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Rights All yKvc áé 2014 Máté, Kovács by ii ————————————————— Signature CEU eTD Collection ewrs rp ooopim rp oe rniiegah,gahretraction graph graphs, transitive core, graph homomorphism, graph Keywords: retractions graph products, graph between described. is interplay cores speculate The graph We and cores. investigated. are is graphs graphs transitive transitive all regarding transitive almost have result that graphs structural transitive a that of out possibility turns It The core. cores. graph class the every called and element relation, minimal equivalence vertex an a is has equivalent homomorphically Being two forth. the and between homomorphism back graph graphs a is there if compatible. equivalent homomorphically are called approaches are two graphs the Two - automorphisms can graph graphs and of homomorphisms symmetry graph and by Structure handled graphs. be of symmetry and structure on results surveys thesis The Abstract iii CEU eTD Collection rp ooopimadgahsrcue22 9 structure graph and homomorphism Graph 4 graphs of groups Automorphism 3 operations Graph 2 Introduction 1 Abstract Copyright Contents of Table 36 . . 28 ...... 22 ...... 27 ...... 15 ...... 12 ...... 10 ...... structure . . . Product ...... 4.4 ...... 7 . . . 7 . Cores ...... 4.3 . . . . ordering . . . . and . . . 6 . . equivalence . . . Homomorphic ...... 4.2 ...... Homomorphisms ...... 4 . . . . 4.1 ...... 4 ...... graphs . . . . . Asymmetric ...... 3.3 ...... graphs . . . . Cayley ...... 3.2 ...... graphs . . Transitive ...... 3.1 ...... product . . Lexicographic ...... 2.5 ...... product . Strong ...... 2.4 ...... product Direct . . . . . 2.3 . . . . product Cartesian . . 2.2 . union and join Graph 2.1 40 ...... 36 ...... 31 ...... product . . direct . The . . . . . 4.4.2 . . product . Cartesian . The . . 4.4.1 ...... symmetry and Cores 4.3.1 iv iii ii 4 1 CEU eTD Collection Bibliography remarks Closing 5 41 ...... graphs Rigid 4.5 40 ...... 40 . 40 ...... join . Graph . . . . . 4.4.5 . . product lexicographic . 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CEU eTD Collection Let product): (Cartesian 2.3 Definition product Cartesian 2.2 Let union): disjoint (Graph 2.2 Definition Let join): (Graph union 2.1 and Definition join Graph 2.1 operations Graph 2 Chapter and 2. 1. 2. 1. 2. 1. iia,ytopst prto swa ilcl h ijituino graphs. of union disjoint the call will I what is operation opposite yet similar, A G G G , , , V V V E E H H H ( ( ( ( ( G G G G G etogah.Terdson no,dntdby denoted union, disjoint Their graphs. two be etogah.Terji,dntdby denoted join, Their graphs. two be etogah.TerCreinpout eoe by denoted product, Cartesian Their graphs. two be ƒ [( ∪ + ∪ + H x H H H H 1 = ) = ) = ) = ) , = ) y 1 V ) E E V V , ( ( ( ( ( ( G G G x G G ) 2 ) ) ) ) , × ∪ ∪ ∪ ∪ y V 2 E E V V )] ( ( ( ( ( H H H H H ∈ ) ) ) ) ) . , E { ∪ , , ( G [ ƒ u , H v ) ] feither if : u ∈ V ( G G + ) 4 , v H ∈ sdfie sfollows: as defined is V ( G H ∪ ) } H G . ƒ sdfie sfollows: as defined is H sdfie sfollows: as defined is CEU eTD Collection hscami reee o o-once graphs. non-connected for if even true property: is cancellation claim the This is theorem the of consequence important An 2.7.1: Corollary graphs of structure the about much too are us graphs tell most not as does sadly theorem factorization prime above The 2.7.1: Note where isomorphism to up graphs, factors. prime the of of product order a the as and representation unique a has graph connected of Every product tensor factorization): the Prime (Sabidussi-Vizing: or 2.7 product Theorem Kronecker the called also is graphs. product Cartesian the this, on Based 2.6.1: Note that Let 2.6: Claim Let 2.5: Claim commutative. and associative is product Cartesian The brevity. for product Box product the 2.4: call Claim also may we “boxes”, generate to tends product the Because 2.3.1: Note fagahi o once hntefcoiainmyntb nqe oneeapeis counterexample A unique. be not may factorization the then connected not is graph a If ( A u | G V , ∪ v ( eteajcnymti ftegraph the of matrix adjacency the be G ) • • stedson no ftogah oegsaecetdbtencopies. between created are edges no - graphs two of union disjoint the is , ) ( | x x y = 1 1 , y n = = ) G ∈ x y and 2 2 V ƒ and and ( -primes. G | V ƒ ( [ [ ( K H H x y 1 1 1 ) ) , , ∪ Then . | x y = K 2 2 d ] ] 2 G n ∈ ∈ ƒ ∪ H H hnteajcnymatrix adjacency the Then . E E K ( ( ( ( 2 2 H G u ) ƒ , ) ) . v or , ( ) K , 1 ( ∪ x , G K y 2 3 = ) ) and ( = ) 5 A d K H G 1 ( eteajcnymti fgraph of matrix adjacency the be ∪ u , K x 2 2 + ) ∪ A K d H G 2 4 ƒ ) ( ƒ H v , ( = y K ) 1 A . ∪ G ⊗ K G 2 I ) ƒ n , H K + = ∼ I n H G ƒ ⊗ K A H then H . Suppose . G = ∼ H . CEU eTD Collection Let Miller): (Imrich, 2.8 Theorem Let product): categorial or product (Direct 2.10 Definition product Direct 2.3 Let 2.9.1: Corollary Let the to isomorphic factors. is 2.9: the graphs Claim of union prime disjoint connected the of of group product automorphism Cartesian the of group automorphism The 2.8.1: Corollary ietpouthsatastv uoopimgopi ahfco a rniieautomorphism transitive a has factor each if group automorphism group. transitive a has product direct has A product the If bipartite. is 2.13: factors Theorem the of are one there most then at factors if bipartite connected is graphs of product direct A 2.12: Claim isomorphism. to up understood are equalities the All 2.11: Claim G 1 ƒ • • • • • G G G G . . . , = ( h ietpoutdsrbtswt h ijitunion: disjoint the with distributes product direct The associative, and commutative is product direct The V G eatastv rp.Te its Then graph. transitive a be eacnetdgah Suppose graph. connected a be H x ƒ ( × 1 G G – – etogah.Tedrc rdc ftetogah,denoted graphs, two the of product direct The graphs. two be G , 1 K x × n ƒ x x 2 1 hnteei permutation a is there Then . 1 2 ) H . . . = ∼ ∼ ∼ ∼ = ) G ƒ ( y y oteoepitgahi unit. a is graph point one the so , G y 1 2 V 1 . n and , . ( y G G 2 ) ) stastv fadol if only and if transitive is × if V ( H φ ) 2 ( , k x − 1 1 , . . . , ijitcomponents. disjoint ƒ fco eopsto sunique. is decomposition -factor φ x n π ( = ) ∈ ∈ Aut S φ n ( 1 G n isomorphisms and ( G 6 x ) i π n h rm atrdcmoiinof decomposition factor prime the and stastv o all for transitive is ( 1 ) ) , . . . , G φ k × ( x ( H π ( 1 k φ ) ∪ )) i : H . i G G = 2 × π ) ( ,..., . . . 1, = ∼ i H ) → G sdfie sfollows: as defined is × G n i H . o which: for 1 ∪ G × H G 2 , is G = k CEU eTD Collection Let product): (Strong 2.14 Definition product Strong 2.4 Let 2.17: Definition product Lexicographic 2.5 connected. is factor every if only and if connected is graphs of product strong A 2.16.1: Corollary factors. multiple for generalized be can this and Let 2.16: Claim isomorphism. to up understood are equalities the All 2.15: Claim follows: • • • • • • • G G G , , , ( ( union: disjoint the with distributes product strong The associative, and commutative is product strong The V G V H H H x x ( ( ‚ 1 1 G G – – – – etogah.Tedrc rdc ftetogah,denoted graphs, two the of product direct The graphs. two be etogah,then graphs, two be , , etogah.Telxcgahcpouto h w rps denoted graphs, two the of product lexicographic The graphs. two be K [ x x × [( [( H x x 2 2 1 s 1 1 ) ) H )= ]) = ∼ x x ∼ ∼ ∼ = 1 1 = ) , , G ( ( x x y y oteoepitgahwt opo ti unit. a is it on loop a with graph point one the so , y y 2 2 V 1 1 V 1 1 ) ) ( or and , , , , ( G y y ( ( G 2 2 ) y y ) ) ) 1 1 × x × , , if if 2 y y V V 2 2 ∼ ( )] )] ( d H H G y ∈ ∈ ) ‚ 2 ) , . , H E E (( ( ( G G a ƒ , × b H ) H , ) ( ) or , c . , d )= )) max 7 { d G ( a , c G ) , ‚ d H ( ( H b 1 , d ∪ ) G H } , 2 ‚ ) H = ∼ sdfie sfollows: as defined is G ‚ H G [ 1 H ∪ ] G sdfie as defined is ‚ H 2 , CEU eTD Collection eiorpi rdc ftogah a rniieatmrhs ru fadol fbt factors both if only and group. if automorphism group transitive automorphism a transitive have a has graphs two of product lexicographic A connected. is factor leftmost the 2.21: if Theorem only and if connected is graphs of product lexicographic A 2.20: Claim l h qaiisaeudrto pt isomorphism. to up understood are equalities the All (Properties): 2.19 Claim of Let 2.18: Definition G pca aeo h eiorpi rdc stegahmlil operation. multiple graph the is product lexicographic the of case special A • • • • G ihthe with n ie u o h other. the not but side, one union: disjoint the with distributes product lexicographic the associative, is product lexicographic The commutative. not is product lexicographic The G eagraph, a be ¯ [ H ¯ = ] n G vre mt graph. empty -vertex [ H n ] . > 0 nitgr The integer. an n mlil of -multiple 8 G stegraph the is G G [ H [ K 1 ¯ n ∪ ] h eiorpi product lexicographic the , H 2 ] = ∼ G [ H 1 ] ∪ G [ H 2 ] from CEU eTD Collection Aut 3.5: Lemma Let 3.4: Lemma Let automorphism): (Graph 3.1 Definition graphs of groups Automorphism 3 Chapter Let 3.3: Lemma group. automorphism an graph a of automorphisms of group any call I text following the In 3.2.1: Note by group Let 3.2: Claim trivial. is ( rp uoopim r pca emttoso h etxsto h rp.Tefloigclaim following The graph. the of set vertex the of permutations special are automorphisms Graph h olwn lispoievr ipetost bev o uoopim c ngraphs. on act automorphisms how observe to tools simple very provide claims following The • • G G G G G ¯ φ φ eagah h e falo t uoopim om ru ihcmoiin ednt this denote We composition. with group a forms automorphisms its of all of set The graph. a be eagraph, a be that say We graph. a be eagraph, a be = ) sabjcino h e fvertices, of set the on bijection a is epce egbuho:if neighbourhood: respectes Aut Aut ( ( G G ) ) n ali h ulatmrhs ru of group automorphism full the it call and , . u u , , v v ∈ ∈ V V ( ( G G ) ) and and φ : G φ φ → ∈ ∈ u Aut Aut ∼ G v sa uoopimof automorphism an is ( ( then G G ) ) Then . uhthat such φ 9 ( u ) d ∼ ( u φ u , φ v ( = ) v = ) G . . v d φ ( G u Then . φ if , v φ ) d . ( u = ) d ( v ) . CEU eTD Collection hti detastv,btntvre-rniie h oka rp sabpriegaho 0vertices. 20 on graph bipartite a is graph Folkman The vertex-transitive. not but transitive, edge is that Proof vertex-transitivity. imply not does Edge-transitivity 3.8.1: Note transitively vertices neighbouring its of vertices any for terms, other In transitively. Let graph): (Vertex-transitive 3.6 Definition Let graph): (Arc-transitive 3.8 Definition vertices that neighbouring such of pairs edges. unordered the two on any for transitively that acts means group This automorphism its if edge-transitive, called is graph A graph): (Edge-transitive 3.7 Definition transitive of classes various of properties elementary following most graphs, and definitons the with deals section This graphs Transitive 3.1 y σ 1 ree ar fvrie r oeie aldac,smtmsflg,smtmsdrce ie nteliterature. the in lines directed sometimes flags, sometimes arcs, called sometimes are vertices of pairs Ordered G G = ( ( oneeapei h oka rp sefiue31,wihi tesals)rglrgraph regular smallest) (the is which 3.1), figure (see graph Folkman the is counterexample A : u V V . , , E E [ ) ) x eagah esyta ti etxtastv,i t uoopimgopat nisvertices its on acts group automorphism its if vertex-transitive, is it that say We graph. a be eagah esyta ti r-rniie fisatmrhs ru cso ree pairs ordered on acts group automorphism its if arc-transitive, is it that say We graph. a be , y ] σ [ [ = 10 ] u . , v ] h oainhr en hteither that means here notation The . x , y iue31 h oka graph Folkman The 3.1: Figure hr exists there 1 . σ 10 ∈ Aut ( G ) uhthat such x σ = ( x x u σ , y and = ) , ( y u . y , σ v ) = hr is there v or , x σ σ ∈ = Aut v and ( G ƒ ) CEU eTD Collection Proof even. be must Let 3.11: Claim Let 3.10: Claim Proof orbits. the inside goes edge no and orbits, two exactly has Let coincidence. a not 3.9: is Claim bipartite is counterexample the that fact The h olwn li sevident. is claim following The • • • • • • • G G G : : ea deadvre-rniiegahwihi o r-rniie hntevlnyo h graph the of valency the Then arc-transitive. not is which graph vertex-transitive and edge an be n ika edge an pick and daetpisimdaeysosta hr r xcl w orbits. two exactly are there that shows immediately pairs adjacent Consequently [ maps Since neighbour a Pick Let are classes colour two the action. and group bipartite, the is of vertex-orbits graph two the the Hence be impossible. can is edge which this orbits, transitivity, two to edge By vertex-orbit. one to in mapped contained is edge an that Suppose edge an orbits. 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( Ω G ƒ 1 ) , CEU eTD Collection Proof transitive. vertex are graphs Cayley 3.15: Theorem rp o which for graph Let 3.14: Definition arcs the taking by arises edness Let 3.13: Definition graphs Cayley 3.2 vertices. non-adjacent of pairs graph a that say We (Nonedge-transitivity): 3.12 Definition graph hr r ayohrntoso rniiiy ecnenorevswt n more. one with ourselves concern we transitivity, of notions other many are There oeta if that Note • • • • • • Γ Γ : eagroup, a be eagopand group a be ris Consequently orbits. Since ih utpiainb any by multiplication Right of set edge the that means all This rp,ado oretesbru sioopi to Cayley isomorphic the is of subgroup group the automorphism course full of the and of graph, subgroup a form this like given permutations The Let E x x Cay ( ∼ utb h ae ec h aec of valency the hence same, the be must Cay V ( y Γ ( ( G Cay , fadol if only and if Γ C , sntarc-transitive, not is ) C C ( sdfie sfollows: as defined is x )) Γ C sivrecoe,te h bandgahi aual nietd tews direct- otherwise undirected, naturally is graph obtained the then closed, inverse is − , 1 = C . ea nes lsdsbe of subset closed inverse an be sx G )= )) { S gh snndetastv fisatmrhs ru cstastvl nunordered on transitively acts group automorphism its if nonedge-transitive is ∈ ea nes-lsdgnrtn e of set generating inverse-closed an be S Γ : g x and , hg o any for Ω ∼ sntsymmetric. not is − 1 ( g y ∈ g g , . C x h ( ∈ } ) ∈ y . where Γ , Γ x sa automorphism: an is , ) G s ∈ / ∈ a ewitnas written be can hg Ω S . sohrieuigvre-rniiiy ecudunite could we vertex-transitivity, using otherwise as − x 1 12 utb even. be must ∈ Γ C o otiigteiett lmn.TeCayley- The element. identity the containing not . Γ . Ω ( g y ∪ Γ omlCye rp saCayley a is graph Cayley normal A . Ω )( T g x nbt ris h u ereof degree out the orbits, both In . ) − 1 = g g y − 1 x − 1 = x y − 1 hence ƒ CEU eTD Collection Proof subset. closed group a If 3.19: Lemma Proof Cayley-graph the of group Let semi- and transitive is action 3.18: this Lemma course Of multiplication. right by regular. regularly itself on acts group A 3.17.1: Note then Let 3.17: Definition size have orbits the all lemma, orbit-stabilizer the By 3.16.1: Note identity. the for except points Let 3.16: Definition Γ • • • • • • Γ Γ | eagroup, a be : 3.15. of proof the by proved is This : Γ eagroup, a be fw dniyeeyvertex every identify we If Hence If vertex other Let aua hiefor choice natural A any For h ugopat rniieyo h alygraph. Cayley the on transitively acts subgroup the eagroup, a be C | . x = , u Γ | y X ∈ csrglryo h etcso graph a of vertices the on regularly acts ∈ | x . V g V ∼ ( , G h ( C G y ) osdrteatmrhs rae by created automorphism the consider X X Since . v ea nes-lsdsbe of subset inverse-closed an be ) fadol if only and if then , . y y Γ Γ fteato ssm-eua n rniiete ecl trglr Clearly regular. it call we then transitive and semi-regular is action the If . esyta h cini eirglri h lmnsof elements the if semi-regular is action the that say We . Γ C x Cay csrglryon regularly acts ∼ ste“egbu maps”: “neighbour the is ( y Γ , fadol if only and if g x C uy ihtegopelement group the with ) g otisasbru smrhcto isomorphic subgroup a contains ux − 1 ∈ C . G g x ehv unique a have we , Γ ux − 13 o otiigteiett.Te h automorphism the Then identity. the containing not 1 G ∼ then , | Γ C g y | . = . ux − x 1 { G 7→ Hence . g g ux saCye rp for graph Cayley a is uv g x hnw e alygahof graph Cayley a get we then , : v − 1 ∼ φ h u uv lal hsmaps this Clearly . u ∼ } uhthat such . u Γ ( . g uy g ux − 1 u ) Γ . smpe osome to mapped is ihsm inverse some with Γ aen fixed no have g to h Hence . Γ with ƒ ƒ ƒ CEU eTD Collection Proof If two. 1964 those (Sabidussi, of 3.21 either Theorem for graph Cayley a be cannot graph elements. exist, Petersen 10 groups have such to two has only group theorems, corresponding Sylow the the hence By vertices, 10 on graph a is graph Petersen The Proof graphs. Cayley not are which graphs transitive vertex are There 3.20: Claim G 2 utpei en ntesneo .8 a 2.18: of sense the in meant is Multiple • • • • • • • • • savre-rniiegahte hr ssm multiple some is there then graph vertex-transitive a is ecntutaCye rp rmteatmrhs ru of group automorphism the from graph Cayley a construct We : counterexamples. small two are dodecahedron the and graph Petersen The : hsCay Thus in neighbours were vertices corresponding the if only elements two that way a such tblzrof stabilizer of transitivity By of definition By σ Let Define Let in contained S vertex a Fix 2 − sivrecoe:let closed: inverse is 1 σ T σ x 1 1 = ∈ S ∈ . : { T S ( = σ Γ x fadol if only and if 1 , { v ∈ S v and S σ ) n let and o any for Γ steeaen op.Cneunl Cay Consequently loops. no are there as = ∼ ∈ T : G σ Γ x G v , all , σ 2 [ : v K ∈ − v ¯ σ n Γ 1 ∼ [ y ] T 1 − T ∼ v = 1 x . with ∈ , y v v 2 ∼ v nteCayley-graph the In . x σ Aut esaennmt n aetesm ie as size, same the have and nonempty are sets ∼ V σ } } [ v h e fatmrhs hs nes maps inverse whose automorphism of set the , ( ] h e fatmrhssta move that automorphisms of set the , 29 v σ G n ( ea de hnapplying then edge, an be σ k σ G 2 − ) mlil of -multiple = ] 2 − Thus . 1 1 ) ): 1 . | and od fadol if only and if holds σ Stab 1 hc stecs fadol if only and if case the is which , σ ( V v 2 ( ) G eogn odfeetpriin r egbusi and if neighbours are partitions different to belonging Cay | . is 14 G ( C [ Γ K 10 ¯ σ , k S ] 1 . and )) ∼ a epriindit eso aesz in size same of sets into partitioned be can 2 σ x G D 1 ( of 2 . Γ 5 σ ∼ fadol if only and if ti tagtowr ocekta the that check to straightforward is It . , G − S x 1 ) hc saCayley-graph. a is which 2 yields sapoel endCayley-graph. defined properly a is . G v v hti h utpeof multiple the is that ∼ oaneighbour. a to v σ σ v 2 − T 1 − σ 1 1 y − − σ 1 ∼ 1 v 1 h dniyi not is identity The . salf oe fthe of coset left a is v to ∈ σ S x 2 − . 1 . . G . ƒ ƒ CEU eTD Collection ubro diin.Ti en hti h opeet applying complement, the in that means This additions. of number that Suppose non-edges). to go edges Proof A 3.24: Lemma symmetry. one least at denoted with graph, graph a of asymmetry created. is of loop degree or The edge double no and connected remains graph the Let 3.23: 3.2). Definition (Fig. graph Frucht’s is graph asymmetric regular smallest The 3.22.1: Example every from different” completely vertex. “look to has graph the property, global a is asymmetry transitivity, Like 3.22.1: Note Let a get to them change 3.22: to Definition have we much symmetry. how some and sources. with are, other graph graphs various such from frequent examples how and describe commentary results added The Erd some of paper with the results follow main We its symmetries. describing no have that graphs with deals section This graphs Asymmetric 3.3 ( G ene h olwn lemmas. following the need We G G = ) ( ymtyo rp sasmer ftecmlmn swl egsg oegsadnon- and edges to go (edges well as complement the of symmetry a is graph a of symmetry A : eagah ecl taymti,if asymmetric, it call We graph. a be V A , E ( G ) ¯ ) eagah nei ftegahi digoeeg,o eoigoeeg nawythat way a in edge one removing or edge, one adding is graph the of edit An graph. a be . G a eeie in edited be can A ( G iue32 rctsgraph Frucht’s 3.2: Figure = ) Aut d + ( G A a ( ) 15 G steps, stetiilgroup. trivial the is ) stemnmlnme feisnee ogta get to needed edits of number minimal the is d en h ubro eein and deletions of number the being d diin and additions a sadRényi and os ˝ eein nthe in deletions a [ the 7 ] , CEU eTD Collection Proof Let 3.26: Theorem authors. original the to due is components. the of group automorphism the of product direct the contains nents Proof Let 3.25: Lemma equality. the get we direction, any in graph done the of complement the get we places, corresponding h olwn sErd is following The • • • • • • G G ( : compo- several with graph a of group automorphism the that fact the from follows claim This : hsi u prah ejs aet n h minimum the it. find on bound to have just graph. we new approach, the our of is automorphism an This is transposition a hence neighbours, common have only that pair any for that Suppose ∆ Let endpoints. their on but Let middle, their on based vertex not share pairs that edge edges the of counts pairs side of hand number the counts side hand right The neighbours: common and degrees between connection following the yields counting Double Let of neighbours that Suppose eagahwt components with graph a be V , j j E ∆ δ d = ) jk jk easml,cnetdgahon graph connected simple, a be jk 0. etenme fcmo egbusof neighbours common of number the be eteidctrfnto of function indicator the be = . d j + n d v k k ≥ . − sadRnisfis hoe ocrigaymti rps h ro recite I proof The graphs. asymmetric concerning theorem first Rényi’s and os ˝ .Lttevrie ftegahbe graph the of vertices the Let 6. 2 d jk − 2 v δ j , jk v G k h ubro o-omnnihor of neighbours not-common of number the , i edelete we , i = X i = [ n ,..., . . . 1, 1 v A j X ( j , n = G n v 1 k etcs Then vertices. ) ∆ ] d ≤ k j i ∈ jk 16 Then . • = E de rmtegah hnw raetovertices two create we Then graph. the from edges n ( X k G − 2 = n v ) 1 1 . j d A ˜ and k ( ( G G v d 1 ) k , . . . , a dtdt.Sneti rcs a be can process this Since to. edited was v ≤ − ∆ k edefine We . 1 min jk ) v (3.1) . n oeta egtanc upper nice a get we that hope and n and , i ∈{ 1,..., k d } k d A = ( j j . G v = deg k i ) k and . 0. o each for v k v h ubrof number the , j gi define Again . k h left the , ƒ ƒ CEU eTD Collection of on graph random a by text, the Throughout graph): (Random 3.27 Definition in exploited rarely be can that structure of type local most proofs. Trans- the or graph. applications imply the hand in structure other large the a imply on to positions tend vertices fixed few neighbourhoods. fairly same have the that sharing Automorphisms vertices two create least to the needed finding into edits editing of by amount automorphism an creating of problem editing the the reduces effectively same that proof the The shows sharing proof vertices The two of be. transposition a to neighbourhood. automorphism, seems of first type it possible as simplest the enlightening creates as not is sadly theorem The 3.26.1: Note n h olwn savr motn hoe:aymti rpsaeubiquituous. are graphs asymmetric theorem: important very a is following The • • • • etcs ecie ymtal needn events independent mutually by described vertices, Since of definition the from deduced be can This ic h iiu lmn fasto ubr sawy tms sbga h average. the as big as that most have at We always is numbers of set a of element minimum the since ecngtteuprbound upper the get can we have clearly We lgigtepeiu nqaiisit h nqaiy32 eaedone. are we 3.2, inequality the into inequalities previous the Plugging 2 2 X X k k v = = n n k 1 1 A ( v v n ( k k G − ( ( X i ) n n = n 1 ≤ 1 − − − X j 1 1 j = n 6= v 1 − − k n k min ∆ ∈{ = ) etcsw enapoaiiysaeo h e fgraphs of set the on space probability a mean we vertices v v 1,..., j i k k ) ) =  ≤ ≤ ∆ n 17 } n 2 ∆ jk n n − 2 X k € ( = n jk n qaiy3.1. equality and n 1 ( 1 n ≤ − 2 ‹ v ([ − k 2 1 ( P i − 2 n 1 ) , 2 ) j i n −  = ] 2 n if v 1 − ∈ ( 1 k n n P − 1 − E − = Š n ( j v = n G if , 1 k 2 1 ) ) )) − 2 t ∆ . ahhvn probability having each , + n 1 j i ‹ = 1, (3.2) , 2 2 , t . 1 2 . ƒ CEU eTD Collection Proof lim in edge-edited be can graph random Let (Erd 3.28 Theorem n • • • Γ →∞ : h oa ubro ylsi h yl representation. cycle the in cycles of number total the is has probability graph this random a that probability the calculate We from S are graphs random the of automorphisms Possible parameter with distributed where ahedge each by graph random the of vertices the denote us Let rbblt faygahapaigon appearing graph any of probability earno rp on graph random a be n P – – – – nterccerpeetto ihccelengths cycle with representation cycle their in n ( hr eangcd remain there lcm the after repeated that Suppose ups that Suppose of choice the after choose to free still if are Clearly edges many how know to need We fteeare there If made. constraints: following the yields edge one Iterating, non-edges. to map ε = ) c a steccelnt fcycle of length cycle the is 0 [ . v j , s éy (1963)): Rényi os, ˝ v π k ] r sa uoopimthen automorphism an is edefine We . ijitcce nteccerpeetto,te hr are there then representation, cycle the in cycles disjoint j j and eog ocycle to belongs ( c n a , k c etcs Let vertices. b ohbln osm cycle some to belong both ( p P ) c a = ( rechoices. free , ξ G ξ c j 2 1 π b j , n , k ) . k ( 1 t em ic hr are there Since term. -th = 2 = = − 1 ε a G ≤ ) ξ ξ X a = ) a nteccerpeetto ftepermutation the of representation cycle the in k tp oyedannrva uoopim hnw have we then automorphism, nontrivial a yield to steps j < n π , > ε j b , and k etcsi 2 is vertices ≤ and π 2 r 18 = gcd ξ P 0 k j 1 π ξ ≤ ξ a , eabtay If arbitrary. be ( eog ocycle to belongs k < j j π c , π b j a ≤ 2 , , r = c = k gcd c 1 π v b , . . . , 2 2 1 .Hnefrevery for Hence 0. ξ − + ) ( , . . . , ( = ( c a j π 2 n a 2 , n ) k , ) hnteeare there Then . c ξ X Let . a b sa uoopim ttrsotthat out turns It automorphism. an as c S segsmpt de n non-edges and edges to map edges as )+ = r j k π n v 1 Let . c P ow osdrpruain from permutations consider we so , 3 n ” a , nti admgahmdl the model, graph random this In . k a r c = π · 2 ξ a 1 c 3 P [ j b — = π , c n 2 k a b hie loehr egtthat get we altogether, choices ( ] l fteecntansare constraints these of All . ε ∈ eteidctn aibefor variable indicating the be . . . . , ) S stepoaiiyta the that probability the is n . ” j c 6= 2 a — k recocst be to choices free , ξ j , k sBernoulli is π π and and ξ r j , k is . CEU eTD Collection eoeby Denote of function in proceed now We n ( 1 2 − • • • • • ε ) htcnb dtdit rp having graph a into edited be can that eceryhave clearly We most that at graphs in of edited number be the can that obtain we formula, Stirling’s and approximations standard Using on inequality the using gcd Since define can with we commute these so and cycles, 1-cycle, other a the to all corresponds point fixed every decomposition, cycle the In Let any for probability the have We yl vrindex over cycle m G de noagahamtiga automorphism an admitting graph a into edges has tp into steps q P admgah have graphs random hence made, be to choices free etenme fpit which points of number the be n π ( ε sa uoopim eaeitrse ntenme fgah htcnb dtdin edited be can that graphs of number the in interested are We automorphism. an as , ( q c a ) , G h rbblt htarno rp forder of graph random a that probability the c h de ob dtdcnb hsnin chosen be can edited be to edges The . b ) P ≤ t n sa es flnt .Fo this From 2. length of least at is ( ε c a ) + 2 1 c ≤ ≤ b X a r , < P 1 n gi yields: again ≤ b ( π 1 X ≤ a q n 2 q − < = r h ubro lmnsntfie yaslce permutation selected a by fixed not elements of number the , ε sa automorphism. an as 2 b gcd ) ≤ P tp antexceed cannot steps r π n gcd  ( ( ε c ∈ m a , 2 n , q S (
c π c ‹ ) n b a . c 2 2 htarno rp has graph random a that + ) , 1 osntfi,and fix, not does π P P c = 2 b sa uoopimcno emr than more be cannot automorphism an as 1 1 + ) ≤ ≤ X a t 2 a a · · · = r < − < 19 4 1 n b b 2 ≤ ≤ X a ” r + r = = r gcd gcd c O 2 1 a π ( c ” n — ( ( t c c c log hc oe exactly moves which 2 a a ≤ = a r , , c c — b n b ≤  )+ )+ .Then 1. ) ≤ . 2 t P P n ‹ t +  2 a r a r ( = = t = + m 2 t 2 n 1 1 . ) n [ ‹ [
n c c ( 2 2 a a a eeie ycagn tmost at changing by edited be can + r as ec h ubro graphs of number the hence ways, ] ] − q . − π n q = 2 t sa uoopim Suppose automorphism. an as etenme fpit fixed. points of number the be )( − 4 2 P n t i r 2 + = (3.4) . t + t q ) 1 elements. (3.3) , c i ≤ 2 ( r − t ) seach as , π . CEU eTD Collection • • • • h aenme fsesb nydltn,o eeigadadn ttesm ie Hence, time. same most at the in at transposition adding in and done deleting be or can deleting, this only and by neighbourhood, steps same transpo- of the a number share since same vertices deleting, the as two editing if take possible to only enough is is sition it Also interesting. are vertex non-fixed 2 yields: approximations trivial taking and Summing most at is fixed as n rnpsto s2 is transposition a ing the for bound upper an Thus edited: be endpoint. to vertex edges for non-fixed choices one of vertices number and non-fixed fixed between one edges with automorphism: edges the and by moved are that edges on concentrate when case the estimate now We fix that permutations of number The etr otasoiin ( transpositions to turn We for and increasing, where For f − en enda rvosy ehv httenme fgah htcnb dtdt il a yield to edited be can that graphs of number the that have we previously, as defined being n t + 2 n O = ≥ ( log α n q = n 2 ) ≥ − hnw dttegaht il rnpsto,ol de hthv xcl one exactly have that edges only transposition, a yield to graph the edit we When . 1 2 p − 2 4 q ε n n p and egtthat get we , 4 n q n
+ q q f ! p m ≥ ( α ≤ < ( X n n 5, = ) n − 2 . m ( 1 1 n 2 ) − < α P + ε p q 4 tp is steps ) n α 1 n  ( = ec h rbblt htagahyed rnpsto sa most at is transposition a yields graph a that probability the hence , ε log aan ySeln’ oml) hnw dttegah ecan we graph, the edit we When formula). Sterling’s by (again, , 2 q 10 ) hr are There 2). 1 m q
p 2 4 = ) < hence , + X q X X q n α = 1 n p 4 = ( n q 1 p m ≥ 2 n 4 ( + − 5 ( 2 ε n n t ) P O q P  n 1 lmnsi tmost at is elements ( − n ( p ≥ n 4 ( − ε f q n ε m − , ( .Tenme fpruain leaving permutations of number The 5. log ) α , q α ‹ q 20 2 = ) ) ) ) n ‹ ≤ log )+ 2 n < ≤
= n 2 n rnpstos n h ubro rpsyield- graphs of number the and transpositions, 2 .7 Thus 0.47. 2 2  − − 2 − 1 ( 0.03 4 n n − f 1 − n ( ( 2 q nq α 1 4 5 ) 1 2 2 − − qn 2 . + + ε ε O ) + f ) ( n ( ‹ n O + n − 2 a oiiedrvtv in derivative positive has ( q O log = p ) ( − n log n t n ( 2 )
) . 2 n ( f , ) n ( n + ) α . 0.47 ) − nq t + nq ) O ! . ( = log 2 n ) O , ( n log n ) . n ( − 0, q 1 2 elements ) oi is it so , CEU eTD Collection • • ieo huh sbfr,w a obtain can we before, as thought of line on graphs most at are there If on depending Since -yl,o hyaemvdb w ijittasoiin.W elwt h w ae separately cases two the with deal that We get transpositions. and disjoint two by moved are they or 4-cycle, For sn h aetcnqe sbfr,w a bantenme fgah htcnb dtdinto edited finally: be and can types, that two graphs these of number of the one obtain can we before, as techniques same the Using q q = = ,w a nyhv -ylsa uoopim.Teeae2 are There automorphisms. as 3-cycles have only can we 3, f ( hr r w osbecngrtos ihrtefu o-xdvrie r oe ya by moved are vertices non-fixed four the Either configurations. possible two are there 4 x ) n < etcsyedn xcl w -ylsa natmrhs ru.Uigtesame the Using group. automorphism an as 3-cycles two exactly yielding vertices ε o every for 1 P . ( G ilstotranspositions two yields P ( G x ilsafour-cycle a yields 6= 2 1 2 ,  P 3 n n P ‹ P ( n ε n ( 2 ε ( 2 , ( ε 4 , n − 3 , 2 ) 2 ) ) ≤ = ) + ≤ 21 1 ) ) − 2 2 ( ≤ ≤ − O − 2 n c ) 0.68 (  6 6 ε = )   € n 4 4 n n n p where , 2 + 2 ‹ ‹ − 3 O 2 2 2 ( Š n ( ( log + n n n − − 2 2 ‹ O 3 2 n ) ) ( . ) log + + . c 2 3 ( − − n ε ) ( ( ) 2 2 n n ) ) ssm oiiecntn only constant positive some is = = 3 n
2 2 − − uhatmrhss thus automorphisms, such 3 2 n n + + O O ( ( log log n n ) ) (3.5) , (3.6) . ƒ CEU eTD Collection Let concepts): related and homomorphism (Graph 4.1 Definition Homomorphisms core. 4.1 a is graph every almost and rigid, is graph every out almost turns it frequent: graphs, very asymmetric are to graphs Similarly rigid framework. that our in possible structure least the exhibit that retracts. of can terms graph in the structure” which “irreducible onto an graph represents smallest it the thus is to, graph back a folded of be core be- the and speaking chapter connections loosely the of the core: concept graph key show The the to is colourings. results on results colouring known from the drawn inner of quickly be the some can examples of to cause refer understanding will better We possibly a graphs. of retracts structure through and retracts, graph of understanding research. of force driving important most the probably is out which turn colourings, Homomorphisms of graphs. generalization of the case be in of mappings to such areas with many work in to natural mapping” is “structure-preserving it and a mathematics as meant is “homomorphism” term the Generally structure graph and homomorphism Graph 4 Chapter nteedn ato h hpe,w ocr usle ihrgdgah.Teeaetegraphs the are These graphs. rigid with ourselves concern we chapter, the of part ending the In better gaining of goal the with homomorphisms approach we as different slightly are goals Our • G , ecall We H egah and graphs be φ ooopimfrom homomorphism a φ : V ( G ) → V ( H G ) najcnypeevn apn.Then mapping. preserving adjacency an to H ihu ute restrictions. further without 22 CEU eTD Collection lmnayfcsadtedefinitions. the and facts elementary 4.1.1: Note ietefloigsalcam ihu rosa oto hmaesrih osqecsof consequences straight are them of most as proofs without claims small following the give I • • • • • • • • • • • • phism. call We call We If ecall We Hom from exists between forth and back momorphism [ call We of If from homomorphism a exists there If call We π Let of fibres The ooopimfrom homomorphism homomorphism natural a is there then in loops enable we If Given φ H a ocnanidpnetvrie o uhahmmrhs oexist. to homomorphism a such for vertices independent contain to has f H ( – – u G : . = ( ) w atto elements partition Two Let v G G , eagraph, a be G φ G and , → φ φ φ φ H φ ( n enlpartition kernel a and V and v ) ietv ooopimi ti ooopimadabjcino h etxset. vertex the on bijection a and homomorphism a is it if homomorphism bijective a opeehmmrhs fi sahmmrhs,i sfihu n ti surjective. is it and faithful is it homomorphism, a is it if homomorphism complete a nioopimi ti ietv ooopimadisivrei loahomomor- a also is inverse its and homomorphism bijective a is it if isomorphism an atflhmmrhs fi sahmmrhs and homomorphism a is it if homomorphism faithful a H ( )] . ulhmmrhs fi sahmmrhs and homomorphism a is it if homomorphism full a G w G sahmmrhs,te h preimages the then homomorphism, a is ∈ . ) f φ to etest ntepartition. the in sets the be eemn atto of partition a determine E sahmmrhs,w ali nendomorphism. an it call we homomorphism, a is H ( H hnw write we then , π ) 1 . G , G π /π /π 2 hnvrnihorn etcsapa ntesm e ftepartition, the of set same the in appear vertices neighbouring whenever etopriin ftevre set, vertex the of partitions two be 1 → v , G π w /π ecndfieagraph a define can we , r daeti hr sa degigbtenayeeetof element any between going edge an is there if adjacent are G 2 6→ fw nbeloops. enable we if G G H to G f and π ednt h e fhmmrhssfrom homomorphisms of set the denote We . hc ecl h enlof kernel the call we which H : 23 G ednt hsrlto by relation this denote we , H → ednt hsby this denote we , G /π h mgsbiggvnby given being images the , f − 1 G ( /π y ) π r h be of fibres the are sfollows: as 1 nrthan finer G φ [ u ↔ ( , G v f ) ] H . G sa nue subgraph induced an is ∈ fn homomorphism no If . → E π ( H 2 G f hnteei a is there Then . fteei ho- a is there If . . ) fadol if only and if π Otherwise . G to H by CEU eTD Collection li 4.2: Claim shmmrhcto homomorphic is two between homomorphism from if a phism Clearly of number. existence chromatic possible the the on decide based to graphs ability the us gives claim last The 4.2.1: Note Let • • • • • • • • • • n nyif only and mapping A well. as transitive The ewe h sets the between Suppose monoid. a form Endomorphisms map can homomorphism no inverse into tree the spanning that Consider out turn homomorphism. could a It not isomorphism. graph is a be not may claim). homomorphism small bijective previous A the of consequence the is (this one larger a to cycle mapping a graph. Similarly a in vertices two of distance the increase never homomorphisms Consequently otayt rus hr ih o eahmmrhs rmagaht nte.Teei no is There another. to graph a from homomorphism from a homomorphism be not might there groups, to Contrary in walk closed a is image the if osqetyif Consequently is graph A G , H → G etogah.Let graphs. two be eaini rniie if transitive: is relation to φ k H f safihu ooopim Pick homomorphism. faithful a is clual fadol fteei ooopimfo tto it from homomorphism a is there if only and if -colourable f ( . H x rma from 0 h ro ftefloigi u oHl n Nesetril and Hell to due is following the of proof The . ) G φ f → K ( − x n 1 1 f ( H stiilyahmmrhs,adi sbjcieo h etxst u clearly but set, vertex the on bijective is it and homomorphism, a trivially is k ) K u rma from ln path -long . . . then , ) n and oiscmlmn.As,teei ohmmrhs rmasalodd small a from homomorphism no is there Also, complement. its to m f ( ( x G K φ χ k k , n ) G ln cycle -long H ( − G oate for tree a to G sawl in walk a is 1 ) . → ( ) P v etesz favre-aia nue ugahof subgraph induced vertex-maximal a of size the be ≤ k ) H . K = χ n and ( x n n fissann re.Teebdigo the of embedding The trees. spanning its of one and H 0 ) e C . H 0 24 k x G n χ oaohrgraph another to → 1 . > ( . . . u G K 2 ∼ ) x . then χ > k v oaohrgraph another to ∈ ( G H H → ) hnteems ea degoing edge an be must there Then . hnteei opsil homomor- possible no is there then K osqetythe Consequently . G sahmmrhs fadonly and if homomorphism a is [ G 18 sahmmrhs if homomorphism a is K ] . k . ↔ eainis relation G which CEU eTD Collection Proof Let 4.3: Theorem Let (1985) Collins Albertson, lemma, (No-homomorphism 4.5 Theorem where h aecmsfo h eain if negation: the from comes name The 4.5.1: Note ro ftelmai aial h aea h ro ftemr eea theorem. general more the of proof the as same the basically is lemma the of proof take we If 4.5.2: Note h neednertoo graph a of ratio independence The 4.4: Definition Lemma’. Homomorphism h rvostermi h oegnrlvrino hti nw nteltrtr ste’No- the as literature the in known is what of version general more the is theorem previous The • • • • • • G G , , : to Suppose Since ytastvt each transitivity By homomorphism a Fix Hence to homomorphic subgraphs Hence H H α , ( K egraph, be K G . K ) egraphs, be H steidpnec ubr h ieo h ags needn e in set independent largest the of size the number, independence the is qm qm ob h n on rp,telmai utaseilcs fTerm43 h original The 4.3. Theorem of case special a just is lemma the graph, point one the be to svre rniie vr etxof vertex every transitive, vertex is H ( ( 1 H G , H , , H K K 2 ) , . . . , = ) etxtastv and transitive vertex H ≥ svre-rniie If vertex-transitive. is p p G H | | G H i q f shmmrhcto homomorphic is | | r l h nue ugah of subgraphs induced the all are obnn hswith this Combining . . : G → K eoeti number this Denote . H n let and i ( m H ( ) G | i G G ( > i → , G ( | K G G i = ) G ( ) H i ) K G = → ≥ ≥ 25 Then . . ) n ahvre of vertex each and H | hnteei ohmmrhs from homomorphism no is there then V α f i qm m H ( − ( ( eog xcl otesm ubro induced of number same the to exactly belongs H ( G then , G 1 | H ( ( H ) ) ) H V . | , | , K , p ( H K . ) = ) i . )) G . fsize of p | H | h li follows. claim the , G m eog to belongs ( [ H 1 ] , ): K ) h r homomorphic are who p G . subgraphs G to H G . i . ƒ CEU eTD Collection hr sn ooopimfo h mle otelarger. the to smaller the from homomorphism no is there then if that fact the Using 4.7.2: Note map- identity an be to co-retraction into the ping fixing and endomorphisms with retracts define authors Some 4.7.1: Note if Let foldings. simple of (Retract): composition 4.7 a Definition is it folding. if simple folding a a homomorphism is this homomorphism call A we others, all fixing while other each from Let (Folding): 4.6 Definition 4.21.1). Corollary. more (c.f. even graphs another, transitive to of graph the case one cases, special from the exist special in can for homomorphism so However, no that verify graphs. to transitive graph, easy it for a makes even lemma of problem, number NP-hard independence an the generally of is which computation the requires lemma no-homomorphism The 4.5.4: Note have would we from homomorphism a is there if Suppose from ones. homomorphism larger no to is there cycles that odd fact smaller the is lemma no-homomorphism the of consequence easy An 4.5.3: Note ΨΦ h olwn li n t ro stknfo osladRyesbook Royle’s and Godsil from taken is proof its and claim following The homomorphisms. special define now We • • G G χ , : Ψ Φ eagah fteei homomorphism a is there If graph. a be H ( H G : : etogah.W a that say We graphs. two be G = ) → H G hntertato sahmmrhs hc a oata h dniyon identity the as act to has which homomorphism a is retraction the Then . → → H χ H G steiett apn of mapping identity the is ( H 2 scle retraction, a called is k scle co-retraction, a called is k ) + h ees sovosyfle n w d ylshv hoai ubr3 but 3, number chromatic have cycles odd two any false: obviously is reverse The . 1 ≥ G 2 → n n + 1 H rmwhich from then , C χ 2 k H ( + G 1 sartatof retract a is ) α k to 2 H ≤ ( ≥ C k . k C 2 χ + n k 2 < ( + n contradiction. a , 1 H + 1 n 1 ) φ ) Since . n take and 26 ≥ eimdaeygtta if that get immediately we htietfiseatytovrie tdsac two distance at vertices two exactly identifies that α G 2 ( C n fteeeittohomomorphisms: two exist there if 2 + α n + ( 1 1 C ) 2 , k + 1 = ) k ytkn vr eodvertex, second every taking by [ 10 G ] a ertatdto retracted be can . 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A classes transitive setup. all this not in and classes, role equivalent special homomorphically a have not do graphs Cayley The Cayley a 4.14.1: contains Note graphs of class equivalent homomorphically a graph. every to words, equivalent other homomorphically is In graph transitive graph. every Cayley that shows 4.13 Claim with 3.21 Theorem 4.14: Claim Let 4.13: Claim Let 4.12: Definition 2 1 nteeryltrtr,crswr locle iia rps sin as graphs, minimal called also were cores literature, early the In 4.21. Theorem See • • G G G G G eagah h graph The graph. a be graph, a be of class equivalent homomorphically the denote We graph. a be ? ? surtatv:n ercineit nit. on exists retraction no unretractive: is of retract a is H eamlil of multiple a be G , G ? scle core a called is G Then . G 2 ↔ ftegahof graph the of 28 H . [ 31 ] and G 1 [ 9 aeamnmleeeta Cayley a as element minimal a have ] if . G with H ( G ) . 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G , : e ieta h te.Ti sacnrdcin oalcrsi ls utb isomorphic. be must class a in cores all so vertex contradiction, bigger a has is them This of other. one the hence than graphs, size two set the Then between isomorphic. retraction be proper would a graphs be the must there surjective, were homomorphisms forth and back both If equal, isomorphic. are are sets cores vertex the the thus and of surjective sizes are the homomorphisms Therefore the retraction. a be cannot homomorphisms forth violated. clearly be would property this subgraph, isomorphic. are cores two of restriction the inclusion. to respect H G H and , G G ↔ egah.Tefloigpoete hold: properties following The graphs. be K ↔ sacr rp hni steuiu rp fsals re pt smrimin isomorpism to up order smallest of graph unique the is it then graph core a is H = G . 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CEU eTD Collection n nsrgraph Kneser the Any exploits it because interesting ( is 4.18 proof Claim The class. the graphs. define that core properties transitive underlying of set large a provides claim Let 4.17.1: Corollary Let (1982), (Fellner 4.17 Claim Proof α 4.19: Claim Proof ( K Let 4. Ceryayodccehscrmtcnme n otisn nue ugah fchromatic of subgraphs induced no contains and 3 number chromatic has cycle odd any Clearly 1. or graph point one the either is graph bipartite a of core The 5. I graph a If 4. number. chromatic smaller Let of and 3. complete are subgraphs induced all graph, complete a Given 2. eeal,oehst okhr oso htagaho aiyo rpsi oe h following The core. is graphs of family or graph a that show to hard work to has one Generally, ehv eyfwncsaycniin ncrs h olwn sdet Fellner to due is following the cores, on conditions necessary few very have We is eoti h neednenmeso h nsrfamily. 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K ? . ) 2 n | n hneither Then . ) L − ( eoeisln graph line its denote .A decluigof colouring edge An 1. K L 2 ( G n ) ) n w etcsof vertices two and → K 2 n K n ene ofind to need we , − 2 1 G n { n ic any since and V oncstwo connects = ∼ 1 , . . . , X 1 ƒ 4 L V ( K Since m G ¯ 2 ) } K are or of 2 ƒ n CEU eTD Collection Proof core Let 4.27: Lemma ro fTerm4.25 Theorem of Proof Proof Let vertex. Let theorem. 4.26: the Lemma prove to lemmas two need will We • • • • • • • • • G G K : : = qiaety hr exist there Equivalently, oethat Note automorphism. Let pair a is there that suppose contradiction By contradiction. a graph, core a of retract a obtained ic h mg of image the Since and et,gvn h eurdpartitioning. required the giving ments, as aetoelements two Take rmtepeiu em,tkn h sets the taking lemma, previous the From | of fibres All Let that contradiction by Suppose eagraph, a be If . 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CEU eTD Collection Let egbus n vr w oajcn etcshave vertices example nonadjacent for two every and neighbours, the of cores two of the product and Cartesian product a of Cartesian core a the is graphs, vertex-transitive it of know case we the in if Even graph known. a rarely are of can factors core we the notions: over compatible control not is are any section retractions gain this and of product point Cartesian main the The that interchangeably. demonstrate products to Box and Cartesian use I subsection this In product Cartesian The 4.4.1 graph understand to try mainly graphs. will core of we terms hence in structure, products graph understand to is however focus main on relies chapter This structure Product 4.4 (2008) Kazanidis (Cameron, 2 Conjecture Let (2008) Kazanidis (Cameron, 4.28 Theorem by described form block a into decomposable is graph a Either the 1: together believe Speculation to This, me leads cores. counterexample as the of graphs following. property complete “core-density” have the on but remark don’t previous who a with those and structure, block disjoint a 5 togyrglrgahi eua rp ecie ytomr aaees vr w daetvrie have vertices adjacent two every parameters: more two by described graph regular a is graph regular strongly A hr r ayrslst hs rmrltdt rp ooopim n rp rdcs Our products. graph and homomorphisms graph to related from chose to results many are There conjecture: following the gave Kazanidis and Cameron yield which those yet: known are graphs vertex-transitive of kinds two only that notes Roberson tsesta opeecrscudhv pca oe hsi nelndb h olwn claim. following the by underlined is this role, special a have could cores complete that seems It • G G eanndetastv rp.Te ihrtecr of core the either Then graph. nonedge-transitive a be Since a easrnl eua graph regular strongly a be ∈ φ C 4 − ƒ X 1 P ( 2 sanra alygah ettasain r uoopim swl,hnefrany for hence well, as automorphisms are translations left graph, Cayley normal a is v te3dmninlcb)i rniiebtntsrnl regular. strongly not but transitive is cube) 3-dimensional (the ) , a − 1 V [ 15 ( G ] ? , ) [ nue nioopi mg ftecore. the of image isomorphic an induces 3 ] , [ 5 14 hnete h oeof core the either Then . ] , [ 23 ] . µ [ omnnihor.Srnl eua rpscnb smercand asymmetric be can graphs regular Strongly neighbours. common 4 [ 4 ] ): ] 36 ): G G scmlt,or complete, is sacmlt rp or graph complete a is (4.1) riscr scomplete. is core its or , G sacore. a is G tefi core. a is itself λ common ƒ CEU eTD Collection esyta oegraph some that say We 4.31: Definition Proof hom-idempotent. are graphs Complete problem. 4.32: the Claim of characteristics some show might separately them Let 4.30: Lemma whether interested are we Consequently, Let 4.29: Claim to surprising. be might cores transitive G h olwn he lisaees osqecso hoe .0 eetees investigating nevertheless, 4.40, Theorem of consequences easy are claims three following The h ais aet oto swe etk graph a take we when is control to case easiest The • • • • • • • G G ntrso retractions. of terms in , : ecamthat claim graph. We the “twisting” while done is collapsing the neighbours to map bours the to graph the “collapses” map The modulo is fal o every for all, of First map the Define coordinates stetre rp sa is graph target the as Clearly in that Note manner. sistent Label vertex. same the of copies graph The and H egah.Then graphs. be H K etogah.Then graphs. two be ( n n k ihnmes1 , . . . 1, numbers with . , s K ) ( n ∼ ,. 1, ƒ f K ( t ) f k n t ,adteefr a form these and ), sahomomorphism. a is , : stastv yCam29 ob ymtyw nycnenorevswith ourselves concern only we symmetry by so 2.9, Claim by transitive is G r V ) G hs e apdto mapped get These . shmieptn if hom-idempotent is ( ( t → K , K n s ƒ ) G n , . K ƒ s n χ H = ) n ( → ,..., . . . 1, K and G n ups that suppose and , n ƒ V ƒ H ( K H G K = ) n K n ƒ n hr r exactly are there , → t , n ) G t ae ntefis oriaeadt nueta neigh- that ensure to and coordinate first the in layer -th f as swell. as max G → csa h identity. the as acts ƒ G f ( 37 t G H t ƒ { ( , χ k . G s holds. , +( ( l G → ( = ) G ) k K , n netgt o h product the how investigate and G − χ n . ƒ t ( t )) H , K l ) n and ( + } n slbldwt ree ar nacon- a in pairs ordered with labeled is . k oiso h etx1(aeythe (namely 1 vertex the of copies ( − t , t r )) +( hr diini understood is addition where k − t )) biul neighbours obviously , G ƒ G relates n CEU eTD Collection Proof Let Let li .6(an el ojk(1995) Poljak Hell, (Hahn, 4.36 again. Claim role important an have might graphs Cayley (1985), Collins (Albertson, 4.35 Theorem true. not is of this subgraphs even (proper) when two of product Cartesian a is core h eesngahi o o-dmoet nfact in hom-idempotent, not is graph Petersen The (1985), Collins (Albertson, 4.34 Claim hom-idempotent. are cycles Odd 4.33: Claim Also immediately. 4.30 Lemma from follows claim The being claim. edges following 4.32.2: of the Note image by the underlined about is worry remark to simple have This also graph. edges. we a as colour complicated are we more we how is that, exactly do retracts is can other this we Checking since If surprising vertex. very same not the is to This mapped done. get basically neighbours two no to that retracted way a be in can vertices graph the a whether checking when idea main The 4.32.1: Note G y x ƒ G Cay enwtr otegnrlqeto fhmieptny h olwn eut it htnormal that hints results following The hom-idempotency. of question general the to turn now We h rvosrsl a egnrlzdfraypwro n nsrgraphs. Kneser any of power any for generalized be can result previous The h he rvoscsswudsgetta if that suggest would cases previous three The • • • sahmmrhs,hence homomorphism, a is s > sisl oe ttrsotta h iuto smc oecmlctd tcudhpe htthe that happen could it complicated: more much is situation the that out turns It core. a itself is : h rp sanra alygah tmasthat means it graph, Cayley normal a is graph the auto- φ are translations right the and left the them. of both morphisms that property the have graphs Cayley Normal Take ( r Γ csa ih rnlto ntelf atrada ettasaino h ih atr Since factor. right the on translation left a as and factor left the on translation right a as acts etopstv integeres. positive two be , S ( ) l , eanra alygah Then graph. Cayley normal a be s ) ∼ ( k , s ) h mg fteeis these of image The . Cay ( ƒ Γ , s i = S 1 ) K [ shom-idempotent. is [ ( 12 1 m ] ): ] [ , φ ): 1 n ( ] t ) Cay : ): , → s 38 +( K G ƒ ( l ( i r Γ − ,2 5, = sacr,te either then core, a is , 1 K t S K φ )) ) n ) ( G ƒ ƒ ƒ , m sahmmrhs.Tecamfollows. claim The homomorphism. a is o icnetdgah,teeaecases are there graphs, disconnected For . ( Cay K K t , m ( , n ,2 5, s ) → ( +( . Γ , ) K k S score. is − n ) K if → t n )) n st e hte ecnmap can we whether see to is gi,teeaeneighbours. are these Again, . Cay > m G ( Γ ƒ . , G S ) has endby defined G siscr,or core, its as φ ( x , y = ) ƒ CEU eTD Collection Let graph): (Homomorphism 4.37 Definition oolr 4.38.1: Corollary Let 4.38: Claim rp shmieptn fadol fi shmmrhclyeuvln oanra Cayley-graph. normal a to equivalent homomorphically is it if only and if hom-idempotent is graph A (1995) Poljak Hell, (Hahn, 4.40 Theorem Let 4.39: Claim Proof K shmieptn fadol if only and if hom-idempotent is Let 2. Let 1. h rvoscam oehrpoetemi euthere. result main the prove together claims previous The • • • • • G G K , , : φψφ [ φ φ ene oso that show to need We of elements the Clearly, φ hc ersnsHom represents which If orn pair: bouring automorphisms, Two V eacr.Then core. a be H H u ( u u K φ , ∼ Hom ( ( etogah.Tehmmrhs graph homomorphism The graphs. two be K ψ v v φ , sacr,then core, a is ψ ) ) u egraphs. be − : : φσ = sahmmrhs where homomorphism a is 1 . fadol ffrevery for if only and if G ( G ∈ G φ ƒ → ] , S ( H H u for sw needed. we as Hom , )= )) → v [ u ) σ φ Hom K sahomomorphism. a is ( ∈ { , H φ eahmmrhs.Te h map the Then homomorphism. a be u V S ψ , ( : K ( and φ Hom ] K G ( ) φψφ K ∈ , , S eahmmrhs.Te h map the Then homomorphism. a be → ψ K , S E φ r h hfsof shifts the are K ) ( = ( r daeti n nyi hymptesm etxt w neigh- a two to vertex same the map they if only and if adjacent are H K . ) sanra Cayley-graph. normal a is K ∈ − = ∼ , , K { ) 1 φ u K σ Consequently . Aut ↔ Cay ∈ )= )) ∈ ∈ ∈ φ V ( S Hom Aut Hom K ( u ( Aut opoeta h alygahi oml esrl have surely We normal. is graph Cayley the that prove to Aut ) G = ec o any for Hence . ( [ ) K ψ 12 ( ( ( φ ( G K K ) K ( ( ] , u K ) : , ) u ): H , ) K 39 . [ . ) S . u ) ) ∼ ) , } Hom . [ : u , u ψ σ , u ( ] ψφ u ( ∈ G ) . Aut , u − H 1 ψ ∈ ) ] ( sdfie sfollows: as defined is K ∈ V : ) G ( E , φ K ( ∀ → ) K : u , ) G [ ∈ Hom ow a n alygraph Cayley a find can we so , u ƒ , K H u } φσφ . ( → H , K K − endby defined ) 1 ] endby defined ∈ E ( K ) n thus and , φ ( ψ u ( , u v = ) = ) ƒ ƒ CEU eTD Collection Let concept. 4.43: core Claim the with well meshes product lexicographic The product lexicographic The 4.4.4 Let 4.42: Theorem that Let products. lexicographic on Klavžar): results the (Imrich, to 4.41 contrasted Theorem if interesting are statements two following The product strong The 4.4.3 he rpsta r omdb nodcceadoemdl on r oe,a hycnb repre- be can they as cores, are point middle one and cycle as odd sented an by formed are that graphs Wheel 4.45.2: Corollary Let 4.45.1: Corollary Let (1982), (Fellner 4.45 Claim join Graph 4.4.5 that Suppose 4.44: Claim product. Cartesian the for result the to that note to important is It 4.43.1: Note Let conjecture): (Hedetniemi’s 3 product Conjecture direct The 4.4.2 H G ? 0 [ ≤ H G G G G G G R , , , , ? H , eagraph. a be ] H H H H = ∼ H . uhthat such ecnetdgraphs, connected be ecnetdgah.Te h oeof core the Then graphs. connected be egraphs. be Then graphs. two be G ecnetd ragefe graphs, free triangle connected, be C 0 2 ‚ k + G H 1 and 0 + . R K G 1 = ∼ H G . scr fadol if only and if core is , r ohcnetdgah and graphs connected both are G H 0 ‚ r oe fadol if only and if cores are H G 0 0 Consequently . [ sasbrp of subgraph a is χ R 9 ( ] erc of retract a G ): × H = ) G min + G ( K R ‚ G { n G erc of retract a χ H ‚ G G o eesrl nidcdsbrp.Ti ssimilar is This subgraph. induced an necessarily not , sacr swell. as core a is ( 40 [ hnteeaesubgraphs are there Then . G H + H ) ) ] H G , ? χ is = osntcnanaytinls Then triangles. any contain not does sacore. a is ( H G G 0 ) ? [ } H G ‚ . ? ‚ H ] where , ? H . hnteeeitretracts exist there Then . G 0 ≤ G G 0 and ≤ G G and 0 sacore. a is H 0 ( G ≤ G [ H H 0 ]) ≤ such ? G = , CEU eTD Collection oeta n uoopimhsa dmoetpwra el n ti xcl h dniy ec no Hence identity. the exactly is it to and belong well, automorphisms as power idempotent an has automorphism any that Note such one exactly is 2 if possible as Choose Since 4.48.1: Note apnsof mappings Let ecl a call We automorphisms. endomorphisms”: “fake of rid the get doing 4.48: to basically Definition have are We first We automorphism. endomorphisms. of for type just certain same, a admit that graphs of number the counted (2009), (Kötters 4.47 Theorem cores. asymmetric the are graphs Rigid endomorphism. as identity the only has it 4.46.1: if Note rigid, simply or retract-rigid called is graph A 4.46: Definition proof Kötters’ proofs, two Babai [ recite by are I at graphs hinted all graphs. idea almost an asymmetric and on about based rigid result are the graphs or to all automorphisms analogously almost of - that terms cores show in we described section be this can In that symmetry endomorphisms. or structure no exhibit graphs Rigid graphs Rigid 4.5 22 φ ] ehv hsestablised: thus have We opoeteterm ed oepeiiaywr.I h ro fErd of proof the In work. preliminary some do we theorem, the prove To R a on kthdin sketched found + etegahpoet httegahi nercie hnams l rpsaeunretractive: are graphs all almost Then unretractive. is graph the that property graph the be n r hr 0 where , a sfiie o every for finite, is ob h iia uhnme.I hr sa dmoetpwrof power idempotent an is there If number. such minimal the be to φ ( : a [ n [ + n ] ] r noitself. into = ) → ≤ r hsoeieptn oe o any for power idempotent one thus [ r a n + ≤ [ ] 18 r apn dmoetif idempotent mapping k ` + ] ? φ n o nieptn oe,w aethat have we power, idempotent an For . . . mk : φ X ∈ [ ` n o oeinteger some for ? n [ ] 21 |{ → G ] [ ): 3 G ∈ [ ] |G n |G n ro ae nteoiia ro fKue n Rödl and Koubek of proof original the on based proof a and ] R ( ( apn hr r powers are there mapping n n n | ) ) | : φ = φ 41 m ∈ O 2  Consequently . End = n 2 € φ ( G 4 3 φ Let . ) Š . |≥|G ≥ }| n ‹ . ` ? n R eoetenndniyidempotent nonidentity the denote ( a n + ) | a . φ r , = a a + r mk + = sadRni(.8,we (3.28), Rényi and os ˝ k φ ic 0 Since . tcnb represented be can it , φ uhthat such 2 ( a + r ) hc sonly is which , ≤ φ r ≤ a = k there , φ [ a 21 + k ] . CEU eTD Collection ro fTerm4.47 Theorem of Proof Proof have: We of points fixed Let 4.49: Lemma 6 h omls ih o ecmltl la here: clear completely be not might formalism The • • • • • • • • φ : hs loehrteeaereally are there altogether Thus, between going anywhere edge an is there if ert h rvoslmaas lemma previous the Rewrite 2 are there Consequently, Obviously h ubrof number the Let q the list we when φ that generality) of loss (without Suppose define Then graphs, any simple inside over going vertex edge no same be the must there to hence vertices neighbouring map cannot endomorphism An eanndniyieptn noopimo h graph the of endomorphism idempotent nonidentity a be φ obt ossigo oeta n lmn.W aefu ae ae ntesz of size the on based cases four have We element. one than more of consisting -orbits respectively. t φ etenme of number the be φ E eoeby Denote . ( C G j i ) φ |{ = ∩ . obt ossigo oeta n lmns thus elements, one than more of consisting -orbits G C {{ Let : G ∈ |{ i , u i G , = v C ( G ∈ } n ; i φ | φ : ) C the sntdbfr.Since before. noted as i u : obt ossigo nyoeeeet(nyfie ons and points) fixed (only element one only of consisting -orbits || ( ea dmoetnndniyedmrhs on endomorphism nonidentity idempotent an be C φ n ∈ j ) |− φ ∈ C : 1 pemg of -preimage i φ End , recocst emd if made be to choices free Q v E ∈ 1 ∈ ( ≤ G ( End G C i = ) < C ) j j j i } }| ≤ ( 6 k G ossso nree ar feeet from elements of pairs unordered of consists n eimdaeyget: immediately we and , 1 = ( ≤ ) 1 42 [ }| i C 2 ≤ c + i P i j = ecl hs the these call we , φ ≤ and 2 1 k ≤ -orbits. | 1 ( C i < φ ≤ E i Y || j i ≤ ( < C C k sa noopim hr sa is there endomorphism, an is G j | j |− j C ≤ . ) i 1 k || ∩ ( ) C 1 j C osbecocsfreg sets. edge for choices possible | + 1 j i E G ≤ Y ) ( 2 i C . < G on | i C j ) ≤ i es h first the sets, || ∩ k n C φ € C j |− etcs Let vertices. 2 j i obt of -orbits t 1 −| φ ) 6= C . + i || ; C as , q j | φ + = [ n G 1 2 c q C k i φ . Š vertices. , i . (4.2) . v c and r xcl the exactly are 1 j ] , . . . , ∈ C [ c j E . i , v ( c G k j t ) ] ethe be φ q ∩ edge φ and C be j i ƒ . CEU eTD Collection • • • • ecnoti h bound the obtain can we nontrivial one is there Either cases. two are that have we one, by bounded is (4.2) product of part second the since Thus, If 2 for bound above the of instead on, here From If sn siain(.) eget we (4.7), estimation Using then one, only is there if and If q n t φ φ − ≤ ≥ 3 ( n ≥ n 0.6 − t h rdc emi 42 contains (4.2) in term product the , φ n ≥ 0.9 1 1 n ) ≤ ≤ X X − and i i < < j j n ≤ ≤ 0.9 1 k k X i q ≤ q = | | φ φ Y hnteeare there then , C C i 1 < i i ≤ | 1 || || j C ≤ ≤ |{ |{ C C Y i n k i | G < G j j 2 € 0.6 ≤ | ≤ | j | 2 ≤ ≥ G ∈ C G ∈ 1 −| ehv that have we , k 1 ≤ € ≥ | C 1 1 2 q 2 Y i 1 2 i < φ ( || (  ( −| n C j n n  ≤  ,hence 3, j ) 2 ) C | k X 2 X i i + : i − : q || = € = k P φ C φ φ 2 q 1 1 p 1 2 j 1 φ | −| | ≤ | ∈ C + ∈ Š C i n t C < φ i i i ) ≤ j End 43 | End ≤ || | 1 2 φ (4.5) . ‹ ‹ k C atr n(.)where (4.2) in factors Š | € 2 j obt rteeaemr.I hr r tlattwo, least at are there If more. are there or -orbit, 2 C | 2 i ≤ = + − ( P ( || − G q C G 2 1 4 φ X € j 2 1 ≤ i ( ) | ) =
i k + n |≤ }| 16 < |≤ }| Š ≤ 1 9 atr fsz tlat2 hence 2, least at size of factors j ≤ − ≤ | 2 1 q k 2 C Š | φ Š t C ( t i € 2 2 φ φ ( 2 n | i || ) 2 q 5 8 ( ( ) 2 φ C ≤  (4.7) . 2 n 2 n 2 Š ) ) j ) | 2 euetetiilbound: trivial the use we , n ≤ € € ≥ ≤ − − 8 5 16 n 9 1 2 2 1 p € 0.9 Š ( p 16  n Š n 9 (4.10) . − n n n (4.4) . n − − Š 2 0.9 n n | n − ) 0.9 C . 1.2 (4.6) . i (4.9) , X − i 2 ≥ | = n k 0.6 1 | (4.8) . and 2 C i | 2  (4.3) , | C j | = .There 1. CEU eTD Collection rirrl ml)tefloigproperties: following the small) arbitrarily h admgahon graph random The 4.50: Theorem 7 Aldgesaea least at are degrees All 1. ec e of set each 4. bt h ags lqeadtelretidpnetsthv ee than fewer have set independent largest the and clique largest the least both at is 3. vertices two any of neighbours common of number the 2. ..Dfiiin3.27. Definition c.f. h ro fKue n öldpnso obnto fresults. of combination a on depends Rödl and Koubek of proof The • • • siaina ntepeiu ae eget we case, previous the in as estimation trmist estimate to remains It If apnsw aefrec ae eoethese denote case, each for have we mappings h li olw as follows claim The least at are there case, third M G G U 4 t ( ( ( φ n n n ) = ) ) = = ≤ ≤ = n n O n P M − ( n n m  1  φ ,te sn h atta hr seatyone exactly is there that fact the using then 2, n ( − 2 ∈ > n 2 ` − ) 1 € ? n 0log 30 2 1 ) · |{ 4 3 n ( 2 p stenme ftefie onsis points fixed the of number the as Š G vertices ( n n 2 n − − ) G ∈ n 2 · ) n ‹ P 2 M + 2 n . − etcsidcsasbrp iha most at with subgraph a induces vertices ( 2 ( φ 3 € n 1 2 ( 1 ( ∈ 7 ( 2 n ) 16 n p |{ ) ` 9 − a ihprobabiltiy with has ? : n ) n n G |{ φ − ε Š ≤ − n ) G n G ∈ ) 1.2 ∈ l n tmost at and n + > G ∈ − 2 0.9 End X n n 0.6 − M ( n n xdpit,hence points, fixed 2 ‹ 0.9 ( ) ( ( n G : n +  ) φ ) ) n l n : }| · ‹ 2 φ ∈ 2 n l 44 ≤ ( 0.9 n End 2 n ∈ − 2 n ) + ( l · M End 1 1 ≤ € € ( 1 i + 16 G ( 8 5 9 l n > − ( ε ) Š Š X ) n G |≤ }| n ) o h rttocases, two first the For . − o n − , 1.2 ) n ( n n }| 0.9 c 2 − 0.9 2 − − n ( 2 0.6 o hc ene oko o many how know to need we which for n 2 n log + 2. ( ) 2 2 n n + ) n n 0.9 € φ ) ( M 3 4 n ( ≤ obto ieto ihasimilar a with two, size of -orbit c Š 3 − ( n n > n − 4 n 1 2 3 4 2 ) ( n ) 1 . 1 0.9 € · m 2 2 4 3 − sacntn,and constant, a is +
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) y uhta hneach when that such , n de mns the amongst edges ∈ .Aseilcs of case special A 0. etcsof vertices G P , z j X ∈ j i H onsthe counts − uhthat such A if 1 a be can j i a ƒ is i CEU eTD Collection Aanw s h hrofieult.Let inequality. Chernoff the use we Again 4. Erd of result classical the of reformulation a just is This 3. Let 2. W li htchoosing that claim We 5. umns pligteCenf nqaiy ehv that have we inequality, Chernoff the Applying summands. ec,h rbblt eaeloigfris for looking are we probability Hence,the P log 2 size of co-cliques or cliques guarantee to graph the where that obtain to now inequality Chernoff and where now can We is. there if 0 and four, the of out bound: edge Chernoff no the is use there if 1 value takes which pair, index holds: relations neighbouring following the of one least at where see numbers, Ramsey 1 ≤ i X i 1 < , i 2 j i c c D ≤ , 2 j , , m ( c c = efie etcs Define vertices. fixed be x 2 2 X , P > > ,ad0ohrie Clearly otherwise. 0 and 1, j i y  = ) r osat as constants are 0 rate. desired the with 0 to tends term last the thus constants, are 0 P onsalteegsta a cu ntesbrp of subgraph the in occur can that edges the all counts P  1 1 x ≤ ≤ i X log i < < m 2 j j ≤
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P 3 4 1 4 x  D  ) n j = m € 2 log ≤ 1 Y 4 1 ] n ‹ i Y h on asta ene tleast at need we that says bound The . , xp ex 1  , i 1 1 16 1 i 1 2 , − − ≤ , i 2 P j Š x y ,  j [ exp sterno aibeta ae au if 1 value takes that variable random the as and , > Y − X − p i 46 1 j i 0. , D  1 4 = i 2 eterno aibea ntefis on.Then point. first the in as variable random the be € ,

− j D 1 4 1 ≥ = € ( − 2 , 3 4 x 4 ε 16 1 € , 1 m 2 = ]  m 2 1 2 y

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n j − ( , , 1 m ) b − = i 1 m ) ∼ = n 4 3 ons hr are there points, − b m 2 c n ( j 2 n
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K eeaigagahfo tb aigtepeiu lc structure, block previous the taking by it from graph a generating , 47 [ 8 ] and [ 20 ] . [ 18 ] and [ 14 ] A . CEU eTD Collection Bibliography [ [ 10 11 [ [ [ [ [ [ [ [ [ 1 2 3 4 5 6 7 8 9 ] ] ] ] ] ] ] ] ] ] ] ihe .AbrsnadKrnL oln.Hmmrhsso -hoai graphs. 3-chromatic of Homomorphisms Collins. mathematics L. Karen and Albertson O. Michael oaAo n olH Spencer. H. Joel and Alon Noga mtra,1995. Amsterdam, Lovász, L. Graham, editors, L. Grötschel, R. M. In and reconstruction. isomorphism, groups, Automorphism Babai. László ee .CmrnadPicl .Kznds oe fSmercGraphs. Society Symmetric Mathematical of Cores Kazanidis. A. Priscila and Cameron J. Peter emnCenf.Amaueo smttcefiinyfrtsso yohssbsdo h sum the on based hypothesis a of observations. tests of for efficiency asymptotic of measure A Chernoff. Herman á Erd Pál etme 1963. September Erd Pál 49 2009. 5419, survey. graphs—A of symmetry Generalized Fan. Suohai itrW ele.Nt nMnmlGraphs. Minimal on Note Fellner. W. 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