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Journal of Combinatorial Theory, Series B
www.elsevier.com/locate/jctb
The Kelmans-Seymour conjecture IV: A proof
✩Dawei He1, Yan Wang,Xingxing Yu2
SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,GA30332-0160, USA
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received21December2016 Availableonline19December2019
Keywords:
K5-subdivision Independentpaths Separation Connectivity Discharging Contraction
Awellknown theoremof Kuratowski in1932 statesthat a graphisplanarif,andonlyif,itdoesnotcontainasubdivision ofK5or K3,3.Wagner provedin1937 thatifa graphother thanK5 doesnot containanysubdivision ofK3,3 thenitis planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graphdoesnotcontainanysubdivisionofK5thenitisplanar oritadmitsacutofsizeatmost4.Inthispaper,wegivea proof of theKelmans-Seymour conjecture. We also discuss severalrelatedresultsandproblems.
©2019ElsevierInc.Allrightsreserved.
1. Introduction
ForagraphG,weuseT Gto denoteasubdivisionofG,andtheverticesinT Gthat correspondto thevertices ofGaresaidto beitsbranch vertices. Thus,T K5 denotesa subdivisionofK5, andthevertices inaT K5 ofdegreefour areitsbranchvertices. For
✩ PartiallysupportedbyNSFgrantsDMS-1265564andDMS-1600738.
E-mailaddresses:dhe9@math.gatech.edu(D. He),yanwang@gatech.edu(Y. Wang), yu@math.gatech.edu(X. Yu).
1 ThisworkwasstartedwhenDHwasastudentatEastChinaNormalUniversity.
2 PartoftheworkwasdonewhileXYwasashort-termvisitoratEastChinaNormalUniversitywhich wasmadepossiblebySTCSMgrantNo.13dz2260400awardedtothemathematicsdepartmentofECNU.
https://doi.org/10.1016/j.jctb.2019.12.002 0095-8956/©2019ElsevierInc. Allrightsreserved.
graphs H andK, we saythatH contains T K ifH contains asubgraph isomorphicto aT K.
The well known resultof Kuratowski [18] statesthata graphis planar if, and only if,itdoesnotcontainT K5 orT K3,3.AsimpleapplicationofEuler’sformulaforplanar graphs shows that, for n ≥ 3, if an n-vertex graph has at least 3n−5 edges then it must be nonplanar and, hence, contains T K5 or T K3,3. Dirac [5] conjectured that for n ≥ 3, if an n-vertex graph has at least 3n−5 edges then it must contain T K5. This conjecture wasalso reportedbyErdősand Hajnal[7]. Kézdy andMcGuiness[15]
showed thataminimal counterexampleto Dirac’sconjecture mustbe 5-connected and containsK4−,whereK4−isthegraphobtainedfromthecompletegraphK4bydeletingan edge. (However,Kelmans[14] andSeymour(see[22])knewinthe1970sthataminimal counterexample to Dirac’sconjecture must be 5-connected.) After somepartial results in[28,30,33,34],Dirac’sconjecturewasprovedbyMader[22],wherehealsoshowedthat every5-connectedn-vertexgraphwithatleast3n−6 edgescontainsT K5 orK4−.
Seymour[26] (alsosee[22,34])and,independently,Kelmans[14] conjecturedthatev- ery5-connectednonplanargraphcontainsT K5.Thus,theKelmans-Seymourconjecture impliesMader’stheorem.Thisconjectureisalsorelatedtoseveralinterestingproblems, whichwe willdiscussinSection7.
Theauthors[9–11] producedlemmasneededforresolvingthisKelmans-Seymourcon- jecture,andwe arenowreadyto proveitinthispaper.
Theorem 1.1.Every5-connectednon-planar graph containsT K5.
The starting point of our work is the following result of Ma and Yu [20,21]: Every 5-connectednonplanargraphcontainingK4−hasaT K5.Thisresult,combinedwiththe resultof KézdyandMcGuiness[15] on minimalcounterexamplestoDirac’sconjecture, gives analternative proof of Mader’s theorem.Alsousing this result, Aigner-Horev [1]
proved thatevery5-connectednonplanarapexgraphcontainsT K5. Asimplerproofof Aigner-Horev’sresultusingdischargingargumentwasobtainedbyMa,Thomas,andYu, and, independently,byKawarabayashi,see[13].
WenowbrieflydescribetheprocessforprovingTheorem1.1.Foramoredetailedver- sion,werecommendthereadertoreadSection6first,whichshouldalsogivemotivation to someofthetechnicallemmas listedinSections 2,3,4and5.
Suppose G is a 5-connected non-planar graph not containing K4−. We fix a vertex v∈V(G),andletM beamaximalconnectedsubgraphofGsuchthatv∈V(M),G/M (thegraphobtainedfromGbycontractingM)isnonplanar,G/M containsnoK4−,and G/M is5-connected(i.e.,M is contractible).Note thatV(M)={v}is possible.Letx denote the vertex of H := G/M resulting from the contraction of M. Then, for each subgraph T of H withv ∈V(T) and with T ∼=K2 or T ∼=K3,H/T isplanar,or H/T containsK4−,or H/T isnot5-connected.If, forsomeT,H/T isplanaror containsK4− then we can find aT K5 inG using resultsfrom [9–11]. Thus, inthis paper, our main work is to dealwith the final case: for any subgraph T of H with x∈ V(T) andwith
T ∼=K2 orT ∼=K3,itfollowsthatH/T isnonplanar,H/T containsnoK4−,andH/T is not5-connected.Inthiscase,thereexistsST ⊆V(H) suchthatV(T)⊆ST,|ST|= 5 or
|ST|= 6,and H−ST isnotconnected.Wewill beusing suchcutstodividethegraph intosmallerparts andusethem tofindaspecialT K5 inH. Thereasonto alsoinclude the case T ∼= K3 is to avoid the situationwhen T ∼= K2, |ST| = 5, and H −ST has exactlytwo components,oneofwhichistrivial.Thissituationdoesnotcauseproblems when T ∼=K3, asthe graphH would thencontain K4−,and we coulduse resultsfrom [9–11].
Wewillneedanumberofresultsfrom[9–11],whicharegiveninSection2.InSection3, we derive asimplified version ofa result ondisjoint pathsfrom [39–41], whichwill be used several times in Section 4. For each subgraph T of H with v ∈ V(T) and with T ∼= K2 or T ∼= K3, we will associate to it aquadruple (T,ST,A,B), where,roughly, A∩B=∅,H−ST =A∪B,andH hasnoedgebetweenAandB.(Aprecisedefinition of a quadrupleis given in Section4.) In Section 4, we provesome basic properties of quadruples,andtakecareoftwospecialcasesinvolvingquadruples(usingdisjointpaths resultsfrom Section3).In Section5, wetake careof othercases involvingquadruples.
WecompletetheproofofTheorem1.1inSection6,anddiscussseveralrelatedproblems inSection7.
Weendthissectionwithsomenotationandterminology.LetGbeagraph.ByS⊆G wemean thatS isasubgraph ofG. WemayviewS ⊆V(G) asasubgraph ofG with vertexsetSandnoedges.ForS⊆G,weuseG[S] todenotethesubgraphofGinduced byV(S). Forany x∈V(G) weuse NG(x) to denote theneighborhood of xinG, and forS⊆GletNG(S)={x∈V(G)−V(S):NG(x)∩V(S)=∅}. Whenunderstood,the referencetoGmaybedropped.ForS⊆E(G),G−S denotes thegraphobtainedfrom GbydeletingalledgesinS;andforK,L⊆G,K−Ldenotesthegraphobtainedfrom K bydeletingV(K∩L) andalledgesofK incidentwithV(K∩L).
A separation in a graphG consists of apair of subgraphs G1,G2 of G, denotedas (G1,G2),suchthatV(G)=V(G1)∪V(G2),E(G1)∪E(G2)=E(G),E(G1∩G2)=∅, E(G1)∪(V(G1)−V(G2))=∅,and E(G2)∪(V(G2)−V(G1))=∅. Theorder ofthis separationis |V(G1)∩V(G2)|, and (G1,G2) issaid to be ak-separation ifits order is k. Aset S ⊆V(G) isak-cut (or acut of size k)inG, where kis apositiveinteger,if
|S|=kandGhasaseparation(G1,G2) suchthatV(G1)∩V(G2)=S,V(G1−S)=∅, andV(G2−S)=∅.(Thus,foraseparation(G1,G2) inagraphG,V(G1)∩V(G2) need notbeacutinG.)Ifv∈V(G) and{v}isacutofG, thenv issaidto be acut vertex of G. For A ⊆ V(G) with G−A = ∅ and for a positive integer k, we say that G is (k,A)-connected if, forany cut S with |S| < k, everycomponent of G−S contains a vertexfromA.Thus,ifGisak-connectedgraphand(G1,G2) isaseparationinGsuch thatV(G2)−V(G1)=∅,thenG2 is(k,V(G1∩G2))-connected.
GivenapathP inagraphandx,y∈V(P),xP y denotesthesubpathofP between xandy (inclusive).Theends of thepathP are thevertices ofthe minimumdegreein P, and allother vertices of P (ifany) are its internal vertices. A path P with ends u andv(or anu-vpath)isalsosaidtobefromutov orbetweenuandv.Acollectionof
paths issaidto beindependent ifno vertexof anypathinthis collectionisaninternal vertex ofanyother pathinthecollection.
LetGbe agraph.LetK ⊆G,S⊆V(G),andT acollectionof2-elementsubsetsof V(K)∪S.ThenK+ (S∪T) denotesthegraphwithvertex setV(K)∪S andedgeset E(K)∪T, andifT ={{x,y}}wewriteK+xy insteadofK+{{x,y}}.
For any positiveinteger k, let[k] := {1,. . . ,k} (and let [0] =∅). A 3-planar graph (G,A) consistsofagraphGandasetA={A1,. . . ,Ak}ofpairwise disjointsubsetsof V(G) (possibly A=∅whenk= 0)suchthat
(a) fordistincti,j∈[k],N(Ai)∩Aj=∅, (b) fori∈[k],|N(Ai)|≤3,and
(c) ifp(G,A) denotesthegraphobtainedfromGby(foreachi)deletingAi andadding edgesjoiningeverypairofdistinctverticesinN(Ai) thatarenotalreadyadjacentin G,thenp(G,A) maybedrawninacloseddiscDintheplanewithnopairofedges crossingsuchthat,foreachAiwith |N(Ai)|= 3,N(Ai) inducesafacialtrianglein p(G,A).
If, inaddition,b1,. . . ,bn arevertices ofGsuchthatbi∈/Aj forany i∈[n] andj ∈[k]
and b1, . . . ,bn occur on the boundary of the disc D inthat cyclic order, then we say that (G,A,b1,. . . ,bn) is 3-planar or, simply, (G,b1,. . . ,bn) is 3-planar (if there is no needtomentionA).Ifthereisnoneedtospecifytheorderofb1,. . . ,bn thenwesimply say that(G,A,{b1,. . . ,bn}) or (G,{b1,. . . ,bn}) is3-planar. WhenA=∅, wesay that (G,b1,. . . ,bn) or(G,{b1,. . . ,bn}) isplanar(inthis case,Gisactually aplanargraph).
Note that if (G,{b1,. . . ,bn}) is 3-planar and G is (4,{b1,. . . ,bn})-connected, then (G,{b1,. . . ,bn}) is in fact planar and G has a plane drawing in a closed disc with b1,. . . ,bn ontheboundaryofthedisk.
2. Previousresults
In this section, welist anumber of previousresultswhich we will useas lemmas in ourproofof Theorem1.1.Webeginwith thefollowing resultofMaandYu[20,21].
Lemma 2.1.Every5-connectednonplanar graph containingK4− has aT K5.
We also need the main result of [10], to takecare of the casewhen the vertex x in H =G/M (seeSection1)isadegree2vertexinaK4−(whichisy2inthelemmabelow).
Lemma 2.2.Let G be a5-connected nonplanar graph and {x1,x2,y1,y2}⊆ V(G) such that G[{x1,x2,y1,y2}]∼=K4− with y1y2∈/E(G).Thenoneof thefollowingholds:
(i) Gcontains aT K5 inwhichy2 isnota branch vertex.
(ii) G−y2 containsK4−.
(iii) Ghas a5-separation(G1,G2)suchthat V(G1∩G2)={y2,a1,a2,a3,a4},andG2
isthegraph obtainedfrom the edge-disjointunion of the8-cycle a1b1a2b2a3b3a4b4
a1 andthe4-cycle b1b2b3b4b1 by addingy2 andtheedges y2bi fori∈[4].
(iv) Foralldistinctw1,w2,w3∈N(y2)− {x1,x2},G− {y2v:v /∈ {w1,w2,w3,x1,x2}}
containsT K5.
Todealwithconclusion(iii) ofLemma2.2,weneedProposition1.3from[9] inwhich aplaystheroleofy2inLemma2.2.
Lemma 2.3. Let G be a 5-connected nonplanar graph, (G1,G2) a 5-separation in G, V(G1∩G2)={a,a1,a2,a3,a4}suchthatG2isthegraphobtainedfromtheedge-disjoint union ofthe8-cycle a1b1a2b2a3b3a4b4a1 andthe 4-cycle b1b2b3b4b1 by addinga andthe edgesabi,i∈[4].Suppose|V(G1)|≥7.Then,foranydistinctu1,u2∈N(a)−{b1,b2,b3}, G− {av:v /∈ {b1,b2,b3,u1,u2}}contains T K5.
Nextwelist afew resultsfrom [9–11]. Forconvenience, we statetheirversionsfrom [11]. First, we need Theorem 1.1 in [11] to take careof the case when thevertex xin H =G/M(seeSection1)isadegree3vertexinaK4−(whichisx1inthelemmabelow).
Lemma2.4. LetGbea5-connectednonplanargraphandx1,x2,y1,y2∈V(G)bedistinct suchthat G[{x1,x2,y1,y2}]∼=K4− andy1y2∈/E(G).Thenoneof thefollowingholds:
(i) GcontainsaT K5 inwhich x1 isnot abranchvertex.
(ii) G−x1 contains K4−,orGcontains aK4− inwhich x1 isof degree 2.
(iii) x2,y1,y2 maybe chosensothat forany distinctz0,z1∈N(x1)− {x2,y1,y2},G− {x1v:v /∈ {x2,y1,y2,z0,z1}}contains T K5.
Whenapplying thenext threelemmas, the vertex awill correspond to thevertex x inH =G/M inSection1.Thefollowing resultis adirectconsequenceof Theorem 1.1 in[9],whichdealswith5-separationswithanapexside.
Lemma2.5. LetGbe a5-connectednonplanar graph andlet (G1,G2)be a5-separation inG.Suppose|V(Gi)|≥7fori∈[2],a∈V(G1∩G2),and(G2−a,V(G1∩G2)− {a}) isplanar.Then oneof thefollowingholds:
(i) for any a∗ ∈ V(G1−G2)∪ {a}, G contains a T K5 in which a∗ is not a branch vertex.
(ii) G−acontainsK4−,or GcontainsaK4− in whichaisof degree 2.
Thenextresult isLemma 2.8in[11],which will be used totake careof 5-cutscon- tainingtheverticesofatriangle.
Lemma 2.6.LetGbea5-connectedgraph and(G1,G2)bea5-separationinG.Suppose that |V(Gi)|≥7fori∈[2]andG[V(G1∩G2)] containsatriangleaa1a2a.Thenoneof thefollowingholds:
(i) Gcontains aT K5 inwhichais notabranch vertex.
(ii) G−a containsK4−,orG containsaK4− in whichaisof degree 2.
(iii) For any distinct u1,u2,u3 ∈ N(a)− {a1,a2}, G− {av : v /∈ {a1,a2,u1,u2,u3}}
contains T K5.
Thefollowing isLemma2.9in[11].
Lemma 2.7. LetG be a graph, A ⊆ V(G), and a ∈ A such that |A| = 6, |V(G)| ≥8, (G−a,A− {a})isplanar,and Gis(5,A)-connected.Thenoneof thefollowingholds:
(i) G−a containsK4−,orGcontains aK4− in whichthedegreeof ais2.
(ii) Ghas a5-separation(G1,G2)suchthat a∈V(G1∩G2),|V(G2)|≥7,A⊆V(G1), and(G2−a,V(G1∩G2)− {a}) isplanar.
We need Theorem 1.4 in [9]. This will be used to show that, for a quadruple (T,ST,A,B) in H = G/M with x ∈ V(T) (see Section 1), x has a neighbor in A (whichcorresponds toG1−G2inthestatement).
Lemma 2.8.Let G be a 5-connected graph and x ∈ V(G), and let (G1,G2) be a 6-separationinGsuchthatx∈V(G1∩G2),G[V(G1∩G2)]containsatrianglexx1x2x, and |V(Gi)| ≥ 7 for i ∈ [2]. Moreover, assume that (G1,G2) is chosen so that, sub- ject to {x,x1,x2} ⊆ V(G1 ∩G2) and |V(Gi)| ≥ 7 for i ∈ [2], G1 is minimal. Let V(G1∩G2)={x,x1,x2,v1,v2,v3}.ThenN(x)∩V(G1−G2)=∅,oroneof thefollow- ing holds:
(i) G containsaT K5 inwhichxisnot abranchvertex.
(ii) Gcontains K4−.
(iii) There exists x3 ∈ N(x) such that for any distinct y1,y2 ∈ N(x)− {x1,x2,x3}, G− {xv:v /∈ {x1,x2,x3,y1,y2}} containsT K5.
(iv) Forsomei∈[2]andsomej∈[3],N(xi)⊆V(G1−G2)∪ {x,x3−i},andanythree independentpaths inG1−xfrom {x1,x2}tov1,v2,v3,respectively,with twofrom xi andone fromx3−i,must containapath fromx3−i tovj.
Weremarkthatconclusion(iv) inLemma2.8willbedealtwithinSection4,usinga resultondisjoint pathsfrom[39–41].WealsoneedProposition4.1from[9] todealwith the casewhen H/T is planar(seeSection1)forsomeT ⊆H withx∈V(T) and with T ∼=K2 orT ∼=K3.
Lemma2.9. LetG be a5-connected nonplanar graph, x∈V(G),and T ⊆G,suchthat x ∈ V(T), T ∼= K2 or T ∼= K3, and G/T is 5-connected and planar. Then G−T containsK4−.
We conclude this section with three additional results, first of which is a result of Seymour[25];equivalent versionsareprovedin[31,24,27].
Lemma 2.10.Let G be a graph and let s1,s2,t1,t2 ∈ V(G) be distinct. Then either G containsdisjointpaths froms1 tot1 andfrom s2 tot2,or(G,s1,s2,t1,t2) is3-planar.
Thesecondresultisdueto Perfect[23].
Lemma 2.11.Let G be a graph, u∈ V(G), and A ⊆ V(G−u). Suppose there exist k independentpathsfrom utodistinct a1,. . . ,ak∈A,respectively,andinternally disjoint fromA.Thenforany n≥k,ifthere existn independentpaths P1,. . . ,Pn inG fromu tondistinct verticesinA andinternally disjointfromA thenP1,. . . ,Pn maybechosen sothat ai∈V(Pi) fori∈[k].
The thirdresult is due to Watkinsand Mesner [38], which gives acharacterization ofgraphsGwith nocycle throughthree givenverticesy1,y2,y3.Roughly,Ghas2-cuts separatingthese threevertices. SeeFig.1foranillustration.
Lemma 2.12.Let G be a 2-connected graph and let y1,y2,y3 be three distinct vertices of G. Then G has no cycle containing {y1,y2,y3} if, and only if, one of the following holds:
(i) Thereexistsa2-cutSinGandthereexistpairwisedisjointsubgraphsDyi ofG−S, i∈[3],suchthat yi∈V(Dyi) andeach Dyi isaunion ofcomponentsof G−S.
(ii) There exist 2-cuts Syi in G, i ∈ [3], and pairwise disjoint subgraphs Dyi of G, suchthat yi∈V(Dyi),each Dyi is aunionof componentsof G−Syi,thereexists z∈Sy1∩Sy2∩Sy3,andSy1− {z},Sy2− {z},Sy3− {z}arepairwise disjoint.
(iii) Thereexistpairwisedisjoint2-cuts Syi inGandpairwisedisjointsubgraphsDyi of G−Syi,i∈[3],suchthat yi ∈V(Dyi),Dyi is aunionof componentsof G−Syi, andG−V(Dy1∪Dy2∪Dy3)has preciselytwocomponents,each containingexactly onevertexfrom Syi fori∈[3].
3. Obstructiontothreepaths
Inordertodealwith(iv) ofLemma2.8,weneedaresultofthethirdauthor[39–41], whichcharacterizesgraphsGinwhichanythreedisjointpathsfrom{a,b,c}⊆V(G) to {a,b,c}⊆V(G) mustcontain apathfrom b to b.The objective ofthis section isto deriveamuchsimplerversion ofthatcharacterizationbyimposingextraconditionson
Fig. 1.No cycle containing{y1, y2, y3}.
G. Thisresultwill beused severaltimes intheproofsofLemmas 4.4and 4.6.To state theresultfrom [39–41], weneedtodescriberungsand ladders.
LetGbeagraph,{a,b,c}⊆V(G),and{a,b,c}⊆V(G).(Here, a,b,c arepairwise distinct, and a,b,c are pairwise distinct.) Suppose {a,b,c}={a,b,c}, and assume that Ghasno separation(G1,G2) such that|V(G1∩G2)|≤3,{a,b,c}⊆V(G1),and {a,b,c}⊆V(G2).(So{a,b,c}and{a,b,c}areindependentsets inG.)Wesaythat (G,(a,b,c),(a,b,c)) isarung ifoneofthefollowingholds:
(1) b=b or {a,c}={a,c}.
(2) a=a and(G−a,c,c,b,b) is3-planar, orc=c and(G−c,a,a,b,b) is3-planar.
(3) {a,b,c}∩ {a,b,c}=∅and (G,a,b,c,c,b,a) or (G,a,b,c,a,b,c) is3-planar.
(4) {a,b,c}∩ {a,b,c}=∅,G hasa1-separation(G1,G2) such that(i){a,a,b,b}⊆ V(G1),{c,c}⊆V(G2),and(G1,a,a,b,b) is3-planar, or(ii){c,c,b,b}⊆V(G1), {a,a}⊆V(G2),and(G1,c,c,b,b) is3-planar.
(5) {a,b,c}∩ {a,b,c}=∅,and Ghasaseparation(G1,G2) suchthatV(G1∩G2)= {z,b} (or V(G1∩G2) = {z,b}), and (i) (G,a,a,b,b) is 3-planar, {a,a,b,b} ⊆ V(G1), {c,c} ⊆ V(G2), and (G2,c,c,z,b) (or (G2,c,c,b,z)) is 3-planar, or (ii) (G,c,c,b,b) is 3-planar, {c,c,b,b}⊆V(G1),{a,a}⊆V(G2), and(G2,a,a,z,b) (or(G2,a,a,b,z))is3-planar.
(6) {a,b,c}∩ {a,b,c}=∅, andthere are pairwise edgedisjoint subgraphs Ga,Gc,M of G such that G = Ga∪Gc ∪M, V(Ga∩M) = {u,w}, V(Gc ∩M) = {p,q}, V(Ga∩Gc)=∅,and(i){a,a,b}⊆V(Ga),{c,c,b}⊆V(Gc),and(Ga,a,a,b,w,u) and (Gc,c,c,b,p,q) are 3-planar, or (ii) {a,a,b} ⊆ V(Ga), {c,c,b} ⊆ V(Gc), (Ga,b,a,a,w,u),and (Gc,b,c,c,p,q) are3-planar.
(7) {a,b,c}∩ {a,b,c}=∅, andthere are pairwise edgedisjoint subgraphs Ga,Gc,M ofG suchthatG=Ga∪Gc∪M, V(Ga∩M)={b,b,w}, V(Gc∩M)={b,b,p}, V(Ga∩Gc)={b,b},{a,a}⊆V(Ga),{c,c}⊆V(Gc), and(Ga,a,a,b,w,b) and (Gc,c,c,b,p,b) are3-planar.
LetL beagraphandletR1,. . . ,Rmbeedge disjointsubgraphsofLsuchthat
(i) (Ri,(xi−1,vi−1,yi−1),(xi,vi,yi)) isarungforeachi∈[m],
(ii) V(Ri∩Rj)={xi,vi,yi}∩ {xj−1,vj−1,yj−1}fori,j∈[m] with i< j,
(iii) for any i,j ∈[m]∪ {0}, ifxi =xj then xk =xi foralli ≤k ≤j, ifvi =vj then vk=vi foralli≤k≤j,andifyi=yj thenyk =yi foralli≤k≤j,and
(iv) L= (m
i=1Ri)+S, where S consists of those edges of L each of which hasboth ends in{xi,vi,yi}forsomei∈[m]∪ {0}.
Then (L,(x0,v0,y0),(xm,vm,ym)) is aladder with rungs (Ri,(xi−1,vi−1,yi−1), (xi,vi, yi)),i∈[m],or simply,aladderalong v0. . . vm.
By definition, for any rung (Ri,(xi−1,vi−1,yi−1), (xi,vi,yi)), Ri has three disjoint paths from {xi−1,vi−1,yi−1} to {xi,vi,yi}. So for any ladder (L,(x0,v0,y0),(xm,vm, ym)),L hasthree disjointpathsfrom {x0,v0,y0}to {xm,vm,ym}.
For asequence W, the reduced sequence of W is the sequence obtainedfrom W by removingallbutoneconsecutive identicalelements.Forexample,thereducedsequence ofaaabccaisabca.Wecannow statethemain resultin[41].
Lemma 3.1. Let G be a graph, {a,b,c} ⊆ V(G), and {a,b,c} ⊆ V(G) such that {a,b,c} = {a,b,c}. Assume that G is (4,{a,b,c}∪ {a,b,c})-connected. Then any threedisjointpathsinGfrom{a,b,c}to{a,b,c}mustinclude onefrombtob if,and onlyif, oneof thefollowingstatementsholds:
(i) G has a separation (G1,G2) of order at most 2 such that {a,b,c} ⊆ V(G1) and {a,b,c}⊆V(G2).
(ii) (G,(a,b,c),(a,b,c))isaladder.
(iii) G has a separation (J,L) such that V(J ∩L) = {w0,. . . ,wn}, (J,w0,. . . ,wn) is planar, {a,b,c}∪ {a,b,c}⊆V(L), (L,(a,b,c),(a,b,c)) isa ladder along ase- quence v0. . . vm, where v0 =b, vm =b, and w0. . . wn is thereduced sequence of v0. . . vm.
Remark1.Wemayremovetheassumptionthat,foranyT ⊆V(G) with|T|≤3,every component of G−T contains some element of {a,b,c}∪ {a,b,c}. When we do, the conclusionof Lemma3.1 holds by simply replacing “(J,w0,. . . ,wn) is planar” in (iii) with“(J,w0,. . . ,wn) is3-planar”.
Remark2.We mayview(ii) asaspecial caseof(iii) by lettingJ be asubgraph ofL.
In theapplications of Lemma3.1 inthis paper, we will consider rungs and laddersin a 5-connected graph without T K5. With such extra conditions, the rungs have much simplerstructure,asgiveninthenextthreelemmas.SeeFig.2.Thisfirstlemmafollows fromasimpleinspectionofthedefinitionofrungs.
Lemma3.2. Let(G,(a,b,c),(a,b,c))bearung.If{a,c}∩ {a,c}=∅andaandc have thesameset ofneighborsin G,then b=b.
Fig. 2.The simple rungs as in Lemma3.4.
Lemma 3.3.LetGbe a5-connectedgraph and(R,R) aseparationin Gsuchthat R− R = ∅, V(R∩R) = {a,b}∪ {a,b,c}, a = b, {a,b} {a,b,c}, and a,b,c are pairwise distinct. LetR∗ be obtained from R by adding the new vertex c and joining c to each neighbor of ain R with an edge, and assume (R∗,(a,b,c),(a,b,c))is arung.
Then b=b,V(R)={a,b,a,c}and E(R)={aa,ac}.
Proof. Notethatif (R∗,(a,b,c),(a,b,c)) isarung oftype(3)–(7) thena andc must have different sets of neighbors in R∗. For otherwise, by checking each of these five types, wesee thatR∗ wouldadmit aseparation (H1,H2) such that|V(H1∩H2)|≤3, {a,b,c}⊆V(H1),and{a,b,c}⊆V(H2).
Hence,sinceaandchavethesamesetofneighborsinR∗,(R∗,(a,b,c),(a,b,c)) isof type(1)or(2).Thus,|V(R∩R)|=|{a,b}∪ {a,b,c}|≤4 and,sinceGis5-connected and R−R=∅,itfollowsthatV(R)={a,b}∪ {a,b,c}.
Suppose (R∗,(a,b,c),(a,b,c)) is of type (2). Then, since c = c, we have a = a and (R∗−a,c,c,b,b) is 3-planar.Hence, cb ∈/E(G) orcb∈/E(G). Thus,{a,b,c}or {a,b,c}wouldbeacutinR∗ separating{a,b,c}from {a,b,c}, acontradiction.
So (R∗,(a,b,c),(a,b,c)) is of type (1). Then b =b, as c∈ {/ a,c}. Since {a,b} {a,b,c},wehavea=a.Hence,sinceR∗hasnoseparationoforderatmost3separating {a,b,c}from{a,b,c},wededucethatE(R)={aa,ac}. 2
Note thattheconclusionofLemma3.3isaspecialcaseof(i) ofthenextlemma.
Lemma 3.4. Let G be a 5-connected nonplanar graph and (R,R) a separation in G such that |V(R)| ≥ 8, V(R ∩R) = {a,b,c}∪ {a,b,c}, {a,b,c} = {a,b,c}, and (R,(a,b,c),(a,b,c)) is a rung. Then for every x ∈ V(R −R), G contains T K5 in which xisnotabranch vertex;orGcontains K4−;or oneof thefollowingholds:
(i) b=b.
(ii) {a,c}={a,c},V(R)={a,c,b,b},andE(R)={bb}.
(iii) V(R)−({a,b,c}∪ {a,b,c})={v}andNG(v)={a,b,c}∪ {a,b,c}, andeither a=a andE(R−v)={bb,cc}or c=c andE(R−v)={bb,aa}.
(iv) {a,b,c}∩ {a,b,c}=∅,V(R)− {a,a,b,b,c,c}={v},NG(v)={a,a,b,b,c,c}, andE(R−v)={aa,bb,cc}.
Proof. Bythedefinitionofarung,Rhasthreedisjointpathsfrom{a,b,c}to{a,b,c}, with one path from b to b. So by the symmetry between a and c and the symmetry between a and c, we may let A,B,C be disjoint paths in R from a,b,c to a,b,c, respectively. First, we consider the case when {a,b,c}∩ {a,b,c} = ∅. If b = b then (i)holds; so wemay assumeb =b.If a=a and c =c then, since Gis 5-connected, V(R) = {a,b,b,c}; so bb ∈ E(R) (because of the paths A,B,C), and we have (ii).
Thus by symmetry between {a,a} and {c,c}, we may assume c = c. Suppose a = a. Then by the definition of a rung, R−a has no disjoint paths from b,c to c,b, respectively.So by Lemma 2.10, (R−a,c,c,b,b) is 3-planar. Since G is 5-connected, R−ais(4,{b,b,c,c})-connected;so(R−a,c,c,b,b) isinfactplanar.If|V(R)|≥7 then GcontainsT K5orK4−(byLemmas2.5and2.2,usingtheseparation(R,R)).IfV(R)= {a,b,b,c,c}then,since(R−a,c,c,b,b) isplanar,either{a,b,c}or{a,b,c}isa3-cut in R separating {a,b,c} from {a,b,c}, contradicting the definition of a rung. Thus, we mayassume |V(R)| = 6 andlet v ∈ V(R)− {a,b,b,c,c}. Since G is 5-connected, NG(v)={a,b,b,c,c}. Therefore, since(R−a,c,c,b,b) isplanar, bc,cb ∈/ E(R). So bb,cc∈E(R),asotherwise{a,v,c}or{a,v,b}wouldbea3-cutinRseparating{a,b,c} from{a,b,c},contradicting thedefinitionofarung.Hence,(iii)holds.
Thus,wemayassumethat{a,b,c}∩ {a,b,c}=∅. Weneedtodealwith(3)–(7)in thedefinitionof arung. Wedealwith (4)–(7)inorder,and treat(3) last(whichis the mostcomplicatedcasewhereweusethedischargingtechnique).
Suppose (4) holds for (R,(a,b,c),(a,b,c)). By symmetry, assume that R has a 1-separation (G1,G2) such that {a,a,b,b} ⊆ V(G1), {c,c} ⊆ V(G2), and (G1,a,a,b,b) is 3-planar. Let V(G1 ∩G2) = {v}. Note that v /∈ {a,b,c,a,b,c}; otherwise, {a,b,c}or {a,b,c} would be cut inR separating{a,b,c}from {a,b,c}. Since G is 5-connected, V(G2) = {v,c,c}. Again, since G is 5-connected, G1 is (5,{a,a,b,b,v})-connected; so (G1,a,a,b,b) isplanar. Moreover, vc,vc,cc ∈E(G);
for otherwise{a,b,c} or {a,b,c}or {a,b,v}would be acut in R separating {a,b,c} from{a,b,c}.If|V(G1)|≥7 thentheassertionfollowsfromLemmas2.5and2.2,using theseparation (G1,G2∪R). Sowe may assume|V(G1)| ≤6. If |V(G1)| = 6 thenlet t∈V(G1)−{a,a,b,b,v};nowNG(t)={a,a,b,b,v}and|(NG(v)−{c,c})∩NG(t)|≥2 (sinceGis5-connected),andhenceR(andthereforeG)containsK4−.Sowemayassume V(G1)={a,a,b,b,v}. Thenva∈E(G);otherwiseNG(v)={a,b,b,c,c}and, hence, ab ∈/ E(G) (as (G1,a,a,b,b) is planar), which implies that {a,b,c} is a cut in R separating{a,b,c}from {a,b,c}, acontradiction.Similarly,va,vb,vb ∈E(G). Then by planarity of (G1,a,a,b,b), wehave ab,ba ∈/ E(G).So aa,bb ∈ E(G) as {b,c,v} and{a,v,c}arenot3-cuts inR separating{a,b,c}from {a,b,c}.Thuswehave(iv).
Suppose(5)holdsfor (R,(a,b,c),(a,b,c)),andassumebysymmetrythat(R,a,a, b,b) is 3-planar, and R has a separation (G1,G2) such that V(G1 ∩G2) = {z,b}, {a,a,b,b} ⊆ V(G1), {c,c} ⊆ V(G2), and (G2,c,c,z,b) is 3-planar. Since G is 5-connected,V(G2)={b,z,c,c}.Thencz,cc ∈E(G) as,otherwise,{a,b,c}or{a,b,z} wouldbea3-cut inRseparating{a,b,c}from {a,b,c}.Hence,since(G2,b,z,c,c) is planar, bc ∈/ E(G). Since (R,a,a,b,b) is 3-planar, (G1,a,a,b,b) is 3-planar. Thus,
