On a problem of R. Halin concerning infinite graphs

Discrete Mathematics 23 (1978) l-15. @ North-Holland Publishing Gmpany

ON A PROBLEM GRAPHS

OF R. HALIN

CONCIIERNI[NG INFINI’lF1G

Thomas ANDREAF §eminar der Universitiit Hamburg, 55, Federal Repddic of Germany

Mafhemafisches

2000

Hamburg

13, Bundesstrasse

Received 17 February 1977 Revised 28 June 1977 Let a be an arbitrary cardinal f 0 and let A be the graph that arises from u disjoint one-way infinite paths by identifying their initial vertices. For this graph A we shcw the following. (*) Let C be an arbitrary graph which contains for every positive integer n a system of n disjoint graphs each isomorphic to A; then G also contains infinitely ma1.y disjoint subgraphs each isomorphic to A. ‘L&Ssharpens two theorems of Halin, who proved the conesponding r~ ult for the case that A is a one-w%y or two-way infinite path. Furthermore we show that (*) holds, if A is an arGIualJ countable tree with a finite diameter.

In [7] the following problem has been posed: (1) Let A ire an arbitrary graph and assume that a graph G contains for every positive integer t2 a system of 12 disjoint graphs each isomorphic to A. Does then G necessarily contain infinitely many disjoint ismorphic copies of A ? In [Sj and [6] this question has been answered affirmatively, if A is a one-way or two-way infinite path. In Cl], [9] and [ 11) it was independently shown by counrerexamples that the answer to the above question is negatk. The papers [I], [2] and [7] deal with the folloCng analogue of (1) : (2) Let A be an arbitrary graph and assume that a graph G contains for every posifiit: integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Does then G necessarily contain infinitely many disjoint subgraphs each isomorphic to ;1 subdivision of A? In [7] this question was answered affirmatively, if A is a graph, such that a subdivision of A is isomorphic to a subgraph of S, where S is the graph of Fig. 1. This class of graphs includes every tree in which each vertex has degree nort greater than 3. This result was sharpened in [2] by showing that the above result holds for all locally finite trees. Furthermore in [l J an examDIe ws given showing that the answer to (2) is negative in the general case of an arbitrary graph A. In addition, it is easy to see th,at the answer to (1) or (2) is afirmatke, it -4 is a finite graph (see [7]). The purpose of the present paper is to prove the l:->llowing two theorem\ concerning problem (1): Let Q be an aabitrar!~ carc’lirral f 0 md let 14 Est.the gr:ll’h

The g.tzphs fansidered in this paper are undirected and do not Cc;r,lait!loops or multiple edges. By V(G) and E(G) we denatethe set of vertices and edges of the graph &;, r~~ctiveIy. A pa& (from 21~to u, , of Iength p1(reE NO)‘9is a &graph haloing ZEN%@12+ 1 different vertices .uo, . . . , dn and it edges e, = (t)i,U& IZ- 1). Let A, -B be graphs; if P = (Q, . . . p v,) is a path, such that (i=O,..., P 1”1 A = q, and P fl B = a,, then P is called a (A, R)-pat)i. A ON-way infinite path (kiefiy: P-p&z) is a graph that consists of a sequence flu,.*Jo,v2, . . . of different vertices and the edges ei = (Ok,Ui+r)(i = &I, 2, . . .). If P = (u,, ul, . . .) is a l-path, a gath of the form fu,, . . . , u,,) is ealIed a commencement of P and a l-path of the form (u,, tt,+1, &+2, .) is raIled a rest of P. TWOl-paths Uz%U, in a graph G are r=aIIedeqtliuaknt in G, if and only if there is a third l-path V in G, such that V and U; have infiniteIy many vertices in ‘common (i = 1,2). In this way a:equivalence re.ation is defined on the set of l-paths of G (see [4]). T) t corresponding equivalence &asses are called ends of G, Let @ be an end of G: ! ml(@) we denote the maximt~m number of disjoint l-paths of 65. (The existence c A~ ~n~f@)was shown in [6].) Let A be a tree (i.e. a connec.ted graph without cycles); for w E V(A) the pul: (A, w) is &If a rooted trpc and w is Ned the mot of A. Let (A, w) be a rcmkd tree and B a subtree uf A. Let t, be the vertex af B that has minimum distance Then v) caUed l

l

is called cozuztczbi~~~ ;J i V(G)}

’ is

isomorphic ts -4, we call A’ 21CQ~Yof /$. We write A g B, if thet*e is a imno ic to A. IIf (A, o) and (S, bj are mated trees and if there

A problem concemint:

infinite graphs

3

is an isomorphic embedding from A into G mapping a on b, then we shall write (A, a) E r (B, b) (or briefly: A z ,B). We say that we subdivide an edge (u, U) of a graph G when we delete (u, U) from E(G) and insert a new vertex w and two new edges (u, w) and (w, v) in G instead of (u, v). G’ is called a subdivision of G, if and omy if G’ may be derived from G by one or morF: successive subdivisions of edges. For graphs A, B we write A c;l:B, if there is a subdivision of A isomorphic to a subgraph of B. Let A be a graph and n a cardinal. By nA we denote the graph that consists of n components each isomorphic to A., The problem treated in this paper can then be re-formulated in the following way: Does n/“i E G for every n EN imply &A c;-;‘r 2. A thearem 021 o-stars De&&ion 2.1. For a cardinal air 0 let A be the graph that arises from a disjoint l-paths by identifying their initial vertices. Then A is called the a-star. For a > 2, ct E V(A) is called main vertex of A, if ~,&a) -- a ; the I-pat’hs of A beginning in a are called main paths of A. Definition 2.2, For n = 1,2, . . . let ‘?I, be a set of n disjoint graphs. Let % = iJEsl 3,. We call ‘5%’G ‘3 a cornplere subsystem of ?I, if for every k E N there is a nk EN such that 1%’n a,,, 13 k. Otherwise %’ is cai‘led incompkte subsystem of 91. Theorem 2.3. For an arbitrary cardinal a # 0 let ‘4 be the a-star and let G be an arbitrary graph, such that G 2 nA for every n EN

Then G 2K,A.

First we shall prove some lemmas. 1 ezma 2.4 will be used to prote Theorem 2.3 for the cases a=&-, and a>rS,. Lemmas 2.5, :!.6 and 2.7 prepare the proof of Theorem 2.3 in the case of CI=&,. hnma

2.4. Let A and G be arbitrary graphs such that nA c G jar every n E N, and let f be a fink subgraph of A. Then fop every n E N one can chose disjokt copies A(nVmj(m = 2, . . . , n) of A in G orzd isomorphisms cp(n*n”: A - A(“*‘“)sr~h that the following holds:

Proof. For every t EIS let us choose arbitrary s,:,tems ‘$1~“’ of t disjoint copiz of A in G. Let ‘?I(“): = UT=, ‘%ja). Further, we choose for every A’ E ?I”“ z certain isomorphism qA’ from A to A’.

Y.mmm 2.5. Let S’ be a subdiu&m of §, whereS is thegraph shown in Fig. 1. For fixed II E X let J$ GG V(S’)(i = 1, . . . , n) be n disjoint S+itc sets. Then there is a dis@int fatMy Vki(k = 1929 of l-~aiksin S’, such that the initial Ve?@X 49fV& iSan ekment QfBie

Fig. I.

Qbvbu~ly ti;lc Mowing wndit ions; (I) and (II) ho!d; %ere is CCsubgraph D of S’ which is isomorphic to a subdivision of the ww (i.e., ehe tree which is re@ar of degree 3). r v?ry 1 = 1, . r , ) n thaz is :‘a in%& suoset ily

‘ .“iZT

d

4 ?c,c ‘i

=I,2 . . . . I,...,n

4 problem concerning itrtin,&graphs

5

(III) Let F:=DlJtJ~,“~ ).. . *. WL , ; then F is a IocalIy finite, iufinite tree. Hence bySatzSin[4]foreveryi=l,..., n the following holds: There is a l-path Pi in F and an infinite subset By = (r>&: k = 1,X _. .}c B: and I! .-qrrespon&,?g sequence of disjoint finite, paths WE,i(ic = 1,2, . . .) such that ‘&‘z,iis a (bE,i, Pi)-path. Foreveq i=I,..., n we chc~losean i&nit{: set Vi of djG;jSint l-paths in F, such that (1) and (2) holds (it is easy to see that this choice :i: possible,. (1) For every P E Qi tile initial -vertex of P is the OC~!Ivertex P and Pi have iI1 common. (2) For every PE!@i and every j(l~j~~) P and ;,;-ivt: only a fink number of vertices in common. Let .

u

T:=.PjU

‘YyLiU u P

keN

(i=l,...,nj

-SPi

(N) ~‘OWwe are ready to define the 1 -paths V,,i( k = 1,2, . . . ; i = 1, . . . , n): (1) Choose the l-path VI I as subgraph of & such that & 1 is the initial verkx of T/z,I, and such that Vl,1 has a rest in common with an arbitrary l-path of 3,. (2) Let (k, i) # (1,l). Let us assume that the l-paths V,, have already been defined for every pair (v, p) < (k, i) (with respect to the lexicographic order), such that the following conditions hold: V,,, is a subgraph of F; the V;,, are pairwise disjoint; the initial vertex of VP+ is element of Bi; V,,, has a rest with a l-path of @P in common. These V_, 1om 3 finite system of n-paths, e&l of which has no KS’ .kGi Pi in common (becauar of (2)); nence: There k a be B’_: and a l-path I’._,, which is subgraph of IF,,such that k is the ;mtial vertex of Vk,i and that Vk,i has a rest with a l-path of vi in common, and such that V~,@r! V& = 6 hoIds for every (v, P) < (k, i). This compietes the proof. Lemma

2.6. Let A be the &,-star. Let G be a graph corztaining n-b 1 copies . . ., Al, A,+1 ofA (rvzNo) such that Al,. . . , A, are disjoint. For the main wrtices

# a,(i ==1, . . . , n). Then G also cont(Jins n + 1 of A such that ai is the main vertex of /I,Ci=

Qi of A,(i =: 1,. . . , .‘2+ 1) let a,,,

disjoint copies A,, . . . , AntI 1 , * > n + I). l

l

Proof. The proof is carried out by induction. For n = 0 Lemma 2.6 is trivial. !,G l’iE !+Iand let ~6 assume that the assertion of Lemma 2.6 is true for el~ery ?I’,< ‘” Let 83, =(a/, 1, Vi,~,. . .) be the set of mairl paths of A,. Without loss clr’rlr:!zr-i-lii we can assume that (I) and (II) hold: can be obtz+ir..tc’!by dro,np;ng a tir jte (I) Ui+ number of paths of U :z: ‘%?J (II) For every i = 1, , . . , n there ;:e r-0 infinite subsets ‘II: ijrrtf S,‘, I of q:, anti L?J at VT’ >h’=bj for cveiy J/E23: aT2t.i n+1: respectively , s otherwise one sniy has to replace Ai by an appropriate subgrapb A I and A,, r 1b~f A :, +1 and apply the induction hypothesis.) \‘(Aj)(l

G

i,

is

0

+

1;

i#i).

(Tk.js

(1) Let @~:=%$=O, !8&,:=& and %$,:=Bk+l. (2) Assume that for rn&&, @?z={vi,. .:, ~JE%?& Bz={q,. . . , Wz}c &,,, a, G 8!, and F$r, E %L+, have already been defined such that (i), (ii) and [iii) h&k

<%i) Vgn W==flfiY=$3

for every kN(l
and for every Wd@.,,

and v 4, Let us choose

wl*), w’;“) E sm\%$ from different ends of H (possible because of (*I)* it follows tht for a p E {1,2} there is an infEte set @m+l~$3, such that W!J% V= $3for every VE i-8,+I. (For otherwise %+jm)and vv(2”)would be members of the same end of H.) Let Wz+l := Wpm). By a similar argument we obtain &J*m+l~&,+land 583m+lG@& such that @%m+,l = 00ant” Vm+ln W = @ for every WE 8!;m+1. This completes ‘he proof of (IIIj. Remark tu (III): Application of Katz 4’ in [S] gives: There is a subgraph of H ix3morphic to a subdivision of S (see Fig. 1). Let (5 be the set of subdivisions S’ sf S contained in G satisfying 4~~ ,I V(S’) for i = 1, . . . , n + I. Deleting a finite number of vertices from an arbitrary subdivision of S there is stilI another subdivision o,f S contained in the remaining graph; hence there is even a subgraph S, of H suela that SHE @. (IV) Let us distinguish the followtig cases: Guse 1. Fix every S’ E (5 and almost every V E I,Jr$! &, S’ fl W= 0 holds. &se 2. There is an S’ F=@5such t!~t S’ f3 V# @for infinitely many V E U r.2: 23,. Case 2a. 9 n Vf B for infkiitel~ many V%s13,+,. C~S~T 2b. S’n Vf 0 for infinitely naany V’E Id rzl Bia case 1. Iu! (1) and (2) we shall dc fine a family (S,,,)~Z~:::;~+,, Sk,iE (5, and a corresponding family (Pk,i)~Z~:-_:;~+T of finite paths, such that (i) and (ii) hold: (3 Pki IS ik (ai, S&-path. r L&i and a, for i= j k# 1. roof of Lemma 2.6, Case 1.) d subgraph S,,l of U vEra1VU U WEsZ,+l W,

-1.

that vn S1,1 #@

f?j

Let

defined

(k. i)t

for every

W.,

Tuch that

ordc~, of the

txkis

b’n&.,

;r‘dl:U!!t:.

VcPl;

to t lllj

by the Remark

Pk., we chwnc

U’. A\

to

:.

of ;I of \’

illlb that thcrc is an ii, ; 2 uhizh

nbow

;I witahlc

;.

there is an S, , c 2

U wFw..i H’ This l:nplics the ,:.:ktcnic P 0 WC C~OO?;Ca suit.tblc L‘OITI~ICI’L~WN

in lJvce,, VU l_jw,n,.,

n W#e. For

Case 2a: In (1)

tor cvcry

$1:~ Yl, and dl:.! g Y,.

in ._I Yaul, VG

for Pt,. If i = n + I. It ft Ilo*~ is contained

scecs91; c Yt,.

the case i# n . 1 By the Remark

VE~U: such th;t

that s,.,

arc mfirlitc

Vnl=-{a,.a,,h.,}=-ti

is contained

I-path

and S, ,, hrtsx .~Iw~dy bvcn

P_ is finite. and bcausc US,.,). San- U~r.~~Otk.,,

of case 1. there

Fimt WC amsidcr which

that P,,,

.:.. I). where **< *’ means the Icxicographk

pair (v, p

Let F= Uwa.r, hypothesis

and let u., z~swnc

(1, II

thcrc

is a path

c~~~rnrnc’nccmcnt of

and (2) we shall Jc.iinc a family I I’,

jL_;’

\C’+ $i.,

wch

\\'

.;. I of finite path\ in

G, such that (i) (ii)

PL,

k a (a,. S’bpath. and Pk., n P,, - ), for i + 1 and u, for I I 1 k t 1.

Because Lemma pr90f

of S’CL G it f~~llows that cvcrv I&, ha: a length -. 1. lirnL;c WC wn . II t 1 t. \b h:c h cmrplctc~ V On: ,: _, t’r Y, )II 1,

2.F to B,:=

0f Lcmm:i

2.6 for the ~89~ 2a

q);~ib

thtr

3y ap#ication of Leuiiuna 2.5 to Bi = V(S) n UC=1 V(.&,i)(i = m, II , n + 1) we get disjoint #pies A,, . . . 9 ,&,_L1 of A having Ui(i = m, . . . , n -t-1) as their ~&ah vert,ie~. Becaus of (ii), Aj is zgograph of S’ “J Ai J’o~:e>EY i = m, L/. . , 12. Hence, k@ilS42 Of (*), them5 iS %u1 hf?mite Set %iG%jQ=l,* a) m-1) SUChtkai. &X /ii:= u weaf V the fdbwbg -&MS: Ai n Aj z fl for WM~ j ==r1, . . s , in - 1 ti~nd n. Now we .only have to apply the induction hypothesis to i=rn,..., and G’:= U l
l

l

-t of disj&t I-paths in a graph G bea@& arr, infipriteset Of disj&nt Y-p&i3 in G. Then there are V,, . . . , V, in the graph U vem VU u SrvemW and there is a k EN such

-2#7. L,et23=(V,,...,V,} and let 23 = {WI1 W,, . . .} be l-paths Fht

(i) and

(ii) bid: (i) lS =(U1, . . . , V,, Wk+l, Wkf2,. . .} is a set of disjoint l-paths; (ii) V, and V, have the same initial vertex, s = 1, . . , t.

#%&-+ ‘The pm~f is by induction ORt. For t = 0 Lemma 2.7 is trivial. Let t E N and fet us assume that the assertion of Lemma 2.7 is true for t - 1. If there is a Vs, E 23 and Gn no4SJ such that for every n > no, I(.$ Wn = 9 h&is, me .Ay kw to apply the induction hypothesis to %I’:= (V’ : 1 s s s t, s f so) and $8’:= . . .}. Thus we can assume that every V, h?d a vertex in common w ?I&1qtW&).&z, with an fntite number of I-paths of !@. Let q be the initial. vertex of V, (z = a,, . . . , t). ‘For every s (1 s s s t) we choose a commencement c of V, such that there are exactly t l-paths of !I& let us say M@vri(r= 1, + . . , t), having a verte:: in ~m.~on with c. Let H:== !J;al (qu u:=x wqs*‘)).Let pr” be the graph that arises from H tiv addim; a new vertex x and the edges (x, vsj(s = 1, . . . , t> to If. We choose any f-l Vertices Xl,.. . , xt+ E V(H). Since there is a c0 and a w80*ro)Z such that r,& VfU w(*~@(i= 1, . . . , t - l), it follows that x is vertex of an infinite componcnt Of WA(Xi:i=l,..., t- 1). By Satz 3 in [3] it follows that there are t l-paths *,. . . .. vf in w such that x is the initial vertex of fl (s = 1, . . . , t) and yr:n G=x (INS, 1~s t; s# r). To complete the proof one only has to take U$=: G --II (s= 1,. . . , t). rem 2.3. Case 1: a c rSb. ‘ihcn Theorem 2.3 of the present paper is the i,nmediatp, consequence of the msk, tkeorem of [2]. Cave 2: Q= X0. (I) Let u be the main *wtex of A. By application of Lemma 2.4 to the case a = a we get: For every n EK the n disjoint coTlies A r&ml , (m = 1,. . . , n) of A in G call kLt choser.1 such that the following holds: a’n*m9& A’q*p’(n+ q), where a(n*m)is the main vertex of A(“*? (IJ) “r’or rzEN let Sn be the grallh that arises from n disjoint paths 6?“),:rp 1; ) . . . ) 5, Ji zT:1, . . . , n) by ide itifying their initial vertices Ui,Q( i= 1

b-33 m, T-2 3 -3)subgrapS:s L!!wpn)

A problem conci?ming infinile graphs

of (3

9

such that (i), (ii) and (iii) hold:

(i) s nvm)is isomorphic to S,; (ii) S(:sW n ,

p-2)

=

fj

(Q

p

mz);

(iii) 2Frn) is subgraph of §(n+l*‘? It f0l10ws that UEfl U L =1 Sfn-“‘)is isomorphic to NoA. (111) (1) Let S(3*mrbe iz s;r,bgraph of A(3vm) isomlarphic #%m) = AO.m)(m = 1 2 31

:~o S3. Further,

1st

((2)For a certain n‘>‘3 M lit us assume that the graphs S(n*m)(m -- 1,. . . , n) have already been defined as %a, i nt srrbgraphs of G each isomorphic to S,, such that there are disjoint anditions:

copi, *) ,P(n*m)(m = 1, . . . , n) of A sa:isfyirrg the following

(*) S(npm)is subgraph OI D;n*fn)(in = 1, . . . ., 0 j. There is an NE i :mch that U k= 1 D(n-m) is s&graph

(* *)

Ifi

i,

of

A(w)_

1*=1&L=l Let

d(nsnl)

be the main vertex of D(n*m) and %, be the scat of main paths E(t~nEG(“~‘n))% 3J, (5,: =23m\a M’ of WTrn) (m = 1, . . . , n). B, : = (vE23,: 5’ n.m) .. -- U vea, V an6 IPm) : = U VEs, V. For q > N we chose a certain A(q*P),such that A(qvp’n U k =1 S’n*“‘)= $4(possit!: #since IJ $ = I S(n*m)finite!). By .application qf Lemma 2.6 to G’ : = U E,_ Tfi(n*m)U Atq*P)it follows: In G’ th$:re are n + 1 disjoint copies I)(‘l+l*m)(rn = 1, . . . : PZ+ 1) of A, such that darn) is msin vertex of 6(nt1,mJ (rn =I I, . . . , n) and a(q*p’is main vertex of O(n+l*n+l). Consider G”:= IJcT$ !%n+y*m)U UkzI fi(n*m). Let $3 ={V,. . . . , V,) be the set of l-paths in ‘J k -_1 @m) h,a.ving exactly their initial vertex in c;timmon with n I4$ . .) be the set of main paths of the graphs U m=l S(Gm! Let {c, #n+l**s; im = 1, , . , , vz+ 1). Let Wj be the l-path obtained from fl by deleting the initial verte,:: (i = 1,2, . L .) and let !@ - ( Wi : i E W). We apply Lemma 2.7 to 8 and !8; thus we get l.I = {cl, . . . , U,, Wtlfl? Wk+*, . . .) as described in Lemma 2.7. Because of (* *), since q XV and because of the choice of the copies of A in I, a(4.~’ g u ;, = 1 DC-m) holds, al?4 consequently u(~*~)& U,( s = 1, . . . , t). T>et D : -= As an immzdiate consequence of the CO:;lJ”,=, s (n*m)U U fsl U, U (_ITs , ~+i. struction of D we have: D conf-kts of r~.+ 1 compont=lats D(n-+i.m~(no = 1, . . . , n + 1 J satisfying t IL (2. (3) (for a suitable choice of the indices): (n+l.m) IS * isomorphic to A(m .= 1,. . Q -+ 1); (1) D tn+l.m) to S,, t ,, such that contains a subgraph S(‘I+i*m)that is isomorphic (2) fi S(n*m) is subgraph of S(n+l*m); such that D is subgrap+ of (_I r’ , iJ ], 1 A’ ’ ” ’ I ~h~)cv c (3) There is an N’c: N’ = &I. T&is (‘C”

Z&e 3: u > &. Application of Lemma 2.4 gives: For every n GN the n djsjoint (I~Z (9)) copies A(Rm’(m = 1, . . . , n) of A in G can be so chosen that u(~*~)ti A.(q@) is the main vertex of A(“-“). WC shail show: whxe dzfGm) (*) For evergt vr,m 4!V (n 2 m) there is a subgraph kcrPL) of A(“rm), &n,m) isomorph& to A, such that J@*‘)n &%“@‘I = $9for ytf 1. This suffices for the proof of case 3, because by saccessive application of ( 1;)we get a sequence of disjoint copies of k in G. Zzt sfarn) be the set of main paths of A C-m). Let 7 bt an ordinal, such that card I i a2 and assume that the l-path I/$?“) has already been defined for i:very ordinal a c r and for every n, m EN (n 2 m), such that (i) sird (ii) hold: (i) V($sm)E@(%m) ’ 832). : = {‘VE Let S : = Uzz2 \J $ = 1 (J 4e7 I+($?), !@$sm): t= { v2n% c CC7) and BL1**) g-pi>: V,‘\ s = tJ-~\gfp~* Then %$‘*‘)f 8 holds. (From card T CQ, I$-,< QitrId card V(S) = 8, *card 7 we get: card V(S) CQ, Since card @c”*‘) = a and becau& of a’%! 5, it follows that card {V E 23(‘*l): V f7 S = 8) = Q holds. Furthermore, we have card @il*l)= card 7 C a ; hexIce 5!& (‘*‘)f $9.)Thus we can choose a 1-i,ath v&l),= gjyl,l)_

(ii) CL, Vp”)n

U,<,

VGs’)-fl for every ra,m ~N(nam;

and !!!$~m):={V~B(“*% Vfl T=@}\@F”)(n# 1). As above one can :;ho~~E!@*“)# $8.TIms for every ~1,m EN (n 3 m) we can choo:;e a E B$n*m,.This way for every ordinal r (card rC a) and for ctcery 1-path 7rT%m, n, mENN(nam) we have defined a l-path V~*mk!B(lhm! L,ck &n,m):= U CardT-car/s”“‘); for these A (wm)the assertion {*) holds. ?ht-

&

(J_

v/1*1)

This completes the proof of Theorem 2.3. 3, A theorem on wmtable

trees with finite diameter

-rem

3.1. Let A be an infinite tree with fiatite diameter and let G be a gr,%ph such that nA s G for every n EN. Then KoA c G.

Km

3*2. Let x be a quasi-order on a set S (i.e., -X is reflexive and transitive). S is called well-quasi-ordered bq <, if for every sequence sl, s2, . . 1 of elements of S there is a pair i, j(i < j), swch that si< Sj. We need the following simple lemma frsm the theory of well-qua&ordered sets ffor tkt proof see for example Lemma 3 in [lo]). Let S be yell-quasi-ordered by %: and let sl, s2, . . . be art arbitmry ertients of S. tit Mi :=(j: ap:aJv(i = 1,2,. . .). 7’hen there is an i,c N 1= 02 for every i Z iO.

11

111[lo] Nash-Williams has shown that every set of trees is well-quasi-ordered by the relation 5 . By 5&, let us denote the class of rooted t.rees (p3, w) such that s(B, w) G p (for p E NJ. We shall prove the following ico~~ll~y 3.4. (Corollary of the Theorem of Nash-Williams). is well-qfmsi-ordemi by the relation C r.

For every p E N,, %$

E%00f. Let (B, Wi)ESp (i= 1729 * _.) be a sequence of rooted trees (for a fixed ~0~). Let us choose p+l cardinalsor,,...,a,, such that (I$a,
Let (A, w) be a rooted tree rnd B a subtree of A such that b E V(B) be an arbitrary vertex. By &(B, b) we denote the set of of (A, w) which has as its elements thusi_: ;
3.6. Let (A, WY) E%, (for p EN,,) be a countrabk rooted tree. 7’kn there is a finite subtree I3 of A such that: (1) w E V@j ‘(2) Fur every b E V(B) and every < &/II, b) there me infinite!> many C’ E &(B, b): such that Cz ,C Proo%. (by i,nduction on p). For F = 0 let I3 :==A. Ct? T; GX-Q~C that t?x ass,nrtion of Lemma 3.6 is true for a certain p ENS, I>dl s(A ‘V)= F-’ ‘1. Let V,:={uE If(A): yJa) = a} and let A, bc: the union of aJJ (w, VJ-paths ipi ,A. Then A0 is a finite graph. (Otherwise, since AC1is bxIiy k&e and connected, it would follow by a well-known theorem of Ktinig that A,, contains a 1-1~~31 ‘a1 contradiction fo s( A, w) = p + 1.J

i2,

T&&w?

. *

m). Thus applying the +nduetion h~ipothesis to (Cn c,) vve get a finite subtree A, of Cr satisfying the caL-9itions of Lemma 3.6. Let B : = U zzo 4; holds (I= 1, . l

l

9

by construction, IS satisfies (I) 27”~~ 1 ,“i ). ‘I’fiiscompletes tt e proof of Lemma 3.6.

j&j For every b E V(B,) and for every C E &(B,, b) there are infinitely many C E GS(g,, b), such thiatCc_ $2’.

Xn case that A is finite let E,,, : = A (m = 3, i, . . .); then the assertion of a 3.7 is immediate. For infinite A the proof is carried out by induction on p. For p = 0, A is finite; hence the assertion is triviali. Let us assume that the assertion of Lemma 3.7 is true for a certain p EN,. Let $(A, w) = p + 1. Choose BO as a kite subtree of A according to Lemma 3.6. Because of (2) in Lemma 3.6 and since A is in&rite, A A B, consists of infinitely many components Ci (,i= 1,2, . . .). Let C, E V(Ci) be the vertex k which (Ci, \f,j is a rooted subtree of (A, w), and let ei be the edge joining ci in B. (i = 1,2, . . .). Sin= w& Ci, tine can apply the induction hypothesis to (Ci, ci); thus for everi i = 1, 2, . . . we get a x%-ties of Ci satisfying the conditions of scqkence Ci,l (i = 0, 1, . y.j of kite ~11 Lemma 3.7. Let B, :=.&U Ui,jhm Ci,jU{eiZ i = I,. . . , fin>(I?2= 1,2,. . .). For every m := 1,2,..., B, is finite il), (2), (3) are immediate. It remains to show (4): Let b E V(B,,J and CE @(.E&,, b); if b E C& for certain i, j 6 m, then (4) follows by the -hoi= of C& ; if b E BO, then (4) follows by the choice of & and because OF @‘I&. b) = %(B,, b)\{(Ci, ci): i = 1, . . . , m). This completes the proof. 3.8. Let (A, W)E 8, be countable fp EN,). L,et B be a finite subtree of A having prqwty (1) and (2) of Lemma 3 5. Let ~rl/ be an isomorphic embeddkg ofA into a graph G and let F be a finite subgraph of G such that #(B) n F = $9.Then there 1s an isomorphic embedding $’ oi ~4 into G such that (i) and (ii) hold: (ii J/‘(a) = +(a) for every a E V(a) (ii) tCjf(kj n F*=g. Since F finite ansli because of (2) of Lemma 3.6, for every b E V(B) and for eve?; C: c E(rZ, b) we can choose a C’E S( 8, b) such that: ba) if”& ,r@’ ‘k3))&(C')nF=j3 1'~) C, , C, E &(B, b), C, f C2 implies 6 ‘I # Ci. By (2 Z there is an kc?fiorphic embeddi lg rc from C into c’ mapping the root c of C Pfl tke root c’ of C’. Let us now ckfke 3’ as folIows: !d] c ~~~~~~~~if rlE v :qc

’ a ,j12: sj&&-; ’ ;,; ;:

C’E &cm

A problem cionLerning infhite

graph

13

Then, because of (a) and (c), +’ is an isomorphic en-rued&r:; Qf .4 into G. Furthermore because of (d), (i) holds; finally, blecac;~t: of fb; ~5 (e), (ir) hizids. This eompbtes the proof of Lemma 3.8. Proof of llilseorem 3.1. Let us choose an arbitrary vertex w E V(A) as the root of A; then we can find a p EN, such that (A, w j E B,. Hence we can apply Lemma 3.7. Let Bo, B1, . . . be the sequence described in Lemma 3.7. For every t, s E = 9 holds (for N (tas) let A w be a copy of A ia G such that AC’+) r~ Aq(t*s2) from A on .4~‘*‘). s1 # So). For every t, s E N let cpOS) be a certain isomomhism L define for every n cl’++ disjoint subgraphs iii the following we shall W*) (m = 1,. . . , n) of G such that B(“*mJPSisrmorphic to B,, and such that for corresponding isomorphic embedciings P:(‘*~) from B, on Btnvm) the fokwing holds: $I(~**)is the restriction of $ (n+l*m)t 3 I3,. Obviously this is sufficient for the proof of Eueorem 3.1, since A(“’ : = U nam B(“-“) (WI = 1,2, . . .) is ct sequence of disjoint copies of A in G. Let n,m~N(n~m) be fixed and assume that for every u, /.LE N (v 2 p) with raf (v, p)< (n, m) (-where “ c ” means the lexicographic order.) the subgraphs B(V*CLJ G have already been so chosen that (1) B(V+) n B(~+)t=fl for p1 $ F~. from B, onlo EPCL) (2) For every (v, ,z) < (n, m) there is an isomorphism ~~~y*p) such that $(y+) is a continuation of I/P+) if vl S v2. from A into G such (3) For every (v, CL)< (n, m) there is an isomorphism $(y*pL) that @‘*cL)is a continuation of $(“*L’ )c We shall distinguish two cases: Case I: m = n. Choose A(‘g’) (for suitable f, s E N (2 3 s)) such that A(‘*“’n U (v,cr)c(n.*) B(* *) = $4; this is possible, because U (V,P)+ mI I?(“+) is finite. Let -(n.m)

$

. -_ q(t.Sj,$,f%*): = Rest,” $(n,*) and B(n.m):= $(n.m)(13,).

Case 2: mf n. Let F:=

&+~+,m~~p,.m R(Y.w); because of (l), $“-**m’(B,_l I 3 holds. Hence we can apply Lemma 3.8 to Bn+ J/(n-l*m) and F: Then there is an isomorphii;m @n*mzof A into G for which (i) $nm) (a) = $(n-l,m) (a) for every a E V(B,_,)

F=Q)

(ii) $(n.*) (A)nF=@ Let then +(nJ”:.= Restg,$(‘L*m) and B(“*“):

= $(nsrn) (B, ).

Thus in both cases we have: (1’): fP9-7 i!P+)=QJfor pf m; (2’): Ifi(tkm)is a continuation of @n-l*m;; (3’): J+mI is an isomorphism from A into G such that $‘n-m) is a contkuatkm Qf q!P-! This completes the proof of Theorem 3.1.

iesof

A. We -choose the t) (n

j, k EN such that j and k ate in e shall prove that XoA c G does not

15

eorie der unendhcherl unendlicher

Graphen,

Bkm~~e J engerschen

Abh.

Combinatorial

Satz, itiath. AWL

Arm. 157 (1964) 125-13:‘. Wege in Graphen, IMP&h.Plachr. :W (1965) n Craphen,

Math. Nach .. 44

_Math. Sem. Univ. bfamblltg 4.3 (lS75) 83-58. and Vazsonyt’s cor!jecture, T ‘Lugs.Am. ath. SW. 95 (1960) 210-225. iscrete Math. 14 (197f%)343-345. [9] J. Lake, A problem concerning infinite graphs, [IO] C. St. J. A. Nash-Williams, Qn well-quasi-ordering infinite trees, Proc. CIambridge Philos. Sot. 61 (1965) 697-720. [ll] D. R. Woodall, A note on a problem of Walin’s J. Combinatcxiai Theory 21 Ser. B (2976) 132-134.