such that α(G) = O(c) and ω(G) = O(c). Wewillmake a
c
serriesofremovalsofvertices ofGuntilwegetagraphwithn,n 1orn 2vertices. Eachtimewewill
− −
remove three vertices as can be seen in Figure 5. It is not hard to see that removing three by three vertices
thiswaypreserves normalityofthegraph.
0 1
v v v v vv vv vv vv
1 2 3 4 55 66 77 88
Figure5: ThegraphG withthreetriplesofvertices (circled) removed. Ifwewouldalsoremovethefourth
4
triple,wewouldgetG .
3
LetG betheresultinggraphwithn,n 1orn 2vertices. SinceG isaninducedsubgraphofG,itholds
′ ′
− −
α(G)= O(c)andω(G)= O(c). Next,weaddatmosttwonewverticesC andDtothegraphG,together
′ ′ ′
withallpossibleedgesfromC andDexceptoftheedgeCD,togetann-vertexgraphG . Clearly,thiscan
′′
increaseαorωbyatmost2. WeleaveittothereadertoshowthatG isanormalgraph. ByProposition2.1
′′
we know that c = Θ(logn). By our construction of the graph G it follows that α(G ) = Ω(logn) and
′′ ′′
ω(G ) = Ω(logn),henceα(G ) = Θ(logn)andω(G ) = Θ(logn).
′′ ′′ ′′
2.3 Animprovedlowerbound forN(c,c)
WehaveshowninProposition 2.1thatN(c,c) 3 2c 1 2. Hereweimprovethisbound.
−
≥ · −
For a positive integer c, define a red-blue graph F as F = G , F = G and, for c > 2, connect five
c 1 1 2 2
copiesofF asshowninFigure6,whereeachedgebetweenacopyofF andavertexin A,B,C,D,E
c 2 c 2
− − { }
7
indicates the full red/blue join. Using induction on c, we can verify that the graphs F are (c,c)-normal: a
c
(c,c)-normal coverofF consists ofthe union ofnormalcovers ofgraphs F whereweadd twovertices
c c 2
−
toeachcliqueandtwovertices toeachstableset(whichvertices weaddcanbeseenfromFigure6,thereis
onlyoneoption). Weleaveittothereadertoshowthatthisisindeedanormalcover. Wecallitthe standard
normalcoverofF .
c
Fc 2(D) A Fc 2(C)
− −
B E
F (E) F (B)
c 2 c 2
− −
C D
F (A)
c 2
−
Figure6: TherecursiveconstructionofF fromfivecopiesofF . TheblueedgebetweenAandF (C)
c c 2 c 2
− −
represents all possible blue edges between A and vertices in F (C). The same holds for any other edge
c 2
−
showninthepicture.
NowwecanproveTheorem1.2,whichstates:
c
Forallintegersc 6,N(c,c) √5 . Additionally, N(5,5) 55andN(4,4) 25.
≥ ≥ ≥ ≥
ProofofTheorem1.2. If we denote f = V(F ), then we have the recursion f = 1, f = 4, f =
c c 1 2 c
| | c
5f +5. This shows that N(5,5) 55 and N(4,4) 25. For c 6, the inequality f √5 follows
c 2 c
− ≥ ≥ ≥ ≥
easilyfromf = 5f +5byinduction (thebasisisforc= 6andc = 7).
c c 2
−
Toconcludethissection,letusdiscussaspecialpropertythattellssomethingabouttheoptimalityofthe
construction ofgraphs F andprevents asimpleoperation thatwouldimprovethelowerboundofTheorem
c
1.2.
Proposition 2.4. For each positive integer c, if a subset of V(F ) of size at most c does not induce a red
c
edge, but intersects all red cliques of the standard normal cover of F , then it is of size c and has at least
c
c 1commonvertices withsomebluecliqueofthestandard normalcoverofF .
c
−
Note that Proposition 2.4 does not hold for graphs G . Take, for example, G and let X = 0,1 and
c 5
{ }
Y = 00,01,10,11 beverticesofheight1and2,respectively. NotethatX andY bothintersectallredand
{ }
allbluecliquesofthestandardnormalcover,whiletheydonotinduceanyedge. Wecanconnectverticesof
X withablueedgeandverticesofY withrededgesandaddanewvertexvtoG ,connectittoX withblue
c
edgesandtoY withrededges. Thenewred-bluegraphisalso(5,5)-normalandhasmoreverticesthanG .
5
Itis clear that wecould add in asimilar wayeven twomorevertices toincrease theblue clique 0,1,v to
{ }
size5andthegraphwouldstillbe(5,5)-normal.
8
Proof. Forc = 1andc= 2,thepropositionistriviallytrue. Actually,aswewillseebelow,thecasec= 2is
essentially theonlyreasonforwhyitcanhappenthatasetofsizeatmostcthatdoesnotinducearededge,
butintersects allredcliques ofthestandard normalcoverofF ,canhave onlyc 1commonvertices with
c
−
somebluecliqueofthestandard normalcoverofF .
c
Assumec> 2andassumethatthepropositionholdsforF . LetU beasubsetofV(F )ofsizeatmost
c 2 c
−
c that does not induce a red edge, but intersects all red cliques of the standard normal cover of F . Using
c
notation from Figure 6, U contains at most two vertices from the set A,B,C,D,E , because any three
{ }
vertices fromthissetinduceatleastonerededge.
IfU contains no vertex in A,B,C,D,E , then wehave tocover all redcliques ofeach offivecopies
{ }
ofF independently. Hence,thereexistsacopyofF forwhichwecancoverallofitsredcliqueswith
c 2 c 2
− −
a set of size at most c/5 that does not induce a red edge. Because the proposition holds for F , we have
c 2
−
c/5 c 2, which means c 2 (which is false). We conclude that U contains at least one vertex from
≥ − ≤
A,B,C,D,E .
{ }
If U contains exactly one vertex from A,B,C,D,E , then we see in Figure 6 that there exist at least
{ }
threecopiesofF forwhichU hastocoveralltheirredcliques withatmostc 1vertices, meaningthat
c 2
− −
thereexistsacopyofF forwhichU hastocoverallofitsredcliqueswithatmost(c 1)/3vertices(not
c 2
− −
inducing arededge). Butweassumedthattheproposition holdsforF ,hence(c 1)/3 c 2,which
c 2
− − ≥ −
meansc 2,whichisfalse.
≤
It follows that U contains exactly 2 vertices from A,B,C,D,E . Because of the symmetry we may
{ }
assume that U contains A and B. By Figure 6, U A,B has to cover all stable sets of F (D) and we
c 2
\{ } −
knowthatitdoesnotinducearededge. Byinductionhypothesis, U A,B isofsizec 2andhasatleast
\{ } −
c 3verticesinsomebluecliqueQofthestandard normalcoverofF (D). ItfollowsthatU isofsizec
c 2
− −
andithasatleastc 1commonverticeswiththebluecliqueQ A,B thatbelongstothestandardnormal
− ∪{ }
coverofF .
c
3 An upper bound on N(c,s)
Thered-bluegraphsconstructedintheprevioussectionalladmit(c,c)-normalcoversforsomec = Θ(logn).
In this section we prove that we cannot do better than that. More specifically, we prove Theorem 1.1 con-
firmingthatN(c,s) 2c+s.
≤
Beforegoingtotheproof,weneedsomepreparation. LetGbea(c,s)-normalgraphwitha(c,s)-normal
cover( , ). ForacliqueC ,wesaythatavertexv C isprivatetoC ifitisinnoothercliquefrom .
C S ∈ C ∈ C
Similarly,forastablesetS ,wesaythatavertexv S isprivatetoS ifitbelongstonootherstableset
∈ S ∈
from . Wesay that a(c,s)-normal cover ( , ) ofGisminimal ifnoproper subset of is aclique cover
S C S C
andnopropersubsetof isastablesetcoverofG. Notethat( , )isminimalifandonlyifeachcliquein
S C S
has a private vertex and each stable set in has aprivate vertex. Clearly, each (c,s)-normal graph has a
C S
minimal(c,s)-normalcover.
If we have a minimal normal cover of G, we can use the algorithm C-reduce to reduce the size of G
withoutaffectingnormality. Notethatthereisnoconditioninthealgorithmontheinputcovertobeminimal.
Claim3.1. Thecover( , )remainsanormalcoveroftheresultinggraphaftertheexecutionof C-reduce.
C S
Proof. If P = 1,thenitisclearthat remainsacoverofG v bycliques. BecausethecliqueC wasthe
| | C −
onlycliquefrom thatincludedv,( , )remainsanormalcover.
C C S
If P > 1,thenbecausev andv wereprivatetoC, remainsacoverofGbycliques. Furthermore,all
′
| | C
stable sets in remain stable sets, since none of them contains both v and v˜, which are adjacent in G, and
S
alledgesfrom v,v tovertices outofC havebeenremoved.
′
{ }
9
Algorithm1:C-reduce
Input: AgraphGwithanormalcover( , ),acliqueC andavertexv thatisprivatetoC.
C S ∈ C
1 LetP bethesetofallprivateverticesofC.
2 if P = 1then
| |
3 Removev fromGandfromallstablesetsin .
S
4 RemoveC from .
C
5 else
6 Becausev P and P = 1,thereexistsv′ P v .
∈ | | 6 ∈ \{ }
7 RemovealledgesinGthatjoinv orv′ withavertexoutside C.
8 Contracttheedgevv′.
Let G be a (c,s)-normal graph with a minimal (c,s)-normal cover ( , ). We say that vertices
C S
v ,v ,...,v V(G)arecliquegeneratorsfor ifthereisabijectivecorrespondence betweenverticesv
1 2 i
andcliquesC|C| ∈ suchthatforeachi,v isprivatCetoC . Wedefinestablesetgeneratorsfor analogously.
i i i
∈ C S
In what follows, we prove Theorem 1.1 by induction on c+s. The strategy is as follows. For a (c,s)-
normalgraphG,wefirsttakeaminimal(c,s)-normal cover( , ). Thenwefindasetofclique generators
C S
andwerunC-reduceforallthesegenerators, i.e.,werepeatthereduction times1. Thenweareleftwith
|C|
a(c 1,s)-normalgraphG. IfG hasatleasthalfoftheoriginalvertices, wecanuseinductionhypothesis
′ ′
−
toshowthatN(c,s) 2N(c 1,s) 2c+s.
≤ − ≤
InorderforGnottolosetoomanyvertices duringthejustdescribed reduction ofclique generators, we
want to prove that in any minimal normal cover wedo not have too many stable sets and cliques. Toprove
this,wewillusethefollowinglemma.
Lemma 3.2. Let( , ) beaminimal(c,s)-normal cover ofa(c,s)-normal graph Gonn > 1vertices. If
C S
U V(G)isasetofcliquegenerators thatisalsoasetofstablesetgenerators of( , ),thenn 2U .
⊆ C S ≥ | |
Proof. From the definition of generators it follows that = = U . If we denote t = U , let U =
|C| |S| | | | |
v ,v ,...,v and let the remaining vertices of G be v ,v ,...,v . Furthermore, let the (unique)
1 2 t t+1 t+2 n
{ }
clique and the (unique) stable set from ( , ) that contains v be denoted by C and S , respectively. For
i i i
C S
i = 1,...,n,wedefinecolumnvectorsc ,s 0,1 t as
i i
∈ { }
1 v C 1 v S
ci(j) = (cid:26) 0 oith∈erwjise and si(j) = (cid:26) 0 oith∈erwjise.
Thisgivesus,foreachi,
1 ifC S = v
ci ·sTi [j,k] = (cid:26) 0 othejrw∩isek. { i}
Because( , )isanormalcover,wehave
C S
n
c sT = J ,
i i t
·
Xi=1
whereJ denotestheall-onest tmatrix. Furthermore,
t
×
t
c sT = I ,
i i t
·
Xi=1
1IntheformalproofofTheorem1.1below,wereducecliquegeneratorsonlyifwehaveatleastasmanystablesetsascliques.
Else,wereducestablesetgenerators.
10