On immersions of uncountable graphs

Journal of Combinatorial Theory,Series B87(2003)130–137/locate/jctbOn immersions of uncountable graphsThomas AndreaeMathematisches Seminar,Universita¨t Hamburg,Bundesstra e55,Hamburg D-20146,GermanyReceived6March2001AbstractIn his paper on well-quasi-ordering infinite trees(Proc.Cambridge Philos.Soc.61(1965) 697),Nash-Williams proposed the conjecture that the class of all graphs(finite or infinite)is well-quasi-ordered by the immersion relation(which is denoted here by p1).In addition,in a subsequent paper,Nash-Williams discussed a weaker version of his original conjecture to the effect that the class of graphs is well-quasi-ordered with respect to a relation p2which, roughly speaking,is obtained by redefining H p1G so that distinct vertices of H can be mapped into the same vertex of G:It is the purpose of the present note to disprove Nash-Williams’two immersion conjectures.r2002Elsevier Science(USA).All rights reserved.Keywords:Immersion relation;Infinite graphs;Well-quasi-ordering;Antichains1.IntroductionFor graph theoreticterminology,we refer to the definitions and notational conventions collected at the end of the introduction or to the textbook of Diestel[1]. The graphs considered in this note do not contain loops or multiple edges.For graphs G;H;we write H p G if G contains a subgraph which is isomorphic to a subdivision of H:A classical result in infinite graph theory states that the class of trees is well-quasi-ordered by the subdivision relation p:Theorem A(Nash-Williams[2]).If T1;T2;y is an infinite sequence of trees,then there exist positive integers i;j such that i o j and T i p T j:The following well-known conjecture of E.Va zsonyi apparently dates back to the 1930s.Forfinite graphs,the truth of the conjecture is an immediate consequence of E-mail address:andreae@math.uni-hamburg.de.0095-8956/02/$-see front matter r2002Elsevier Science(USA).All rights reserved.doi:10.1016/S0095-8956(02)00030-8T.Andreae/Journal of Combinatorial Theory,Series B87(2003)130–137131the graph minor theorem of Robertson and Seymour[4];cf.also[1].In the general case,however,the conjecture remains far from proved to this day,although partial results on the infinite case can be obtained as consequences of results of Thomas[6].Conjecture(Va zsonyi).If G1;G2;y is an infinite sequence of graphs in which every vertex has degree p3;then there exist positive integers i;j such that i o j and G i p G j: Theorem A and Va zsonyi’s conjecture deal with restricted classes of graphs, namely,trees and graphs with maximum degree at most3,respectively.This prompted Nash-Williams[2]to suggest another conjecture for the class of all graphs which,if true,would have both Theorem A and Va zsonyi’s conjecture as corollaries. For a graph G;let PðGÞdenote the set of nontrivial paths of G:For graphs G;H;an immersion j:H-G is a mapping j:VðHÞ,EðHÞ-VðGÞ,PðGÞsuch that(i)if v A VðHÞ;then jðvÞA VðGÞ;(ii)if v;v0A VðHÞ;v a v0;then jðvÞa jðv0Þ;(iii)if e¼vv0A EðHÞ;then jðeÞA PðGÞand the path jðeÞconnects jðvÞwith jðv0Þ; (iv)if e;e0A EðHÞare distinct,then jðeÞand jðe0Þare edge-disjoint and(v)if e¼vv0A EðHÞand v00A VðHÞ;then jðv00Þe VðjðeÞÞ\f jðvÞ;jðv0Þg:Writing H p1G to indicate that there exists an immersion j:H-G;Nash-Williams’conjecture reads as follows.Conjecture A(Nash-Williams[2]).If G1;G2;y is an infinite sequence of graphs,then there exist positive integers i;j such that i o j and G i p1G j:In his paper[3],Nash-Williams subsequently presented a weaker version of Conjecture A which is still strong enough to imply Theorem A and Va zsonyi’s conjecture.Let CðGÞdenote the set of cycles of a graph G:For graphs G;H;an immersion j:H-G in the weak sense is a mapping j:VðHÞ,EðHÞ-VðGÞ,PðGÞ,CðGÞsuch that(i),ðii0Þ;ðiii0Þ;(iv),(v)hold,where (i),(iv)and(v)are as above and the statementsðii0Þ;ðiii0Þare as follows.(ii0)if e¼vv0A EðHÞand jðvÞa jðv0Þ;then jðeÞA PðGÞand the path jðeÞconnects jðvÞwith jðv0Þand(iii0)if e¼vv0A EðHÞand jðvÞ¼jðv0Þ;then jðeÞA CðGÞand jðvÞis a vertex on cycle jðeÞ:Writing H p2G if there exists an immersion j:H-G in the weak sense,the modified version of Conjecture A reads as follows.Conjecture B(Nash-Williams[3]).If G1;G2;y is an infinite sequence of graphs,then there exist positive integers i;j such that i o j and G i p2G j:In[4],Robertson and Seymour announced a proof of Conjecture A for the case of finite graphs G1;G2;y(which,of course,implies the truth of Conjecture B forfinitegraphs).It is the purpose of the present paper to show that,in the general case of arbitrary graphs,counterexamples to Conjectures A and B exist.The question whether or not the conjectures hold for countable graphs remains open.(Actually,the paper also leaves open the question whether Nash-Williams’conjectures hold for graphs of cardinality less than the first limit cardinal greater than the cardinality of the continuum.)The paper is organized as follows.In Section 2,we establish a result which,to some extent,clarifies the relationship between the immersion relations p 1and p 2:we show that for each family of graphs G i ði A I Þthere exists a family of graphs H i ði A I Þsuch that,for all i ;j A I ;G i p 1G j if and only if H i p 2H j (Theorem 1).In particular this means that,in order to disprove Conjecture B,it is enough to find a counterexample to Conjecture A.In Section 3,we modify some of the ideas of Thomas [5]in order to make them compatible with the immersion relation p 1:This,when taken together with Theorem 1of the present note,implies Theorem 2,which states that there exists a sequence H 1;H 2;y of uncountable graphs such that H i 4/2H j for all i ;j with i o j :Theorem 3is a sharpened version of Theorem 2showingthat the sequence H 1;H 2;y can be modified to obtain an antichain H 01;H 02;y ;thatis,H 0i 4/2H 0j holds for all i ;j with i a j :Our terminology is standard.For graph theoreticterms not defined here,we refer to Diestel [1].The vertex set and edge set of a graph G are denoted by V ðG Þand E ðG Þ;respectively.If e is an edge joining vertices v ;w ;then e is denoted by vw :For v A V ðG Þ;the degree of v in G is denoted by d G ðv Þ:An edge is said to be pendant if one of its ends has degree one.A family of graphs G i ði A I Þis edge-disjoint if E ðG i Þ-E ðG j Þ¼|for all i ;j A I with i a j :A path is a graph P consisting of n þ1distinct vertices a 0;y ;a n ðn X 0Þand edges a i a i þ1ði ¼0;y ;n À1Þ:For paths,we use notations like P ¼ða 0;y ;a n Þ:A path is trivial if it consists of just one vertex.For a path P ¼ða 0;y ;a n Þ;the vertices a 0;a n are called endvertices of P :A path P is an x ;y -path (or likewise P is said to connect x with y )if x and y are its endvertices.In order to stress the difference of the above defined notions of an ‘immersion’and an ‘immersion in the weak sense’,the former will also be referred to as an immersion in the strong sense .A binary relation %on a set Q is a quasi -order if it is reflexive and transitive.We say that Q is well -quasi -ordered by %if the relation %is a quasi-order on Q and if for each infinite sequence q 1;q 2;y of elements of Q there exist positive integers i ;j such that i o j and q i %q j :Given a quasi-order %on Q ;a family q i ði A I Þof elements of Q is called an antichain (with respect to %)if q i I q j for all i ;j A I with i a j :The set f 0;1;y g of natural numbers is denoted o :The symbol c denotes the cardinality of the continuum.Cardinals are identified with their initial ordinals.2.A result linking the two versions of the immersion relationIn this section,we establish a result (Theorem 1)on the relations p 1and p 2which in particular shows that,in order to disprove Conjecture B,it is enough to findT.Andreae /Journal of Combinatorial Theory,Series B 87(2003)130–137132a counterexample to Conjecture A.Most likely,Theorem1can also be applied in other situations to reduce a problem for p2to the analogous problem for p1: Theorem 1.For each family of graphs G iði A IÞthere exists a family of graphs H iði A IÞsuch that,for all i;j;A I;G i p1G j if and only if H i p2H j:In particular,G iði A IÞis an antichain with respect to p1if and only if H iði A IÞis an antichain with respect to p2:Proof.Let G iði A IÞbe a family of graphs.We may assume that G i-G j¼|for all i;j A I;i a j:Let m;a;b;g be cardinals such thato p m o a o b o g and d GiðvÞo m for all i A I and v A VðG iÞ:ð1ÞFor the purpose of defining the graphs H i;we introduce a graph F as follows.Let VðFÞbe the union of disjoint sets X0;X1;y;X5where X0consists of three elements a;b;c and X1;y;X5are infinite sets with j X1j¼m;j X2j¼j X3j¼a;j X4j¼b;j X5j¼g:Let further x:X2-X3be a bijection and define the edge set of F by EðFÞ¼f ax:x A X1,X2,X3g,f bx:x A X1,X4,f c gg,f cx:x A X5g,f xx0:x A X2;x0¼xðxÞg:The graph F is displayed in Fig.1.In particular,observe that in F there are*m edge-disjoint paths of length two connecting a with b;*a edge-disjoint triangles attached at a;*b pendant edges attached at b and*g pendant edges attached at c:Let V¼SVðG iÞ;where the union is taken over all i A I:For each v A V;let F v be acopy of F such that F v-F w¼|whenever v a w and such that F v-G j¼|for all v A V and j A I:For each v A V;let f v:F-F v be an isomorphism and put a v:¼f vðaÞ;b v:¼f vðbÞ;c v:¼f vðcÞ:For each i A I;let H i be the graph that results from G i and the graphs F vðv A VðG iÞÞby identifying each vertex v A G i with vertex a v A F v:Since the so-defined graphs H i are disjoint,we may write dðxÞrather than d Hi ðxÞto denote the degree of a vertex x in H i:For each i A I;we put A i:¼VðG iÞð¼Fig.1.The graph F occurring in the proof of Theorem1.T.Andreae/Journal of Combinatorial Theory,Series B87(2003)130–137133f a v :v A V ðG i ÞgÞ;B i :¼f b v :v A V ðG i Þg ;and C i :¼f c v :v A V ðG i Þg :Further,we write x B y to indicate that x and y are vertices of the same graph F v :Clearly,if G i p 1G j then H i p 1H j and thus H i p 2H j :For the proof of the converse,let i ;j A I and assume that h :H i -H j is an immersion in the weak sense.Let g :G i -H j denote the restriction of h to G i :Clearly,Theorem 1is proved if we show thatg is an immersion G i -G j in the strong sense :ð2ÞFor the proof of (2),first observe thath ðc ÞA C j for all c A C i :ð3ÞIndeed,as a consequence of the construction of the graphs H i and H j ;together with assumption (1),one obtains d ðc Þ¼g 4d ðv Þfor all c A C i and all v A V ðH j Þ\C j :Hence(3).For similar reasons,we have h ðb ÞA B j ,C j for all b A B i :However,for b A B i ;h ðb ÞA C j is impossible since there is no cycle of H j passing through a vertex of C j ;while each vertex of B i is on a cycle of H i :Henceh ðb ÞA B j for all b A B i :ð4ÞNow,let a A A i :Note that there are a edge-disjoint cycles passing through a and,consequently,the same must be true for h ða Þ:From this,together with the fact that a 4m ;one concludes that h ða ÞA A j ;and thus we have found thath ða ÞA A j for all a A A i :ð5ÞWe next show that the following holds.Let a A A i ;c A C i with a B c :Then h ða ÞB h ðc Þ:ð6ÞFor the proof,consider b A B i with a B b B c :We first show h ða ÞB h ðb Þ:Suppose that h ða ÞB h ðb Þdoes not hold.Note that there exists a family ðP n Þn o m of m edge-disjoint paths of H j connecting h ða Þwith h ðb Þsince there are m such paths of H i connecting a with b :For each n o m ;let e n be the edge of P n which is incident with h ða Þ:From the supposition that h ða ÞB h ðb Þdoes not hold,together with (5),it follows that all edges e n are in G j and thus we have d G j ðh ða ÞÞX m ;which contradicts (1).Hence h ða ÞB h ðb Þ:Thus in order to obtain (6)it remains to show h ðb ÞB h ðc Þ:Suppose that this does not hold.Then one concludes from (4)and (5),together with h ða ÞB h ðb Þ;that the path h ðbc Þcontains h ða Þas an inner vertex,which is impossible.Hence h ðb ÞB h ðc Þ:We claim thath ðc Þa h ðc 0Þfor all c ;c 0A C i ;c a c 0:ð7ÞSuppose h ðc Þ¼h ðc 0Þfor distinct c ;c 0A C i :By (3),we have h ðc ÞA C j :Denote by D the component of H j Àh ðc Þcontaining the vertices of B j ;let e be the unique edge connecting h ðc Þwith D :Let b ;b 0A B i with b B c ;b 0B c 0:By (4)we have h ðb Þ;h ðb 0ÞA D and thus we conclude from h ðc Þ¼h ðc 0Þthat e is an edge of both h ðcb Þand h ðc 0b 0Þ;which is impossible.Hence (7).As a consequence of (3),(5)–(7)one obtainsh ða Þa h ða 0Þfor all a ;a 0A A i ;a a a 0:ð8ÞT.Andreae /Journal of Combinatorial Theory,Series B 87(2003)130–137134By(8),g is an immersion G i-H j in the strong sense.Further,by(5),g maps the vertices of G j onto vertices of G j:Moreover,note that it follows from the construction of H j that a path of H j is completely contained in G j if its endvertices are in G j:Hence(2).&3.Counterexamples to Conjectures A and BIn this section,based on ideas of[5],we construct a counterexample to Conjecture A.The existence of a counterexample to Conjecture B then follows from Theorem1. For an ordinal a;2a denotes the set of mappings x:a-f0;1g:We put2o a¼Sb o a2b:If x A2a and b p a;then x p b A2b denotes the restriction of x to b:For x0A2o and n A o;we putU nðx0Þ¼f x A2o:x p n¼x0p n g:The set2o is considered as a topological space endowed with the product topology, i.e.the topology whose basicopen sets are the U nðxÞ:Let k0;k1;y be cardinals with c p k0o k1o?.For each x A2o and n A o;let A x;n be a set of cardinality k n such that A x;n-2o oþ1¼|and A x;n-A y;m¼|whenever ðx;nÞaðy;mÞ:For each X D2o;we define a graph G X byVðG XÞ:¼f x p a:x A X;a A oþ1g,[x A Xn A oA x;n;EðG XÞ:¼[x A Xn A of av:a A A x;n;v A f x;x p n gg:For X D2o and v A VðG XÞ;we denote by d XðvÞthe degree of v in G X:As a consequence of the definitions,one obtains d XðvÞ¼sup f k i:i A o g if v A X;d XðvÞ¼j U nðxÞ-X jÁk n¼k n if v¼x p n for x A X and n A o;and d XðvÞ¼2;otherwise. Hence,for X;Y D2o;we haved XðxÞ4d YðvÞfor all x A X and v A VðG YÞ\Y:ð9ÞLemma1.For X D2o;let x;x0be distinct elements of X:Let further n A o:Then there exists a system of k n edge-disjoint x;x0-paths of G X if and only if x A U nðx0Þ: Proof.Assume x A U nðx0Þ:Then x0p n¼x p n:Let f:A x0;n-A x;n be a bijection.Foreach a A A x0;n;we define a path P a of length four by putting P a¼ðx0;a;x0p n¼x p n;fðaÞ;xÞ:Then these paths form a system of k n edge-disjoint x;x0-paths of G X: For a proof of the‘only if’direction,assume that x e U nðx0Þ:Let P be an x;x0-path of G X:From the definition of G X;together with the fact that x e U nðx0Þ;one readily obtains that P must contain a subpath P0¼ðy;a;v;b;zÞwith y A X\U nðx0Þ; z A U nðx0Þ;v¼y p m¼z p m;a A A y;m;b A A z;m for some m A o:Because y e U nðx0Þand z A U nðx0Þ;we have y p n a z p n:Hence m o n and thus we have proved that each x;x0-path P of G X must pass through a vertex v with v¼y p m for some y A X and T.Andreae/Journal of Combinatorial Theory,Series B87(2003)130–137135m o n:Note that there are onlyfinitely many vertices of this kind and each such vertex has degree at most k nÀ1:Thus there cannot exist k n edge-disjoint x;x0-paths of G X:&Lemma2.For X;Y D2o;let g:G X-G Y be an immersion in the strong sense and denote by j the restriction of g to X:Then j is a continuous injective function X-Y: Proof.By(9),we have jðXÞD Y:Clearly j is injective and thus it remains to show that j is continuous.For this purpose,let x0A X:For some n A o;consider the basic open neighborhood U nðjðx0ÞÞ:We have to show that there exists a natural number a n such thatjðU anðx0Þ-XÞD U nðjðx0ÞÞ:ð10ÞWe show that this is true for a n¼n:Let x A U nðx0Þ-X:If x¼x0;then jðxÞA U nðj x0ÞÞclearly holds.Hence let x a x0:By Lemma1,x0and x are joined by k n edge-disjoint paths of G X;and thus the same must hold for jðx0Þand jðxÞin G Y:Consequently,again by Lemma1,jðxÞA U nðjðx0ÞÞ:Hence(10).&By a result of Thomas[5,Theorem2]there exists a sequence X1;X2;y of subsets of2o(each of cardinality c)such that for all i;j with i o j there is no continuous injective function X i-X j:By combining this with Theorem1and Lemma2of the present paper,we obtain the following result.Theorem 2.There exists a sequence H1;H2;y of uncountable graphs such that H i4/2H j for all positive integers i;j with i o j:Proof.Let X1;X2;y be as given above.Put G i:¼G Xi ði¼1;2;yÞ:Then,byLemma2,we have G i4/1G j for all i;j with i o j and thus Theorem2follows by application of Theorem1.(It immediately follows from the proof of Theorem1that the resulting graphs H1;H2;y are uncountable.)&4.AntichainsThe following is a sharpened version of Theorem2.Theorem 3.There exists a sequence H01;H02;y of uncountable graphs such thatH0i 4/2H0jfor all positive integers i;j with i a j:Proof.Let G iði¼1;2;yÞbe as in the proof of Theorem2.Then G i4/1G j for all i;j with i o j:Note that a graph G X(as defined in Section3)is infinite and connected if X is non-empty.Consequently,the G i are infinite connected graphs.Let l1o l2y be cardinals with l14j VðG iÞjði¼1;2;yÞ:For i¼1;2;y;let G0i result from G i byadding l i new vertices of degree0.Then j VðG0j Þj o j VðG0iÞj for all j;i with j o i;andT.Andreae/Journal of Combinatorial Theory,Series B87(2003)130–137 136thus G0ip1G0j is impossible if j o i:On the other hand,G0i p1G0j is also impossible if i o j since this clearly would imply G i p1G j:Hence G0i4/1G0j for all i;j with i a j: Theorem3then follows from Theorem1.(By the proof of Theorem1,the resulting graphs H01;H02;y are uncountable.)&Note that the above proof of Theorem3yields disconnected graphs H01;H02;y.Inorder to obtain connected graphs showing the same,one can proceed as in the proofof Theorem3with just one slight modification when defining G0i :just pick one of thel i new vertices and join it by edges to all other vertices of G0i :Let a i A VðG0iÞbe thisvertex which now has degree l iði¼1;2;yÞ:Then any immersion G0i-G0j(in the strong sense)would map a i to a j(because of their degrees),and hence G i to G j:But the former is impossible for i4j;while the latter is impossible for i o j:Application ofTheorem1then yields the desired connected graphs H01;H02;y.(Note that,by theproof of Theorem1,the connectedness of the graphs H0i is an immediateconsequence of the connectedness of the graphs G0i:)References[1]R.Diestel,Graph Theory,Springer,New York,2000.[2]C.St.J.A.Nash-Williams,On well-quasi-ordering infinite trees,Proc.Cambridge Philos.Soc.61(1965)697–720.[3]C.St.J.A.Nash-Williams,A glance at graph theory—Part II,Bull.London Math.Soc.14(1982)294–328.[4]N.Robertson,P.D.Seymour,Graph Minors XX.Wagner’s conjecture,preprint.[5]R.Thomas,A counter-example to‘Wagner’s conjecture’for infinite graphs,Math.Proc.CambridgePhilos.Soc.103(1988)55–57.[6]R.Thomas,Well-quasi-ordering infinite graphs with forbiddenfinite planar minor,Trans.Amer.Math.Soc.312(1989)279–313.T.Andreae/Journal of Combinatorial Theory,Series B87(2003)130–137137。