The guessing number of undirected graphs - Index of

Theguessingnumberofundirectedgraphs 

DemetresChristofidesandKlasMarkström 

Abstract 

In[Rii07],Riisintroducedaguessinggameforgraphswhichisequivalent 

tofindingprotocolsfornetworkcoding.Inthispaperweproveupperand 

lowerboundsforthewinningprobabilityoftheguessinggameonundirected 

graphs.Wefindoptimalboundsforperfectgraphsandminimallyimperfect 

graphs,andpresentaconjecturerelatingtheexactvalueforallgraphsto 

thefractionalchromaticnumber. 

Finally, we discuss extensions of our results to directed graphs and 

graphswithnon-uniformprobabilities. 

1 Introduction 

Anactiveareaofresearchincommunicationtheoryduringthelasttenyears 

hasbeenthedevelopmentofprotocolsfornetworkcoding[ACLY00].Innetwork 

codingthereareseveralsendersandreceiverswhowishtopassmessagesbetween 

eachother,howevertheroutersinthenetworkcanonlysendonemessageat 

atime. Inordertoavoidbottlenecks,networkcodingallowstheroutersto 

computeanddistributenewmessages,aslongasthereceiverscancomputethe 

senders’originalmessagefromthecollectionofnewmessagestheyreceivedin 

itsstead. 

In[Rii07],Riisconnectednetworkcodingtothemucholderproblemoffinding 

an optimalBooleancircuitforaBooleanfunction. In thatpaper, Riis 

disprovedaconjectureofValiantincircuitcomplexityandshowedthatboth 

networkcodingandthisformofcircuitcomplexitywereequivalenttoacertain 

typeofmultiplayerguessinggameonagraph. 

Theguessinggamecanbedescribedasfollows:EachvertexvofagraphG 

isassignedaplayer,alsodenotedbyv,andauniformlyrandomintegerfrom 

{1...s}. Aplayervcanseethenumbersassignedattheneighboursofv,but 

notthenumberatv. Withoutcommunication,eachplayermustnowmakea 

guessastothevalueofitsownrandomnumber. Theteamofplayerswins 

ifallofthemguesscorrectlytheirownvalue,andlosesifanyoneisincorrect. 

Theobjectiveistonowfindastrategywhichmakesthewinningprobabilityas 

largeaspossible.Atfirstitmightseemlikethereisnowayofgettingahigher 

winningprobabilitythans −n onannvertexgraph,butRiisobservedthatthis 

isfarfrombeingthecase. IfGisthecompletegraph,theplayerscanusethe 

1

followingsimplestrategy: Eachplayerpickstheuniquenumbersuchthatthe 

sumofthatandallothernumbersis0modulos. Ifeveryplayerfollowsthis 

strategytheywillallbecorrectwhenthesumofallnumberis0modulos,which 

happenswithprobabilitys −1 ,avaluewhichdosnotevendependonn. 

In[Rii07],Riisconsideredthisgameongeneraldirectedgraphs,whereplayer 

vcanseethenumberatvertexuifandonlyifthereisanedgefromvtou. 

Motivatedby thisgeneralquestion, inthispaperwe studytheproblemfor 

undirectedgraphs. Asweshallsee,theproblemoffindingoptimalguessing 

gamestrategiescanbetranslatedintoaquestionregardingthesizeofthelargest 

independentsetinanauxiliarygraph,andusingthisgraphwecanfindgood 

boundsforthewinningprobabilities. 

Hereisanoutlineofthepaper.Inthenextsectionwegivesomeformaldefinitionsandprovesomebasicboundsforthewinningprobabilityoftheguessing 

games. InSection3,weexplainhowtheguessingnumberscanbedetermined 

bycomputingthefixedpointsofsomespecificmaps. Wealsousethemethods 

developedtheretoshowthatthereisanaturallydefinedlimitoftheguessing 

numberswhichwecalltheasymptoticguessingnumber. InSection4,wefind 

lowerboundsontheguessingnumbersusingfractionalcliquecovers.InSection 

5,wedefinethecodegraphoftheguessinggameandprovethattheguessing 

numbercanbecomputedbydeterminingtheindependencenumberofthisgraph. 

InSection6,wefindupperboundsontheguessingnumbersusingentropyinequalitiesandposeaconjectureregardingtheasymptoticguessingnumberof 

eachgraph.FinallyinSection7weconsidersomegeneralisationsofthemethods 

ofthepaperinsimilarcontexts. 

2 Definitionsandsomebasicbounds 

Westartoutbymakingamoreformaldefinitionoftheguessinggame. 

Definition2.1.IntheguessinggameonagraphG,eachvertexvisassigned 

anintegerxv∈{1,...,s}uniformlyatrandom. 

GivenavertexvofagraphG,astrategyforplayervfortheguessinggame 

onGisafunctionfv :{1,...,s} N(v) →{1,...,s}. Thevalueoffviscalled 

v’sguess. AstrategyFfortheguessinggameonGisasequenceoffunctions 

(fv) v∈V(G)suchthateachfunctionfvisastrategyforplayervfortheguessing 

game. 

WesaythattheplayerswinifallguessesarecorrectandwewriteCor(F) 

todenotethisevent. 

Wenowdefinetheguessingnumberofthegraphviathewinningprobability 

intheoptimalstrategyforthatgraph. 

Definition2.2. Theguessingnumbergn(G,s)ofagraphGwithrespectto 

thepositiveintegersisthelargestβsuchthatthereexistsastrategyFforthe 

guessinggameonGsuchthatwithprobability 1 

s n−β everyplayervguessesits 

ownvaluexv,i.e.Pr(Cor(F))= 1 

s n−β. 

2

Ingeneral,onecouldconsiderstrategieswhereeachplayermakesarandom 

choicebasedontheavailableinformation. Howeverthereisalwaysanoptimal 

strategywhichisdeterministic,soweonlyconsiderdeterministicstrategies. 

Lemma2.3. EveryrandomisedstrategyfortheguessinggameonagraphG 

haswinningprobabilityatmost 1 

s n−gn(G,s). 

Proof.ArandomisedstrategycanbedescribedbyassigningaprobabilityPr(F) 

toeachdeterministicstrategyF. Howeverthewinningprobabilityofsucha 

strategyis 

 

F 

Pr(F)Pr(Cor(F))max 

F Pr(Cor(F))= 

1 

s n−gn(G,s). 

Usingthisterminologywecannowgiveaformalversionofthestrategyfor 

thecompletegraphwhichwedescribedintheintroduction. 

Example 2.4. LetGbethe completegraphKn on nvertices. Wedefine 

a strategyforthe guessinggameon G by definingfv tomap thesequence 

(xu) u∈N(v)totheuniqueintegerx ′ v∈{1,...,s}suchthatx ′ v+ 

u∈N(v) xuis 

divisiblebys.Wewillcallthisthecliquestrategy.Observethatalltheplayers 

guesscorrectlyifandonlyifx ′ v=xvforeveryv∈V(G)whichholdsifandonly 

if 

. Since 

v∈V(G) xvisdivisiblebys. Theprobabilityofthiseventisthus 1 

s 

theprobabilitythatasingleplayerguessesitsownvaluecorrectlyisalso 1 

s ,this 

cannotbeimprovedandwefindthatgn(Kn,s)=n−1. 

Fromthecliquestrategywecandefineanaturalstrategyforgeneralgraphs 

bypartitioningthevertexsetintocliques. 

Definition2.5.Acliquecover,orcliquepartitionofagraphGisapartition 

ofV(G)intovertexdisjointcliques.Thecliquecovernumbercp(G)ofGisthe 

minimumcardinalityofacliquecoverofG. 

NotethatthecliquesinacliquecoverofGinduceapartitionofthecomplementofGintoindependentsets,i.e.apropervertexcolouringofG. 

Hence 

cp(G)=χ G . 

Lemma2.6.ForeverygraphGandeverypositiveintegers,gn(G,s)n− 

cp(G) 

Proof.Astrategygivingthisboundcanbeconstructedbytakingaminimal 

cliquecoverandlettingtheplayersineachcliquefollowthecliquestrategyfor 

thatclique.Thisgivesprobabilityatleast 1 

s cp(G)thatallplayersguesscorrectly. 

Wecallthestrategyusedintheproofoftheabovelemma,thecliquecover 

strategy. 

Intheotherdirection,wecanproveanupperboundonthewinningprobabilitybyconsideringindependentsetsinthegraph.Asusual,weletα(G)denote 

thesizeofthelargestindependentsetinG. 

3

Lemma2.7.ForeverygraphGandeverypositiveintegers,gn(G,s)n−α(G) 

Proof.LetIbeamaximumindependentsetinGandletF =(fv) v∈V(G)be 

astrategy. Wechoosetherandomnumber(xu) u∈V(G)intwostages. Inthe 

firststage,wegenerateallnumbersxu withu /∈I. ObservethatsinceI is 

independentallfunctionsfuwithu∈Icannowbedetermined. Inthesecond 

stage,wegenerateallnumbersxuwithu∈Iandobservethattheprobability 

thatplayervguessescorrectlyisexactly1/swiththeeventsbeingindependent. 

ItfollowsthatPr(Cor(F))s −α(G) asrequired. 

Eventhesetwosimpleboundsareenoughtodeterminetheguessingnumbers 

exactlyforlargeclassesofgraphs. Infact,thesetwoboundsdeterminethe 

guessingnumberofGpreciselywhenα(G)=cp(G). Oneparticularlynatural 

classwhichsatisfiesthispropertyistheclassofperfectgraphs,introducedby 

Berge[Ber63].Thisistheclassofgraphssuchthatχ(H)=ω(H)forallinduced 

subgraphsofG. AclassicaltheoremofLovász[Lov72]tellsusthatagraphis 

perfectifandonlyifitscomplementisperfect. Inourcontextthismeansthat 

foraperfectgraphGwehaveα(G)=ω(G)=χ G =cp(G) 

Corollary2.8.IfGisperfectthengn(G)=n−α(G). 

ThistellsusamongotherthingsthatifGisbipartitethengn(G,s)=n− 

α(G).OnestrikingconsequenceisthatadisjointunionofnK2’s,i.e.amatching 

ofsizenhasthesameguessingnumberasthecompletebipartitegraphKn,n, 

despitebeingasubgraphofithavingafactorofnfeweredges.Inotherwords, 

allthisextrainformationthattheplayershaveinplayingtheguessingnumber 

forKn,ncontributesnothingtothewinningprobabilityfortheguessinggame 

onthisgraphcomparedwiththeguessinggameonthematchingofsizen. 

3 Fixedpointsandtheasymptoticguessingnumber 

Anotherwayofdescribingtheguessingproblemisintermsoffixedpointsofthe 

mappingsgivenbydifferentstrategies.NotethatastrategyFcanbeviewedas 

amappingfromAs={1,...,s} V(G) intoitself.ThestrategyFguessescorrectly 

onagivenoutcomeofrandomnumbersx=(xv) v∈V(G)ifF(x)=x. Thusthe 

problemoffindingagoodstrategyfortheguessinggamecanbeinterpretedas 

findingamappingF:As→Aswithasmanyfixedpointsaspossible,wherefv 

onlydependsonxu∈N(v).Wecallamappingwiththedependencestructurea 

strategymappinganddenotethesetofallstrategymappingsonAsbyS(G,s). 

IfweletFix(F)denotethenumberoffixedpoints,thenwecancomputethe 

guessingnumberas 

gn(G,s)= max 

F∈S(G,s) log sFix(F) 

LetusdefineFix(G,s)=max F∈S(G,s)Fix(F). 

Whiletheguessingnumberisgivenbythemaximumnumberoffixedpoints 

foramappingofthistypetherearealsomappingsattheotherextremewithno 

fixedpoints,foreverynon-trivialgraph. 

4

Example 3.1. LetG=K2 andtakes=2. Letthefirstplayerguessthe 

samevalueashesees(i.e.theoutcomeoftherandomexperimentofthesecond 

player),andletthesecondplayerguesstheoppositeofthevalueitsees.Forthis 

strategy,Fix(F)=0. Thusforeverygraphwithatleastoneedgethereexists 

astrategywhichneverguessesallvaluescorrectly,andthiscanbeextended 

tolargersaswell. Thisisofcoursetobeexpectedastheaveragenumberof 

fixedpointsoverallstrategymappingsis1. Soifweareabletofindstrategy 

mappingswithmorefixedpoints,thentheremustalsoexiststrategymappings 

withnofixedpointsatall. 

Sofar,foralltheexamplesofgraphswehaveseen,theguessingnumberwas 

independentofs. Ourearlierresultsshowthatinordertoseeadependence 

onswemustconsidernon-perfectgraphs. Thestrongperfectgraphtheorem 

[CRST06]tellsusthatagraphisperfectifandonlyifneitherGnoritscomplementcontainsaninducedoddcycleoflengthatleastfive. 

Soweturnour 

attentiontothefivecycleC5,whichisthesmallestnon-perfectgraph. 

Example3.2.LetG=C5,bethecycleon5vertices. Sinceα(C5)=2and 

χ(C5)=3,Lemmas2.6and2.7give2gn(G,s)3foreverys. 

Fors=2,letFbethestrategywhereavertexguesses2ifbothitsneighbours 

havevalue1,andguesses1otherwise.Thisstrategyguessescorrectlyonthefollowingx:{11212,12112,12121,21121,21211}.Thuswehavegn(C5,2)log 

25, 

andaquickcomputercheckshowsthatthisisindeedoptimal. 

Fors=3,letFbethestrategywhereavertexguesses2ifbothitsneighbours 

havevalue1,guesses3ifatleastoneneighbourhasvalues3andnoneighbourhas 

value2,andguesses1otherwise.ThisstrategyhasFix(F)=11andacomputer 

checkshowsthatitisthebestsymmetricstrategy.Howeverbycomputercheck, 

usingthemethodsofthenextsection,weknowthatthereexistsamorecomplex, 

vertexdependent,strategywhichisoptimalandhasFix(F)=12.Thuswehave 

gn(C5,3)=log 312

leastonu∈N(v),thenf ′ v(x)=s+1,otherwisef ′ v(x)=fv(x ′ ),wherex ′ is 

obtainedfromxbydefiningx ′ utobeequaltoxuifxu∈{1,...,s}andequalto 

1otherwise.ObservethatF ′ isindeedastrategymapping,everyfixedpointof 

FisalsoafixedpointofF ′ andmoreoverF ′ has(s+1,...,s+1)asafixed 

pointaswell. 

Lemma3.4.IfHisasubgraphofGthenFix(H,s)Fix(G,s) 

Proof.GivenastrategymappingFfortheguessinggameonH,weextendit 

toastrategymappingF ′ fortheguessinggameonGbydefiningf ′ v(x)tobe 

equaltofv({xu:u∈V(H)})ifv∈V(H)andtobeidentically1otherwise.It 

isimmediatethatF ′ isastrategymappinghavingatleastasmanyfixedpoints 

asF. 

Thelastinequalityisveryfarfrombeingstrict,asshownbyourearlier 

examplewiththebalancedcompletebipartitegraphon2nverticesversusthe 

matchingofsizen. 

Wecanalsoboundthenumberoffixedpointsforcompositevaluesofs 

Lemma3.5.Fix(G,s1s2)Fix(G,s1)Fix(G,s2) 

Proof.Thisfollowsbysimplywritingeachrandomnumberxvas(a−1)s1+b, 

withb∈{1,...,s1}anda∈{1,...,s2},andusingtheoptimalmappingsfors1 

ands2toguessaandbindependently. 

Theorem3.6.Thelimitlims→∞log sFix(G,s)exists,andisatmost|V(G)|. 

Moreover,itisequaltosup s∈Nlog sFix(G,s). 

Proof.SupposeGhasnverticesandletuswriteas=Fix(G,s).BythedefinitionofFix(G,s)itfollowsthatass 

n foreverysandsoifthelimitexiststhen 

itisatmostn. ByLemma3.3wehavethatas+1asforeverys∈Nandby 

Lemma3.5wehavethatastasatforeverys,t∈N.Ouraimistoshowthat 

lims→∞log sasexistsandisequaltosup s∈Nlog sas.Givens,t∈Nweclaimthat 

log sas k 

k+1 log tat, 

wherek=⌈logs/logt⌉. Theresultimmediatelyfollows. Indeed,givenε>0, 

letℓ=sup s∈Nlog sasandtaketlargeenoughsuchthatlog tat>(1−ε)ℓ.With 

thistfixed,wecannowpickslargeenoughsuchthatk/(k+1)>(1−ε).We 

deducethatlog sas>(1−ε) 2 ℓ,thusliminflog sasℓandsolimlog sasexists 

andisequaltoℓ.Toprovetheclaim,observethat 

logsas= logtas logts logtatk logttk+1log tak t 

k+1 

= k 

k+1 log tat. 

Hencemayintroducethefollowingasymptoticversionoftheguessingnumber. 

6

Definition3.7.Theasymptoticguessingnumbergn(G)ofGis 

gn(G)= lim 

s→∞ log sFix(G,s) 

Example3.8.LetusreturnbacktoC5.Wehavealreadyseenthatgn(C5,2 k ) 

gn(C5,2)=log 25.Itfollowsthatgn(C5)log 25.Inparticulargn(C5)>2.In 

thenextsectionwewillimprovefurtheronthisbound. 

4 Lowerboundsviafractionalcliquecovers. 

Sofarwehavedeterminedexactlytheguessingnumbersofperfectgraphsand 

foundanon-triviallowerboundfortheasymptoticguessingnumberofC5. In 

fact,amuchbetterboundcanbederivedforbothC5andothernon-perfect 

graphsbymakinguseofafractionalcoverings.Beforedefiningfractionalcoveringsweintroducethet-foldblowupofgraphsandtheblow-upstrategy. 

Definition4.1.GivenagraphGwedefinethet-foldblowupofG,denotedby 

G t ,asfollows:ForeachvertexvinGtherearetverticesv1,...,vtinG t with 

verticesviandujbeingneighboursinG t ifandonlyifvanduareneighbours 

inG. 

Laterwewillalsoneedthefollowingspecialcaseofthestronggraphproduct. 

Definition4.2.ThestrongproductofKtandG,denotedKt∗Gisthegraph 

obtainedfromG t byaddingalledgesoftheform(vi,vj)foreachvertexvinG. 

Wecanusethet-blowupofGtoobtainbetterboundsfortheguessing 

numberofGwithrespecttovaluesofswhichareperfecttpowers. 

Definition4.3.LetGbeagraph,lettbeapositiveintegerandlets=s t 1 

forsomeintegers1>1. GivenanystrategyF forG t withrespecttos t 1 ,we 

canobtainastrategy,whichwecalltheblow-upstrategy,forGwithrespect 

tosasfollows: IfxisthenumberassignedtoavertexvofG,wewritex= 

t i−1 

i=1 (xi−1)s1 ,wherexi∈{1,...,s1}foreachi.Thenweassignthenumbers 

x1,...,xttotheverticesv1,...,vtofG t .WethenfollowthestrategyFonG t . 

Ify1,...,ybarethenumbersguessedbyFatverticesv1,...,vb,thenthenumber 

guessedbytheblow-upstrategyatvertexvwillbey= t 

i=1 

(yi−1)s i−1 

1 . 

Wenotethatasimilarblow-upstrategycanbeusedinthecasethatsisa 

productoft(notnecessarilyequal)integersstrictlygreaterthan1.Wediscuss 

howthiscanbedoneinsubsection7.1. 

Theorem4.4.LetGbeagraphandlett,s1bepositiveintegers.Thengn(G,s) 

gn(G,s1). 

Proof.ItisimmediatethatwecanpartitionG tintotvertexdisjointcopiesofG. LetFbethestrategyonG twithrespecttos1whichfollowsthebestpossible strategyoneachofthistcopiesofG. Wenowfollowtheblow-upstrategy. 

Thewinningprobabilityisatleasts −t(n−gn(G,s1)) 

1 =s−(n−gn(G,s1)) andtheresult 

follows. 

7

WecouldhaveinfactprovedthistheoremusingLemma3.5instead. We 

willseeamorepowerfulapplicationoftheblow-upstrategyafterweintroduce 

fractionalcoveringsofgraphs. 

Definition4.5. AfractionalcliquecoverofagraphGisafamilyofcliques 

H1,...,HtofGtogetherwithnon-negativeweightsw1,...,wtsuchthat 

{i:v∈Hi} wi 

1forallv ∈ V(G). The maximumvalueof t i=1wi overallcliquecovers 

H1,...,HtofGisknownasthefractionalcliquecovernumberofGandisdenotedbycp 

f(G). Itisalsoknownasthefractionalchromaticnumberofthe 

complementGofG,denotedbyχf(G). 

Itiswellknownthatcp f(G)isalwaysarationalnumberandthatα(G) 

cp f(G)cp(G). Similarly,wehavethatω(G)χf(G)χ(G). Oneofthe 

basicresultsinfractionalgraphtheoryletsusrelatethefractionalchromatic 

numbertothechromaticnumberofasuitableblowupoftheoriginalgraph. 

Fortheseandotherresultsonfractionalgraphtheorywereferthereaderto 

[SU97]. 

Theorem4.6.ForeachgraphGthereexistsapositiveintegert,withsuchthat 

χf(G)= χ(Kt∗G) 

t .Equivalently,consideringthecomplementofKt∗G,wehave 

cp f(G)= cp(Gt ) 

t 

Observethatforanypositiveintegertandanyintegermultiplesoftwe 

havethatχ(Kt∗G)/tχ(Ks∗G)/sχf(G). Inparticular,iftisaninteger 

forwhichtheconsequenceofTheorem4.6holds, thenitalsoholdsforany 

integermultipleoft. NotethatconsideringthecomplementofG,thereisan 

integert(notnecessarilythesameastheonegivenbyTheorem4.6)suchthat 

cp(G t )=tcpf(G).Sowecanuseacombinationofthecliquecoverstrategyand 

theblow-upstrategytoobtainbetterstrategiesfortheguessinggame. 

Definition4.7.LetGbeanygraphandlettbeanypositiveintegersuchthat 

cp(G t )=tcpf(G). Thenthefractionalcliquestrategyisdefinedbytakingthe 

blow-upstrategyofGwithrespecttothecliquecoverstrategyofG t . 

Example4.8. ConsiderthefractionalcliquestrategyforC5. Thefractional 

chromaticnumberofC5 is 5 

2 andafractionalcliquestrategyforC5 canbe 

obtainedbyusingthecliquecoverstrategyonthe2-foldblowupC 2 5 ofC5.This 

graphisdepictedinFigure1,withacliquecoveringgivenbythethickedges. 

Theorem4.9.LettbeanypositiveintegerasinDefinition4.7.Thengn(G,s) 

n−cp f(G)wheneversisaperfectt-power.Inparticular,gn(G)n−cp f(G). 

Proof.Weapplythefractionalcliquestrategy.Wehavecp(G t )=tcp f(G)and 

so,if|V(G)|=nands=s t 1 ,thenthewinningprobabilityis 

s −(tn−cp(Gt )) 

1 

=s −(tn−tcp f (G)) 

1 

=s −(n−cp f (G)) . 

Theresultfollows. Theresultfortheasymptoticguessingnumberalsofollows 

immediatelysincegn(G,s)n−cp f(G)holdsforinfinitelymanys. 

8

Figure1:C 2 5 withacliquecovershownasthickedges. 

Forthefive-cycle,andothersymmetricalgraphs,wecanapplythefollowing 

wellknownlemma(Proposition3.1.1in[SU97]). 

Lemma4.10.IfGisavertextransitivegraphonnverticesthenχf(G)= n 

α(G) 

andcp f(G)= n 

ω(G) . 

Twosimpleclassesofvertextransitivegraphsaretheoddcyclesandtheir 

complements. 

Example4.11.ForC5,Lemma4.10improvesourpreviousboundstogn(C5,s) 

5 

2 foreveryswhichisaperfectsquare. Thisisexactlyhalfwaybetweenthe 

twosimpleboundsgivenbyα(C5)andχ(C5). Fortheoddcyclesingeneral, 

wefindthatgn(C2k+1,s) 2k+1 

2 foreveryswhichisaperfectsquare,andfor 

theircomplementswefindgn(C2k+1,s)(2k+1)− 2k+1 

k 

=2k−1− 1 

k for 

everyswhichisaperfectk-power.Hence,bythestrongperfectgraphtheorem 

[CRST06],wehavefoundimprovedboundsfortheclassofminimallyimperfect 

graphs. 

Fortheoddcycles,theimprovementgivenbyconsideringcp f(G)insteadof 

cp(G)isbounded.Howevertherearefamiliesofgraphswheretheimprovement 

canbearbitrarilylarge. 

Example4.12.Givenpositiveintegersn,rwithn>2r,theKnesergraphG= 

Kn:rhasthefamilyofallr-subsetsof{1,...,n}asitsverticeswithtwovertices 

beingneighboursifandonlyifthecorrespondingsetsaredisjoint.Thesegraphs 

areclearlyvertextransitiveon n 

r vertices.Itisimmediatethatω(G)=⌊n/r⌋. 

TheErdős-Ko-Radotheoremimpliesthatα(G)= n−1 r−1 andsobyLemma4.10 

wegetthatχf(G)=n/r. SotheguessingnumberofGisgn(G,s)= n n − r , 

foreveryswhichisaprefectrthpower.FortheKnesergraphstakingther-fold 

blowupofKn:rwillgiveaworkingcliquecoverstrategy,seeChapter3of[SU97] 

forfulldetailsoftheblowupsofKnesergraphs.. 

9 

r

On the otherhand, by Lovász’solutionto Kneser’sconjecture, we have 

χ(G)=n−2r+2andsothesimplecliquecoverstrategyonlygivesgn(G,s) 

 

−(n−2r+2).Soifn=tr,thenwegetanimprovementof(t−2)(r−1). 

n 

r 

ThefamilyofKnesergraphshassomeadditionalimportanceinconnection 

withfractionalcoverduetothefactthatthefractionalchromaticnumberof 

agraphcanbecompletelydescribedintermsofwhichKnesergraphsGhas 

homomorphismsinto,seeChapter3of[SU97]. 

Othergraphswithlargegapsbetweentheboundscanbefoundbye.g.using 

theMycielskiconstructionorthestronggraphproduct,underwhichfractional 

chromaticnumbersaremultiplicative.See[SU97]formoredetailsontheseand 

otherconstructionsforgraphswithprescribedfractionalchromaticnumbers. 

5 Thecodegraph 

Intheprevioussectionwehaveseenthattheguessingnumbercanbedescribed 

intermsofthenumberoffixedpointsofthestrategymaps,andsincethisisa 

finitesetofmappingsofafinitesetitisinprinciplepossibletofindtheoptimal 

strategyforagivenGandsbyanexhaustivesearch.Wealsosawhowtoprovide 

goodlowerboundsfortheguessingnumberintermsofχf(G),butwedidnot 

findmatchingupperboundsingeneral. Ournextaimistointroducethecode 

graphfortheguessinggamewithagivenGandsand,usingthisgraph,both 

provideamoreefficientcomputationalprocedureforfindingoptimalstrategies 

andaconjectureastowhattheasymptoticguessingnumbergn(G)shouldbe. 

LetΩ(n,s)denotethesetofallstringsoflengthnoverthealphabet{1,...,s}. 

Given a graph G on n vertices we will usually identify Ω(n,s) with As = 

{1,...,s} V(G) . OurultimateaimistodeterminewhichsubsetsofΩ(n,s)are 

fixedpointsetsofsomestrategymapFonG. 

Definition5.1.ThecodegraphX(G,s)hasvertexsetΩ(n,s)withtwovertices 

xandyofX(G,s)beingadjacentifandonlyifthereisavertexvofGsuch 

thatxv=yvbutxu=yuforallu∈N(v) 

Asthenexttheoremshows,thecodegraphcompletelydescribesthesetof 

optimalstrategiesfortheguessinggame. 

Theorem5.2.LetIbeasetofverticesofX(G,s). ThenIisthesetoffixed 

pointsofanoptimalstrategymappingFifandonlyifitisamaximalindependent 

set. 

Proof.LetFbeastrategymapping(notnecessarilyoptimal)havingIasitsset 

offixedpoints. Letx,y∈IandsupposethattheyareneighboursinX(G,s). 

ThenthereexistatleastonevertexvofGsuchthatxv=yvbutxu=yuforall 

u∈N(v).Butx,yarefixedpointsofFandsoxvandyvcanbedeterminedby 

thevaluesofxuandyuwithu∈N(v).Thusxv=yv,acontradiction. 

Conversely,givenamaximumindependentsetI,wedefineamappingF: 

As→Asasfollows:ForeachvertexvofGandeachvertexxofX(G,s),ifthere 

10

isanx ′ ∈Isuchthatx ′ u=xuforeachu∈N(V)thenwedefinefv(x)=x ′ v. 

Otherwise,wedefinefv(x)=1. NotethatsinceIisanindependentsetthen 

eachfviswell-definedandthemappingF={fv} v∈V(G)isastrategymapping. 

WeclaimthatF hasIasitsthesetoffixedpoints. Indeed,byconstruction 

everyelementofIisafixedpointofF.Moreover,sincethesetoffixedpoints 

ofF isanindependentsetcontainingthemaximumindependentsetI,then 

itmustbeequaltoI. ItnowfollowsthatF isoptimalasnootherstrategy 

mappingcanhavemorefixedpoints. 

ItfollowsthatwecancomputetheguessingnumberofGbycomputingthe 

independencenumberofX(G,s) 

Corollary5.3.ForeverypositiveintegersandeverygraphGwehavethat 

gn(G,s)=log s(α(X(G,s))). 

Oursimpleboundsfortheguessingnumbercaneasilybetranslatedinto 

propertiesofthecodegraph. 

Theorem5.4. 

1. Ifα(G)=t,thenα(X(G,s))s n−t . 

2. Ifχ(G)=k,thenα(X(G,s))s k . 

Proof. 

1. LetIbeanindependentsetofsizeα(G)=tinG. Givenx,y∈Ω(n,s) 

wesaythattheyareequivalentifandonlyifxu=yuforeveryu /∈A. 

Observethatthisisanequivalencerelationwitheachequivalenceclass 

beingacliqueinX(G,s)andhavingsizeexactlys t . Everyindependent 

setofX(G,s)cancontainatmostoneelementfromeverysuchcliqueand 

thereforeitcanhavesizeatmosts n−t . 

2. LetH1,...,HkbeacliquecoverofGofsizekandletIbethesetofall 

elementsofx∈X(G,s)suchthat 

v∈Hi xv≡0modsforeach1ik. 

ThenitiseasilyseenthatIisanindependentsetofX(G,s)ofsizes k . 

TheboundfromTheorem4.9canalsobetranslatedintothissettingby 

consideringthecodegraphofG t ,forasuitablevalueoft. 

Thecodegraphhass n vertices. Givenanypairofstringsoflengthnwe 

candecideintimen 2 iftheyareadjacentinthecodegraphornot. Havingconstructedthecodegraphwecannowusestandardalgorithmsforfindingmaximumindependentsetstofindboththeguessingnumberandoptimalguessingstrategies.Usinge.g.themaximumindependentsetalgorithmfrom[TT77] 

wegetthefollowingupperboundonthecomplexityofdeterminingtheoptimal 

strategy. 

11

Corollary5.5.Thereexistsanalgorithmwhichconstructstheoptimalguessing 

strategyintimeatmost2 1 

3sn Theupperboundhereisofcourseprohibitivelylarge,howeveritcanbe 

reducedbyusingthelargeautomorphismgroupofX(G,s),e.g.usingmethods 

similartothoseof[RAMS04]. 

Lemma5.6.X(G,s)isvertextransitiveandAut(G)×Z n s⊆Aut(X(G,s)) 

Proof.LetφbeanautomorphismofGandlety∈Z n s.Thentheautomorphism 

(φ,y)ofAut(X(G,s))isdefinedby 

(φ,y)(x)v=(x+y) φ(v) 

wheretheelementsofΩ(n,s)areviewedasvectorsinZ n swithadditiondefined 

componentwise.ItisstraightforwardtoverifythatthisisindeedanautomorphismofX(G,s)andthat(φ,y)◦(φ 

′ ,y ′ )=(φ◦φ ′ ,y+y ′ )asrequired. 

6 Upperboundsviaentropyinequalities 

Ournextaimistoshowthatourboundontheguessingnumberforoddcycles 

andtheircomplementsgiveninExample4.12issharp. Asatoolwewilluse 

entropyinequalitiesofdiscreterandomvariable. Recallthatgivenadiscrete 

randomvariableXwithoutcomeslabelled1,...,N,itsbinaryentropyisdefined 

as 

N 

H(X)= Pr(X=i)log 2Pr(X=i), 

i=1 

Itwillbemoreconvenientforwhatfollowstoworkwiththes-entropyinstead 

H(X)= 

N 

Pr(X=i)log sPr(X=i). 

i=1 

Fromnowon,wewilldropthesubscriptsandfollowtheconventionthatallour 

logarithmsarewithbases.TheconditionalentropyofXgivenanotherrandom 

variableY istheentropy 

H(X|Y)= 

Pr(X=i,Y=j)logPr(X=i|Y=j). 

i,j 

Wewillusethefollowingpropertiesoftheentropyfunction. 

Theorem6.1.LetX,Y,X1,...,Xnbediscreterandomvariables 

1. H(X)0withequalityifandonlyifXisdetermenistic. 

2. IfXtakesvaluesinasetS,thenH(X)log|S|withequalityifandonly 

ifXisuniformlydistributedonS. 

12

3. H(X,Y)=H(X|Y)+H(Y). 

4. H(X|Y)0withequalityifandonlyifXisdeterminedbyY. 

5. H(X|Y)H(X)withequalityifandonlyifXandY areindependent. 

6. ForA⊆{1,...,n},letXAdenotestherandomvectorofXi’swithi∈A. 

Withthisnotation,Hisasubmodularfunction,i.e. 

H(XA∪B)+H(XA∩B)H(XA)+H(XB). 

Fortheseandotherpropertiesoftheentropyfunctionwereferthereader 

to[CT06,AS08]. Recently,entropyinequalitieshavebeenusedusedtoderive 

boundsforotherproblemsincombinatorics,see[Rad03]forasurvey. 

Theorem6.2.Foreachk,s∈Nwehavethatgn(C2k+1,s) 2k+1 

2 . 

Proof.AssumethattheverticesofC2k+1arenumberedas1,...,2k+1with 

N(i)={i−1,i+1}whereadditionandsubtractionaredonemodulo2k+1.LetI 

beamaximumindependentsetofX(C2k+1,s)andletX=(X1,X2,...,X2k+1) 

bethecharacteristicvectorofanelementpickeduniformlyatrandomfromI. 

Byproperty(2),wehavethatH(X)=log|I|. Wewillnowproceedtobound 

H(X)fromabove. Whenestimatingentropieswewillneedtoconsidermany 

quantitiesoftheformH(XB)andinordertosimplifyournotationwewillwrite 

themasH(B)instead. 

SinceXisafixedpointofastrategymapping,weknowthatXiisdetermined 

byXi−1andXi+1.Inparticular,Xisdeterminedjustbytherandomvariables 

X2,X3,X5,X7,...,X2k+1andsobyproperties(3)and(4)wehavethat 

H(X)=H(X|2,3,5,...,2k+1)+H(2,3,5,...,2k+1) 

=H(2,3,5,...,2k+1) 

Applyingnowproperties(3)and(5)wegetthat 

andbyproperty(2)itfollowsthat 

Likewise,wehave 

H(X)H(2,3)+H(5,7,...,2k+1), 

H(X)H(2,3)+log(s k−1 )=H(2,3)+k−1. 

H(X)H(1,2,3,4)+H(6,8,...,2k)H(1,2,3,4)+k−2. 

Addingtheseinequalitiesandusingproperty(6)wegetthat 

2H(X)H(2,3)+H(1,2,3,4)+2k−3 

H(1,2,3)+H(2,3,4)+2k−3. 

13

ButsinceX2isdeterminedbyX1andX3wehavethatH(1,2,3)=H(1,3) 

log(s 2 )=2andsimilarlyH(2,3,4)2.Puttingallthesetogetherweget 

log|I|=H(X) 2k+1 

2 

andsobyLemmas5.2and5.3wegetthatgn(C2k+1,s) 2k+1 

2 

Usingalargerfamilyofindexsetswecansimilarlyfindaboundforthe 

guessingnumberofthecomplementofanoddcycle. 

Theorem6.3.Foreachk,s∈Nwehavethatgn(C2k+1,s)2k−1− 1 

k . 

Proof.AssumethattheverticesofC2k+1arenumberedas1,...,2k+1with 

N(i)=[2k+1]\{i−1,i+1}whereadditionandsubtractionaredonemodulo 

2k+1. LetI beamaximumindependentsetofX(C2k+1,s)andletX = 

(X1,X2,...,X2k+1)bethecharacteristicvectorofanelementpickeduniformly 

atrandomfromI. Byproperty(2)wehavethatH(X)=log|I|. Wewillnow 

proceedtoboundH(X)fromabove. 

Foreachi∈[2k+1],letJi=[2k+1]\{i−1,i+1}andKi=[2k+1]\{i}. 

ObservethatXiisuniquelydeterminedfromthevaluesofallXjwithj=iand 

soH(X)=H(Ki).ObservealsothatforanyS⊆Jiwehavethat 

H(X)=H(Ki−1)H(Ji)+H(S∪{i+1})−H(S)2k−2+H(S∪{i+1})−H(S). 

Indeed, thefirstinequalityfollowsfromproperty(6)bytakingA = Ji and 

B=S∪{i+1},andthesecondinequalityfollowsfromproperty(2).Similarly, 

wehave 

H(X)2k−2+H(S∪{i−1})−H(S). 

LetT={3,5,...,2k+1}.ApplyingthesecondinequalitywithS=Tandi=2 

weget 

H(X)2k−2+H(T∪{1})−H(T) 

ApplyingthefirstinequalityrepeatedlywithS=T∪{1},T∪{1,2},T∪{1,2,4},...,T∪ 

{1,2,4,...,2k−6}andi=1,3,5,...,2k−5respectivelyweget 

H(X)2k−2+H(T∪{1,2})−H(T∪{1}) 

H(X)2k−2+H(T∪{1,2,4})−H(T∪{1,2}) 

··· 

H(X)2k−2+H(T∪{1,2,4,...,2k−4})−H(T∪{1,2,4,...,2k−6}) 

andsummingallthesetogetherwiththepreviousinequalityupweget 

Sincealso 

(k−1)H(X)2(k−1) 2 +H(J2k−1)−H(T)2k(k−1)−H(T). 

H(X)=H(3,4,...,2k+1)H(T)+H(4,6,...,2k)H(T)+(k−1), 

14

weget 

andtheresultfollows. 

kH(X)(2k+1)(k−1) 

HerewehavefoundthatifGisaminimallyimperfectgraphthenitsguessing 

numberisatleastgn(G,s)n−χf(G)foreveryvalueofs. Wealsoknow 

fromtheprevioussectionthatequalityholdsforinfinitelymanyvaluesofs.In 

particularitholdsforeveryswhichisaperfectb-powerforsomebwhichcan 

bedeterminedfromG.WealsoknowfromSection2thatgn(G,s)n−χf(G) 

foreveryperfectgraphG,whereinfactweevenhaveequalityforeveryvalueof 

s.Weconjecturethatthisinequalityistrueingeneral. 

Conjecture6.4.ForeverygraphGandeverypositiveintegerswehavegn(G,s) 

n−χf(G).Inparticulargn(G,s)=n−χf(G)foreveryvalueofsforwhichthe 

fractionalcliquestrategycanbeusedandsogn(G)=n−χf(G). 

Itwouldalsobeofinteresttofindtheguessingnumberforvaluesswhere 

thefractionalcliquestrategycannotbeused. EvenforC5weknowonlythe 

valuefors=3,andwecanobtainsomeboundsforothernon-perfectsquares 

viaLemmas3.3and3.5. 

Forvaluesofsotherthanperfectsquareswedonotevenhaveafullunderstandingoftheguessingnumbersforoddcycles. 

Example6.5.Fors=2wehavecomputedtheindependencenumbersofthe 

codegraphsforsmalloddcyclesusingastandardlinearprogrammingsolver. 

Wefoundthatα(X(C7,2))=8,α(X(C9,2))=16andα(X(C11,2))=32,i.e. 

exactlythevaluesattainedbythecliquestrategyonthesegraphs. ForC7,we 

alsofoundthatα(x(C7,3))=29>3 3 ,soherethecliquestrategyisnotoptimal. 

Weclosethissectionwithtwoproblems. 

Problem6.6.Isthecliquestrategyoptimalforgn(C2k+1,2)forallk3? 

Problem6.7.Isthecliquestrategyoptimalforgn(C2k+1,3)forallk4? 

7 Generalisations 

7.1 Nonuniformrandomnumbers 

Intheresultssofarwehaveassumedthatallrandomnumbersxvwerepicked 

uniformlyatrandomfromthesingleset{1,...,s}. Howevertherearetwoextensionswhichcouldnaturallyappear. 

Thefirstistoassumethattherandom 

numbersareindependentasbeforebuteachxvcomesfromsomenon-uniform 

distributionPv,whichmaydependonv.Thesecondistoallowthedifferentxv 

tocomefromsetsSvwhichmayvaryinsizewithv.Thecodegraphapproach 

fromtheprevioussectioncaneasilybeadaptedtobothofthesemodifications. 

15

InordertodealwithdistinctPvwecangiveeachstringinΩ(n,s)aweight 

vPv(xv)anduseastrategygivenbyamaximumweightindependentset.With thisweightingthemaximumweightwillcorrespondtothemaximumwinning 

probability. 

LikewisethecodegraphcaneasilybeadaptedtodifferingSvbyreplacingthe 

uniformΩ(n,s)withtheproductset 

vSv,usingthesameneighbourrelation asbefore. 

7.2 Directedgraphs 

In[Rii07]theemphasiswasondirectedgraphs,asthisiscaseusedinnetwork 

codingandbooleancircuits. Whileourdiscussionhasmainlybeenintermsof 

undirectedgraphsthereareseveralmethodsandboundswhichcanbegeneralizedtodirectedgraphsaswell.Wewillheregiveonlyafewexamples. 

ThecodegraphX(G,s)canbedefinedforadirectedgraphsbyreplacingthe 

neighbourhoodofvusedininDefinition5.1withthesetofout-neighboursof 

v,i.e.thesetofverticestowhichthereisadirectededgefromv.Asbeforethe 

guessingnumberisgivenintermsofthemaximumindependentsetsofX(D,s). 

Recallthatadirectedgraphisacyclic ifitisnotpossibletowalkalong 

acyclebyfollowingthedirectionsofitsedges. GivenadirectedgraphDlet 

α(D)denotethesizeofthelargestinducedsubgraphofDwhichisacyclic. 

Thisparametercanbeusedinplaceoftheindependencenumbertoboundthe 

guessingnumberofD. 

Theorem7.1.LetDbeadigraphonnvertices.Thengn(D,s)n−α(D) 

Proof.LetAbeaninducedacyclicsubgraphofGofsizeα(D). LetA0bethe 

setofsinksinA,i.e.thoseverticesinAwhoseonlyout-neighboursareinD\A, 

andrecursivelydefineAi tobethesetofverticesfromA\Ai−1whoseonly 

out-neighboursareinAi−1. 

AsintheproofofLemma2.7givenastrategyF,weexposetherandom 

numbersxuinstages. Atthefirststageweexposeallnumbersxuwithu/∈A. 

Thenallfunctionsfuwithu∈A0areuniquelydetermined.Wenowexposeall 

numbersxuwithu∈A0andweobservethattheprobabilitythatallplayers 

inA0guesscorrectlyiss −|A0| . Nowallfunctionsfuwithu∈A1areuniquely 

determinedandtheprobabilitythatallplayersinA1guesscorrectlyiss −|A1| . 

Proceedinginductivelyweconcludethatthewinningprobabilityisatmosts −|A| 

andtheresultfollows. 

Asinthecliquestrategywemayconstructstrategiesforlargerdirected 

graphsbypartitioningtheirvertexsetintodisjointcopiesofsmallergraphs, 

followingtheoptimalstrategiesindependentlyoneachpartofthepartition. 

Howeverunlikeforordinarygraphs, wherethecliquesplayedapivotalrole, 

theredoesnotseemtobeacanonicalfamilyofdirectedgraphstopartitionthe 

graphinto.Themostnaturalanalogueofthecompletegraphsaretournaments, 

whicharedirectedcompletegraphs,butsincetherearetournamentswhichare 

16

acyclic,someadditionalconditionsseemtoberequired. Givenanyfamilyof 

directedgraphswithknownvaluesoftheirguessingnumberswemayalsouse 

fractionalcoversusingthesegraphsinthesamewayaswedidinTheorem4.9. 

7.3 Infinitegraphs 

Itisalsonaturaltoaskwhathappensifweplaytheguessinggameoninfinite 

graphs.Webeginbyconsiderthecountablyinfinitecompletegraphwiths=2, 

say.Asacomparison,recallthatthewinningprobabilityonanyfinitecomplete 

graphwiths=2is1/2. 

Considertheset{0,1} N ofcountablyinfinite0,1-sequences. Wewouldlike 

tofinda‘large’subsetAofthissetsuchthatanytwosequencesinAdifferinat 

leasttwoplaces.HavingfoundsuchanA,wecanthendefineanaturalstrategy 

FwhichhasAasitssetoffixedpoints. Itisnotdifficulttopartition{0,1} N 

intotwosetsAandBsuchthatbothofthemhavethepropertythatanytwo 

oftheirsequencesdiffernatleasttwoplaces: Onejustdefinesanequivalence 

relation∼on{0,1} N bysayingthattwosequencesx,yareequivalentifandonly 

iftheydifferinafinitenumberofplaces. ForeveryequivalenceclassCof∼ 

wethenpickasequencex∈CanddefineAxtobethesetofallsequencesin 

CwhichdifferfromxinanoddnumberofplacesandBxtobethesetofall 

sequencesinCwhichdifferfromxinanevennumberofplaces. Itiseasyto 

checkthatA=∪xAxandB=∪xBxhavetherequiredproperties. 

Thisisgreatasitseemstoshowthatthewinningprobabilityforthisgraph 

isalso1/2. Howeverthereisonecatch. ThesetsA,Babovewereconstructed 

usingtheaxiomofchoiceandinfactitisnottoodifficulttoseethattheyare 

notmeasurable. Forourpurposeshowever,wewanttocalculatethewinning 

probabilityandsowemustrequirethateachindividualstrategyfv:{0,1} N → 

{0,1}ismeasurable. Sothequestionnowtranslatestofindingameasurable 

subsetAof{0,1} N suchthatanytwosequencesofAdifferinatleasttwoplaces 

andAhasaslargeameasureaspossible. Thenforeachk∈N,thestrategy 

fk:{0,1} N →{0,1}definedbyfk(x)=0ifthesequencex ′ obtainedfromxby 

makingitsk-thdigit0belongstoAandfk(x)=1otherwise,isameasurable 

strategy.MoreoverthesetoffixedpointsA ′ ofthestrategymappingF=(fk)k∈N 

isameasurablesubsetof{0,1} N containingAandsoithasthesamemeasure 

asA. 

WeproceedtoshowthatanysuchAmusthavemeasure0. Observethat 

thisisimmediateifAisanopensubsetof{0,1} N (undertheproducttopology). 

IndeedthesetsUx,n={y:yi=xi∀in}formabasisforthetopologyand 

clearlynosuchsetcanbeasubsetofAasx,x ′ ∈Ux,nwherex ′ isobtainedfrom 

xbychangingits(n+1)-thdigit. IfthereissuchasetAofpositivemeasure, 

thenwecanfindabasicopensetUx,nsuchthatm(A∩Ux,n)51m(Ux,n)/100. 

NowletA0bethesetofallsequencesinA∩Ux,nwhose(n+1)-thdigitis0and 

letB0bethesetofallsequencesobtainedfromA0bychangingthe(n+1)-th 

digitto1. WedefineA1andB1analogously. NowobservethatA0,A1,B0,B1 

aredisjointsubsetsofUx,nwithm(A0)+m(A1)51m(Ux,n)/100andm(A0)= 

17

m(B0),m(A1)=m(B1). Itfollowsthatm(A0∪A1∪B0∪B1)m(Ux,n),a 

contradiction. 

Soeveninthecompletecountablyinfinitegraphitturnsoutthatthewinning 

probabilityfortheguessinggameis0. 

Acknowledgments 

ThematerialinthispaperwasfirstpresentedattheNewtonInstituteworkshop 

“CombinatorialandProbabilisticInequalities”in2008andthesecondauthor 

wouldliketothanktheinstituteforitshospitalityduringhisstaythere. 

References 

[ACLY00] RudolfAhlswede,NingCai,Shuo-YenRobertLi,andRaymondW. 

Yeung. Networkinformationflow. IEEETrans.Inform.Theory, 

46(4):1204–1216,2000. 

[AS08] NogaAlonandJoelH.Spencer. Theprobabilisticmethod. Wiley- 

InterscienceSeriesinDiscreteMathematicsandOptimization.John 

Wiley&SonsInc.,Hoboken,NJ,thirdedition,2008. WithanappendixonthelifeandworkofPaulErdős. 

[Ber63] C.Berge. Perfectgraphs. InSixPapersonGraphTheory.Indian 

StatisticalInstitute,1963. 

[CRST06] Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin 

Thomas. Thestrongperfectgraphtheorem. Ann.ofMath.(2), 

164(1):51–229,2006. 

[CT06] ThomasM.CoverandJoyA.Thomas. Elementsofinformation 

theory.Wiley-Interscience[JohnWiley&Sons],Hoboken,NJ,second 

edition,2006. 

[Lov72] L.Lovász. Acharacterizationofperfectgraphs. J.Combinatorial 

TheorySer.B,13:95–98,1972. 

[Rad03] J.Radhakrishnan. ComputationalMathematics,ModellingandAlgorithms,chapterEntropyandcounting. 

NarosaPublishingHouse, 

2003. 

[RAMS04]ArathiRamani,FadiA.Aloul,IgorL.Markov,andKaremA.Sakallah.Breakinginstance-independentsymmetriesinexactgraphcoloring.InProceedingsoftheconferenceonDesign,automationandtest 

inEurope-Volume1,DATE’04,pages10324–,Washington,DC, 

USA,2004.IEEEComputerSociety. 

18

[Rii07] SørenRiis. Informationflows,graphsandtheirguessingnumbers. 

Electron.J.Combin.,14(1):ResearchPaper44,17,2007. 

[SU97] EdwardR.ScheinermanandDanielH.Ullman. Fractionalgraph 

theory. Wiley-InterscienceSeriesinDiscreteMathematicsandOptimization.JohnWiley&SonsInc.,NewYork,1997. 

Arational 

approachtothetheoryofgraphs,WithaforewordbyClaudeBerge, 

AWiley-IntersciencePublication. 

[TT77] RobertEndreTarjanandAnthonyE.Trojanowski.Findingamaximumindependentset.SIAMJ.Comput.,6(3):537–546,1977. 

DemetresChristofides,InstituteforTheoreticalComputerScience,FacultyofMathematicsand 

Physics,MalostranskéNáměstí25,18800Prague,CzechRepublic.demetres@kam.mff.cuni.cz 

KlasMarkström,DepartmentofMathematicsandMathematicalStatistics,Ume˚aUniversity, 

90187Ume˚a,Sweden.klas.markstrom@math.umu.se 

19

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