Homomorphism Complexes and k-Cores

0.18 MB·

by  Greg Malen

HOMOMORPHISM COMPLEXES AND K-CORES
6
GREGMALEN
1
0
2

b
e Abstract. Weprovethatthetopologicalconnectivityofagraphhomomorphismcomplex
F Hom(G,Km)isatleastm−D(G)−2,whereD(G)=maxδ(H).Thisisastronggeneraliza-
H⊆G
3 tionofatheoremofC˘ukic´andKozlov,inwhichD(G)isreplacedbythemaximumdegree
1 ∆(G).Italsogeneralizesthegraphtheoreticboundforchromaticnumber,χ(G)≤D(G)+1,
asχ(G) = min{m : Hom(G,Km) , ∅}. Furthermore, weusethisresulttoexamineho-
O] mologicalphasetransitionsintherandompolyhedralcomplexesHom(G(n,p),Km)when
p=c/nforafixedconstantc>0.
C
.
h 1. Introduction
t
a
Thestudyofgraphhomomorphismcomplexes,Hom(G,H)forfixedgraphsG and H,
m
grew out of a rising interest in finding topologicalobstructionsto graph colorings, as in
[ Lova´sz’s proof of the Kneser Conjecture [11]. Specifically, Lova´sz used the topologi-

2 cal connectivity of the NeighborhoodComplex, N(G), to provide a lower bound on the
v chromatic numberof a Kneser graph. It turns out that N(G) is a specialization of a ho-
4 momorphismcomplex, as it is homotopyequivalentto Hom(K ,G) (see Proposition 4.2
2
5 in[2]). Thestructureoftheunderlyinggraphshasmanyquantifiableeffectsonthetopol-
8
ogyofthesecomplexes.The0-cellsofHom(G,H)arepreciselythegraphhomomorphisms
7
0 fromG → H,andsothemoststraighforwardsuchresultisthatthechromaticnumberof
. G, χ(G),istheminimummforwhichHom(G,K )isnon-empty.For∆(G)themaximum
1 m
0 degreeofG, BabsonandKozlovconjectured[2], andC˘ukic´ andKozlovlaterproved[6]
6 that if ∆(G) = d, then Hom(G,Km) is at least (m − d − 2)-connected. Note that when
1 m=d+1thisrecoversthewellknownboundχ(G)≤∆(G)+1.
v: In graph theory, there is an improvementof this boundusing the minimum degreeof
i inducedsubgraphs,namelythatχ(G)≤maxδ(H)+1,forδ(H)theminimumdegreeofH.
X H⊆G
ThevalueD(G) ≔ maxδ(H)isknownasthedegeneracyofG,agraphpropertywhichis
r
H⊆G
a
commonlyutilized in computerscience and the study of large networks. Using D(G) in
placeof∆(G),inSection3weproveageneralizationofthisgraphtheoreticresultwhich
willalsospecializetotheC˘ukic´–KozlovTheorem. InSection4wefocusonthecasethat
H = K andshow,withonlyafewnotableexceptions,thatHom(G,K )isdisconnectedif
3 3
itisnon-empty.
Furthermore,inthesettingofrandomgraphstheevolutionofD(G)forasparseErdo˝s–
Re´nyi graph G ∼ G(n,p) has been well studied. For p = c/n for a constant c > 0,
Pittel, Spencer,andWormwaldexhibitedsharpthresholdsforthe appearanceandsize of
subgraphswith minimum degree k [13]. In Section 5 we combine our results with their
worktoexhibitbothphasetransitionsandlowerboundsforthetopologicalconnectivityof
therandompolyhedralcomplexesHom(G(n,c/n),K ) forconstantc > 0. When m = 3,
m
Date:February16,2016.
1
2 GREGMALEN
we are able to use the results from Section 4 to evaluate the limiting probability for the
connectivityofHom(G(n,c/n),K )forallc>0.
3
2. BackgroundandDefinitions
Inthefollowing,allgraphsareundirectedsimplegraphs,andK isthecompletegraph
m
onmvertices. ForagraphGandavertexv∈V(G),whenitisclearfromcontextwemay
writesimplythatv∈G.Inthissectionwedefineagraphhomomorphismcomplex,orhom-
complex,andvariouspropertieswhichwillbeusedtoacquireboundsonthetopological
connectivityofthesecomplexes.
Definition 2.1. For fixed graphsG and H, a graph homomorphism from G → H is an
edgepreserving map onthe vertices, f : V(G) → V(H) such that{f(v), f(u)} ∈ E(H)if
{v,u}∈ E(G).
Definition2.2. ForfixedgraphsGandH,cellsinHom(G,H)arefunctions
η:V(G)→2V(H)\∅
withthe restrictionthatif{x,y} ∈ E(G), thenη(x)×η(y) ⊆ E(H). The dimensionofηis
definedtobedim(η)= (|η(v)|−1),withorderingη⊆τifη(v)⊆τ(v)forallv∈V(G).
v∈XV(G)
The set of 0-cells of Hom(G,H) is then precisely the set of graph homomorphisms
fromG → H, withhigherdimensionalcellsformedoverthemasmultihomomorphisms.
Furthermore,everycellinahom-complexisaproductofsimplices,andhom-complexes
areentirelydeterminedbytheir1-skeletons.Ifηisaproductofsimplicesandits1-skeleton
is contained in Hom(G,H), then η ∈ Hom(G,H). The 1-skeleton of Hom(G,H) can be
thoughtofinthefollowingmanner.Twographhomomorphismsη,τ:G →Hareadjacent
inHom(G,H)ifandonlyiftheirimagesdifferonexactlyonevertexv∈G. Soη(v),τ(v),
η(u)=τ(u)forallu∈G\{v},andthe1-celljoiningηandτisthemultihomomorphismσ
forwhichσ(v)=η(v)∪τ(v)andσ(u)=η(u)=τ(u)forallu∈G\{v}.
Sincethepropertyofbeingahomomorphismisindependentondisjointconnectedcom-
ponentsofagraph,hom-complexesobeyaproductrulefordisjointunions.Foranygraphs
G,G′andH,
Hom(G∐G′,H)=Hom(G,H)×Hom(G′,H)
Thus,whenconvenientwecanalwaysrestrictourattentiontoconnectedgraphs. Herewe
alsointroducethenotionofagraphfolding.
Definition2.3. Denotetheneighborhoodofavertexv∈GbyN(v)={w∈G :w∼v}. Let
v,u ∈ G bedistinctverticessuchthatN(v) ⊆ N(u). ThenafoldofG isahomomorphism
fromG →G\{v}whichsendsv7→uandisotherwisetheidentity.
Lemma2.4. (Babson,Kozlov;Proposition5.1in[2]).IfGandHaregraphsandv,u∈G
aredistinctverticessuchthatN(v)⊆N(u),thenthefoldingG →G\{v}whichsendsv7→u
inducesahomotopyequivalenceHom(G\{v},H)→Hom(G,H).
IfT isatree,forexample,thenT foldstoasingleedge,soHom(T,K )≃Hom(K ,K ).
m 2 m
AndbyProposition4.5in[2],Hom(K ,K )ishomotopyequivalenttoawedgeof(m−n)-
n m
spheres, andinparticularHom(K ,K ) ≃ Sm−2. So Hom(T,K ) ≃ Sm−2 foranytree T.
2 m m
Forathoroughintroductiontohom-complexes,see[2].
HOMOMORPHISMCOMPLEXESANDK-CORES 3
Notation2.5. ForagraphGdefinethemaximumdegreetobe∆(G)≔ max{deg(v)},and
v∈V(G)
theminimumdegreetobeδ(G)≔ min{deg(v)}.
v∈V(G)
Definition2.6. Thek-coreofagraphGisthesubgraphc (G)⊆Gobtainedbytheprocess
k
ofdeletingverticeswithdegreelessthenk,alongwithallincidentedges,oneatatimeuntil
therearenoverticeswithdegreelessthank.
Regardlessoftheorderinwhichverticesaredeleted,thisprocessalwaysterminatesin
theuniqueinducedsubgraphc (G) ⊆ G whichismaximaloverallsubgraphsofGwhich
k
haveminimumdegreeatleastk. Thek-coreofagraphmaybetheemptygraph,andthe
existenceofanon-emptyk-coreisamonotonequestion,asc(G)⊆c (G)whenever j≤l.
l j
Hereweareinterestedinthemosthighlyconnectednon-trivialsubgraph.
Definition2.7. ThedegeneracyofGisD(G)≔maxδ(H),forHaninducedsubgraph.
H⊆G
ThenD(G)isthemaximumksuchthatc (G)isnon-empty.Giventhesedefinitions,we
k
arenowabletostatethemaintheorem:
Theorem2.8. ForanygraphG,Hom(G,K )isatleast(m−D(G)−2)-connected.
m
And D(G) ≤ ∆(G), hencethis willimply the C˘ukic´–KozlovTheorem. Ourinterestis
primarilyinapplyingthisresulttothecasethatG isasparseErdo˝s–Re´nyirandomgraph,
whereD(G)hasbeenstudiedextensivelyandismuchsmallerthan∆(G). Itshouldalsobe
notedthatEngstro¨mgaveasimilarstrengtheningoftheC˘ukic´–KozlovTheoremin[8],re-
placing∆(G)withagraphpropertywhichisindependentofD(G)andwhichmayprovide
abettertoolforstudyingthedenseregime.
3. ProofofTheorem2.8
To provethe theorem,we firstintroducea resultofCsorbawhichfindssubcomplexes
thatarehomotopyequivalenttoHom(G,K )byremovingindependentsetsofG.
m
Notation3.1. LetGandH begraphs.Define
Ind(G)≔{S ⊆V(G):S isanindependentset}
AndforI ∈Ind(G)define
HomI(G,H)≔ η∈Hom(G\I,H): thereisη∈Hom(G,H)withη|G\I =η
n o
So Hom (G,H)isthe subcomplexofHom(G\I,H)comprisedofallmultihomomor-
I
phismsfromG\I →H whichextendtomultihomomorphismsfromG →H.
Theorem3.2. (Csorba;Theorem2.36in[5])ForanygraphsGandH,andanyI ∈Ind(G),
Hom(G,H)ishomotopyequivalenttoHom (G,H).
I
The proof of this is an application of both the Nerve Lemma and the Quillen Fiber
Lemma,andamorethoroughexaminationofthispropertyisgivenbySchultzin[14].Note
thatwhenH = K andI ={v}foranyvertexv∈G,amultihomomorphismη:G\{v}→K
m m
4 GREGMALEN
hasanextensionη:G →K aslongasthereissomevertexinK whichisnotinη(w)for
m m
anyw∈ N(v). Thus
. Hom{v}(G,Km)=η∈ Hom(G\{v},Km):(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)w[∈N(v)η(w)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)≤m−1
Lemma3.3. LetGbeagraphwithavertexvsuchthatdeg(v)≤k. Thenthe(m−k−1)-
skeletonofHom(G\{v},K )iscontainedinHom (G,K ).
m {v} m
ProofofLemma3.3.Letη∈Hom(G\{v},K )\Hom (G,K ). Then
m {v} m
m=(cid:12) η(w)(cid:12)≤ |η(w)|
(cid:12) (cid:12)
(cid:12)(cid:12)w[∈N(v) (cid:12)(cid:12) w∈XG\{v}
(cid:12) (cid:12)
(cid:12) (cid:12)
And (cid:12) (cid:12)
(cid:12) (cid:12)
dim(η)= (|η(w)|−1)= |η(w)| −|N(v)| ≥ m−k
 
Henceifη∈Hom(G\{wv∈X}G,\K{vm})withdim(η)≤wm∈XG\−{v}k−1,thenη∈Hom{v}(G,Km). (cid:3)
So if Hom(G \{v},K ) is (m−k −2)-connected, then Hom (G,K ) is (m−k − 2)-
n {v} n
connected. Combining this with Csorba’s theorem, we have the following corollary and
theproofofTheorem2.8:
Corollary3.4. LetGbeagraphwithavertexvsuchthatdeg(v)≤k. IfHom(G\{v},K )
n
is(m−k−2)-connected,thenHom(G,K )is(m−k−2)-connected.
n
Proof of Theorem 2.8. Let G be a graph with D(G) = k. Then ck+1(G) is the empty
graph,andthereisasequenceG =G , G = G \{v},withdeg (v) ≤ k,terminating
inG =∅. SoG isasinglev0erteix,andi−H1om(iG ,KG)i−1=∆im,them-simplex,
|V(G)| |V(G)|−1 |V(G)|−1 n
whichiscontractible,andthus(m−k−2)-connected. Hence,byinduction,Hom(G,K )
m
isalso(m−k−2)-connected.(cid:3)
4. Hom(G,K )
3
WhenG is a graph such that χ(G) ≤ 3, Hom(G,K ) has a particularlynice structure.
3
IfGdoesnothaveanisolatedvertex,thenHom(G,K )isacubicalcomplex,andBabson
3
andKozlovshowedthatitadmitsametricwithnonpositivecurvature[2]. Noticethatfor
a connected graph G with χ(G) ≤ 3, the bound obtained by Theorem 2.8 when m = 3
provides no new information. When D(G) = 1, G is a tree and folds to a single edge,
so Hom(G,K ) ≃ Hom(K ,K ) ≃ S1. When D(G) = 2, the boundmerelyconfirmsthat
3 2 3
Hom(G,K )isnon-empty,andforD(G) > 2itgivesnoinformationatall. Hereweshow
3
thatHom(G,K )is,infact,disconnectedforalargeclassofgraphsGwithχ(G)≤3.
3
Theorem4.1. IfGisagraphwithχ(G)=3,thenHom(G,K )isdisconnected.
3
Thisresulthasbeenformulatedpreviouslyinthelanguageofstatisticalphysics,where
Glauberdynamicsexaminespreciselythe1-skeletonofHom(G(n,p),K ). See,forexam-
m
ple, the workof Cereceda, vanden HeuvelandJohnson[3]. We providea proofhere in
the contextof workon hom-complexesof cyclesdoneby C˘ukic´ and Kozlov,and offer a
HOMOMORPHISMCOMPLEXESANDK-CORES 5
moregeneralapproachforliftingdisconnectedcomponentsviasubgraphsinthefollowing
lemma.
Lemma 4.2. Let G ⊆ G′ and H be graphs such that Hom(G,H) is non-empty and dis-
connected. Letη ,η ∈ Hom(G,H)be0-cells, i.e. graphhomomorphismsfromG → H,
1 2
suchthattheyareindistinctconnectedcomponentsofHom(G,H). Ifthereareextensions
η1,η2 ∈Hom(G′,H)withηi|G =ηi fori∈{1,2},thenHom(G′,H)isalsodisconnected.
ProofofLemma4.2. Letη ,η begraphhomomorphismsfromG → H suchthatthey
1 2
areindistinctconnectedcomponentsofHom(G,H),withextensionsη ,η ∈Hom(G′,H).
1 2
Wemayassumethatdim(η)=0foreachi,sinceotherwiseany0-cellstheycontainwould
i
also be extensions of η. Suppose that η ,η are in the same connected component of
i 1 2
Hom(G′,H). Thenthereisapathη = τ ∼ τ ∼ τ ∼ ... ∼ τ = η inHom(G′,H),with
1 0 1 2 l 2
dim(τj) = 0forall j. Letτj = τj|G,for0 ≤ j ≤ l. ThenasfunctionsonV(G′),foreach j
thereisonevertexvj ∈G′forwhichτj(vj),τj+1(vj),andtheyagreeonallothervertices.
If vj ∈ G′ \G, then for the restrictionsτj = τj+1. If vj ∈ G, then τj(vj) , τj+1(vj), but
theyagreeonallothervertices,soτj ∼τj+1inHom(G,H).HencethepathinHom(G′,H)
projectsontoapossiblyshorterpathfromη toη inHom(G,H),whichisacontradiction.
1 2
Thereforetheextensionsη andη mustbeindifferentconnectedcomponentsofHom(G′,H),
1 2
whichisthusdisconnected.(cid:3)
ThesubgraphsweexaminetoobtaindisconnectedcomponentsofHom(G,K )willbe
3
cycles. In[7], C˘ukic´ and Kozlovfullycharacterizedthe homotopytype ofHom(C ,C )
n m
foralln,m∈N. Inparticular,theyshowedthatall0-cellsinagivenconnectedcomponent
havethesame numberof whattheycallreturnpoints. Forthese complexes,letV(C ) =
n
{v1,...,vn}suchthatvi ∼v(i+1)modn forall1≤i≤n,andletV(Cm)={1,...,m}suchthat
j ∼ (j+1)modmforall1 ≤ j ≤ m. Forthepurposeofdefiningthereturnnumberofa
0-cellη,wemomentarilydropthesetbracketnotationandwriteη(v)= j.
i
Definition4.3. A return pointofa 0-cellη ∈ Hom(C ,C ) is a vertex v ∈ C suchthat
n m i n
η(vi+1)−η(vi) ≡ −1modm. Thenr(η), thereturnnumberofη, is thenumberofvi ∈ Cn
whicharereturnpointsofη.
Notethatsinceηisahomomorphism,thequantityη(vi+1)−η(vi)isalways±1modm.
ItsimplymeasuresinwhichdirectionCn iswrappingaroundCm ontheedge{vi,vi+1}.
Lemma 4.4. (C˘ukic´,Kozlov, Lemma 5.3in[7])Iftwo 0-cellsofHom(C ,C ) are in the
n m
sameconnectedcomponent,thentheyhavethesamereturnnumber.
For a 0-cell η ∈ Hom(G,K ), one may always obtain another 0-cell by swapping the
m
inverseimagesoftwoverticesinK . Whenthetargetgraphisacompletegraph,werefer
m
totheinverseimageofavertexinK asacolorclassofη. Whenm=3,wehaveK =C
m 3 3
andwecantracktheeffectthatinterchangingtwocolorclasseshasonthereturnnumber.
Notation4.5. Fora0-cellη ∈Hom(G,K ),letη denotethe(l, j)-interchanging0-cell
m {l,j}
obtainedbydefiningη−1 (l)=η−1(j), η−1 (j)=η−1(l),andη−1 (i)=η−1(i)foralli<{l, j}.
{l,j} {l,j} {l,j}
Lemma4.6. Fixapair{l, j}⊂{1,2,3}, l, j. Thenforany0-cellη∈Hom(C ,K ),every
n 3
v ∈C isareturnpointforexactlyoneofηandη . Hencer η =n−r(η).
i n {l,j} {l,j}
(cid:16) (cid:17)
6 GREGMALEN
ProofofLemma4.6. Considertheedge{vi,vi+1}. SIncethetargetgraphis K3, we have
that{η(vi),η(vi+1)}∩{l, j},∅. If{η(vi),η(vi+1)}={l, j},then
η{l,j}(vi+1)−η{l,j}(vi)=−(η(vi+1)−η(vi))
Thusv is a returnpointof η if and onlyif it is nota returnpointof η . Alternatively,
i {l,j}
if|{η(vi),η(vi+1}∩{l, j}| = 1,thenoneofvi andvi+1 hasastationaryimageunderan(l, j)-
interchangewhiletheotherdoesnot. Byfixingtheimageofonevertexandchangingthe
other,thisflipsthedirectionthatCniswrappingaroundK3ontheedge{vi,vi+1}. Soreturns
becomenon-returnsandviceversa. Thereforeeachv ∈C isareturnpointofexactlyone
i n
ofηandη{l,j}foranyfixed{l, j}⊂{1,2,3}.(cid:3)
TheproofofTheorem4.1willthenproceedbyfindinganoddcycleinG andusinga
colorclassinterchangetoproduce0-cellsindistinctconnectedcomponents,whichcanbe
liftedfromHom(C2k+1,K3)toHom(G,K3).
Proof of Theorem 4.1 Since χ(G) = 3, Hom(G,K ) is non-empty and G contains an
3
odd cycle H = C2k+1 for some k ∈ N. Note that we do not require H to be an induced
cycle. Let η ∈ Hom(G,K3) be a 0-cell, and let τ = η|H be the induced homomorphism
on H. Fix a distinct pair {l, j} ⊂ {1,2,3} and consider the (l, j)-interchange η . It is
{l,j}
straightforwardthat η{l,j}|H = τ{l,j} ∈ Hom(H,K3). Thus by Lemma 4.6, r τ{l,j} = 2k+
1 − r(τ) , r(τ). Hence, by lemma 4.4 they are in different connected c(cid:16)ompo(cid:17)nents of
Hom(H,K ), and by Lemma 4.2 their extensions η and η are in different connected
3 {l,j}
componentsofHom(G,K3),whichisthusdisconnected.(cid:3)
Bipartitegraphsareabitmorecomplicated. WehaveseenthatforanygraphGwhich
foldstoasingleedge,Hom(G,K )≃S1. WhenGisbipartiteanddoesnotfoldtoanedge,
3
it must contain an even cycleC for k ≥ 3. So by the same method one can start with
2k
η ∈ Hom(G,K3) and take its restriction η|C2k = τ ∈ Hom(C2k,K3). But if r(τ) = k, then
interchangingtwo color classes no longer guaranteesdisjointcomponents. In particular,
theexistenceoftoomany4-cyclesclosetoC mayforcer(τ) = k. Forexample,for Q
2k 3
the1-skeletonofthe3-cube,Hom(Q ,K )isconnected.NotethatD(Q )=3,sothelower
3 3 3
boundachievedbyTheorem2.8doesnotprovideanyinformation. Restrictingtothecase
thatG omitsthesubgraphsin Figure4.1issufficienttoensurethatthisdoesnothappen,
andthatHom(G,K )isdisconnected.
3
H = H =
1 2
Figure4.1. IfHom(G,K )isconnected,thenGmustcontainH orH .
3 1 2
Theorem4.7. IfG isafinite, bipartite, connectedgraphwhichdoesnotfoldtoanedge
anddoesnotcontainH orH asasubgraph,thenHom(G,K )isdisconnected.
1 2 3
To showthis requiresa greatdealof case-by-casestructuralanalysis whena minimal
evencycle inG has length6, 8, or 10. The methodof the proofwill suggesta stronger,
albeitmoretechnicalresult.
HOMOMORPHISMCOMPLEXESANDK-CORES 7
ProofofTheorem4.7. LetG bea finite,bipartite,connectedgraphwhichdoesnotfold
to an edge and doesnotcontain H or H as a subgraph. SinceG is finite, bipartite and
1 2
does not fold to an edge, it must contain an even cycle of length greater than 4. Let
H = C , k ≥ 3 be a fixed cycle inG which has minimallength over all cycles inG of
2k
lengthgreaterthan4.LabeltheverticesofHasv1,...,v2ksuchthatvi ∼v(i+1)mod2kforall
i. H cannothaveinternalchords,asanysuchedgewouldcreateacycleC for4<l< 2k,
l
orinthecasethatk = 3anantipodalchordwouldcreateacopyofH . For2k ≡ imod3,
1
takeτ ∈Hom(H,K )suchthatτ(v ) = {jmod3}for j ≤ 2k−i,withτ(v ) = {2}ifi = 1,
3 j 2k
andτ(v )={1}, τ(v )={2}ifi=2. Figure4.2depictsτfork ∈{3,4,5}.
2k−1 2k
{1} {1} {1}
{2} {2}
{2} {2}
{3} v1 {2} v1 v v1 v
v v 10 2
v v 8 2
6 2 {3} v v {3}
9 3
{1} v7 v3 {3}
v v v8 v4
5 3 v6 v4 {2} v v {1}
{2} v {3} v 7 v 5
4 {3} 5 {1} 6
{1} {2}
{1} {2} {3}
(a) C , 2k≡0mod3 (b) C , 2k≡2mod3 (c) C , 2k≡1mod3
6 8 10
Figure4.2. ImageofτonH =C
2k
As defined, only the last two vertices of C can be return points of τ. So r(τ) ≤ 2 in
2k
all cases, and hence is not equal to k for any k ≥ 3. Then for any {l, j} ⊂ {1,2,3},
r(τ ) , r(τ). Thus τ and τ are in distinct connected components of Hom(H,K ).
{l,j} {l,j} 3
Soif we canconstructanextensionηto allofG, thenη willextendτ andhenceη
{l,j} {l,j}
andη willbeindistinctconnectedcomponentofHom(G,K ).
{l,j} 3
Defined(u,v)tobetheminimalnumberofedgesinapathconnectingtwoverticesu,v∈G.
LetσbeabipartitionofG,viewedasa0-cellofHom(G,K )withimagecontainedin{1,2}.
3
We willletη = σonG\H andthenmakeadjustmentsasnecessarywhenthedefinition
conflictswithτ. Definethefollowingsets:
B ≔{u∈G :min{d(u,v):v∈H}=i}
i
A ≔{u∈ B :∃v∈ N(u)∩H withσ(u)=τ(v)}
1
A ≔{u∈A:|N(u)∩H|=i,|N(u)∩A|= j}
i,j
SoB isthesetofpointsdistanceifromH,andAisthesetofpointsinG\Honwhichwe
i
cannotdefineη=σ. ToarriveatanappropriatedefinitionforηonthesetA,wemustfirst
makesomestructuralobservationsconcerningthepartitioningofAintothesubsetsA .
i,j
Claim4.8. Foreveryu ∈ B , |N(u)∩H| ≤ 2. Andif|N(u)∩H| = 2, then N(u)∩H =
1
{vi,v(i+2)mod2k}forsomei≤2k.
ProofofClaim4.8. Letu∈ B . ThenconnectingutoanytwopointsinH createsacycle
1
of length at most k +2 < 2k, as in Figure 4.3 (a). Since G is bipartite and H is mini-
malwithrespectto havinglengthatleast6, thisisvalidonlyifitcreatesa4-cycle,with
8 GREGMALEN
N(u)∩H = {vi,v(i+2)mod2k}forsomei ≤ 2k. Similarly,anyfurtherpointsofattachment
would need to be in a 4-cycle with each of the other two vertices in N(u)∩H. But this
isonlypossibleif H = C , inwhichcase it wouldcreate severalcopiesof H , shownin
6 1
Figure4.3(b).(cid:3)
(a) H=C , |N(u)∩H|>2⇒C (b) H=C , |N(u)∩H|>2⇒H
8 6 6 1
Figure4.3. Obstructionsto|N(u)∩H|>2
Thisimposesthati ≤ 2forA ⊆ A,whilethefollowingtwoclaimswillshowthat jisat
i,j
most1,andthat j=1onlyifi=1.
Claim4.9. Letk ≥ 4,andletu ∈ B . Then|N(u)∩B | ≤ 1,andif N(u)∩B , ∅then
1 1 1
|N(u)∩H|=1.
ProofofClaim. Letk ≥ 4, letu ∈ B , andlet w ∈ N(u)∩B , with u ∼ v andw ∼ v .
1 1 i j
G is bipartite, so v , v . If v / v , as in Figure 4.4 (a), there is a cycle of length l
i j i j
with 4 < l ≤ k + 3 < 2k. So v ∼ v . Suppose further that |N(u) ∩ H| = 2. Then
i j
N(u)∩H = {v(j−1)mod2k,v(j+1)mod2k} since both must be adjacent to vj, as in Figure 4.4
(b). But then the induced graphon {u,w,v(j−1)mod2k,vj,v(j+1)mod2k} is a copy of H2. So
if N(u)∩B , ∅ foru ∈ B , then|N(u)∩H| = 1. Nowsuppose|N(u)∩B | > 1. Then
1 1 1
thereis a vertex x ∈ N(u)∩ B with x , w. As with w, the vertexin N(x)∩H mustbe
1
adjacenttov. If x ∼ v , asinFigure4.4(c),then{u,w,x,v,v }isacopyof H ,andif x
i j i j 2
isconnectedtotheotherneighborofv,asinFigure4.4(d),thenitcreatesacopyofH .
i 1
Thus|N(u)∩B1|≤1. (cid:3)
x
u w u
vi v v u v u
j i i
x
v v
vj j w j w
w
(a) v /v; (b) |N(u)∩H|=2; (c) |N(u)∩B |=2; (d) |N(u)∩B |=2;
i j 1 1
Ck+3 H2 H2 H1
Figure4.4. InvalidstructuresinGifClaim4.9isfalse.
Fork =3asimilarresultholdswhenB isreplacedbythesubsetA:
1
Claim 4.10. Let k = 3, andlet u ∈ A. Then |N(u)∩A| ≤ 1, andif N(u)∩A , ∅ then
|N(u)∩H|=1.
HOMOMORPHISMCOMPLEXESANDK-CORES 9
ProofofClaim. Letu∈ Aandletw∈ N(u)∩A. Recallthatfor2k =6, τ(v)={imod3}
i
foralli. Thenwithoutlossofgeneralityassumethatσ(u)={1}andu∼v . Soσ(w)={2},
1
andsincew∈Aitmustbeadjacenttoatleastoneofv andv .ButGisbipartiteandu∼v
2 5 5
createsa5-cycle,asindicatedinFigure4.5(a).Thusw∼v ,andthesameargumentholds
2
to show that u ∼ v if one first assumes that w ∼ v . Furthermore, if w ∼ v or w ∼ v
1 2 4 6
there is a copy of H or H , respectively, depicted in Figure 4.5 (b), (c). By symmetry
1 2
u / v , u / v . Hence,ifu ∈ Awith N(u)∩A , ∅,then|N(u)∩H| = 1. Nowsuppose
3 5
thereisx∈N(u)∩Awithx,w. Thenσ(x)={2},andbytheprecedingargumentwehave
thatx∼v . But,asshowninFigure4.5(d),{u,w,x,v ,v }isthenacopyofH . Hencefor
2 1 2 2
u∈ A, |N(u)∩A|≤1. (cid:3)
w
u
u u
x
u
v v v v w
1 1 v v 1 v 1 v
2 6 2 2
w
v
5 v
4
w
(a) w∼v ; C (b) w∼v ; H (c) w∼v ; H (d) |N(u)∩A|>2; H
5 5 4 1 6 2 2
Figure4.5. InvalidstructuresinGifClaim4.10isfalse.
Then for all k ≥ 3 we have that A = A ∪ A ∪ A , and vertices in A which are
1,0 2,0 1,1 1,1
adjacent to each other have adjacent attaching vertices in H. When possible, we will
define η(v) = {3} for vertices v ∈ A. However, this not possible on all vertices in A ,
1,1
norforverticesinA whichhaveaneighborinH thatisalreadyassigned{3}. Sodefine
2,0
A= u∈A :{τ(v):v∈N(u)∩H}={σ(u),{3}} ∪ u∈ A :σ(u)={2}
2,0 1,1
(cid:8) (cid:9) (cid:8) (cid:9)
Nowdefineη:G →K by:
3
τ(u) ifu∈ H
{1} ifu∈ A∩A ;
 1,1 
η(u)= {{23}} iiffuuu∈∈∈AAA\∩∩AAA;22,,00wwiitthhσσ((uu))=={{12}} 
 σ(v) elseu∈G\(A∪H)withN(u)∩A,∅ 
Noticethatforu∈ A,η(u)={1,2}\σ(u)=σ(w)foranyw∈ N(u).Hencethelastqualifier
assigns{3}toanyneighborsofverticesinAwhichneedtobeswitched,andη = σonall
remainingvertices.
It is clear that this defines a homomorphism on A ∪ H, and on any edges between ver-
tices u ∈ G \ (A ∪ H) for which N(u) ∩ A = ∅. Thus it suffices to check that this is
consistentonedgeswithavertexu ∈ G\(A∪H)forwhichN(u)∩A , ∅. Notethatby
definition,anysuchuwillhaveη(u)={3}. Sotoensurethatη(w), {3}foranyw∈ N(u),
10 GREGMALEN
wemustcheckthatN(u)∩ A\A =∅,andforallw∈ N(u)thatN(w)∩A=∅. Thecases
k=3andk≥4mustagain(cid:16)behan(cid:17)dledseparately.
Claim4.11. Letk≥4,andletu∈G\(A∪H)withx∈ N(u)∩A.ThenN(u)∩ A\A =∅,
andifw∈ N(u)thenN(w)∩A=∅. (cid:16) (cid:17)
Proof of Claim. Let u and x be as in the claim. Since x ∈ A ⊆ B , it must be that
1
u∈(B ∪B )\A. Supposefirstthatu∈ B . Then|N(u)∩B |≤1byClaim4.9,andhence
1 2 1 1
N(u)∩A={x}. SoN(u)∩ A\A =∅.Similarly,Claim4.9impliesthatN(x)∩B ={u}and
1
that|N(x)∩H|=1,sox∈(cid:16) A ,(cid:17)whichcontradictsourassumptionthatx∈A⊆ A ∪A .
1,0 2,0 1,1
Sou< B .
1
Suppose instead that u ∈ B , and thus N(u) ⊆ B ∪ B ∪ B . For w ∈ N(u)∩ B it is
2 1 2 3 3
immediatethatw < A and that N(w)∩A ⊆ (N(w)∩B ) = ∅. Thenassume thatthere
1
is w ∈ N(u)∩ B , w , x. The(cid:16)n σ(w) =(cid:17)σ(x) since G is bipartite, and thus for any at-
1
taching points in H, v ∼ x, v ∼ w, it must be that i and j are either both odd or both
i j
even. Since x ∈ A ∪A we mayassume that eitheri , j or x ∈ A . Recall thatan
2,0 1,1 1,1
adjacentpairinA formsasquarewithitsadjacentpairinH. Soifi= jandx∈ A with
1,1 1,1
{z} = N(x)∩A ,thenzisadjacenttooneofthetwoverticesinN(v)∩H. Asshownin
1,1 i
Figure4.6(a),thiscreatesacopyofH . Sowemayassumethati , j. Butthentheshort
1
pathbetweenv andv inH alongwith{u,w,x}createsacycleoflengthl, 6≤l≤k+4.
i j
z
x x
x v v
i i
vi u u u
v v
w j j
w w
(a) i= j, x∈A ; H (b) i, j; C fork=5 (c) i, j; C fork=4
1,1 1 9 8
Figure4.6. w∈ N(u)∩B , w, x.
1
Ifk ≥ 5thislengthislessthan2k,asinFigure4.6(b),whichcontradictstheminimality
of H. If k = 4, this creates a cycle of length 8 only if j = (i+4) mod 8, as in Figure
4.6(c). Recallthatasdefinedfork = 4 (seeFigure4.2(b)), τ(vi) , τ(v(i+4)mod8)forall
i. But σ(x) = σ(w), and therefore x and w cannotboth be in A. As this is true for any
pair of vertices in N(u)∩ B , it must be that |N(u)∩ A| ≤ 1, and hence by assumption
1
N(u)∩A={x}. Furthermore,foranyw∈ B \A,ifw∼zwithz∈A,thenbyClaim4.9it
1
mustbethatz∈A . ThusN(w)∩A=∅forallw∈ N(u)∩B .
1,0 1
Now suppose that w ∈ N(u) ∩ B . Note that w < A ⊆ B . Then let x = x and let
2 1 u
x ∈N(w)∩B . Notethatσ(x ),σ(x )sincethereisapathoflength3fromx tox ,so
w 1 u w u w
theconnectingverticesforx andx inH mustbedistincttoavoidcreatinganoddcycle.
u w
ThenthepathbetweentheirconnectingverticesinH alongwith x ,u,w,and x createsa
u w
cycleoflengthl, 6≤l≤k+5. Fork >5suchacyclecontradictstheminimalityofH.
Suppose k = 4. Then to avoid creating a cycle of length less than 8, this construction
requiresthatthepathinHbetweentheconnectingpointsofx andx islength3.Suppose
u w
by way of contradictionthat x ∈ A. So both x ,x ∈ A ⊆ A ∪A . Withoutloss of
w u w 2,0 1,1