On locally k-critically n-connected graphs

Discrete Mathematics North-Holland

120 (1993) 1833190

183

On locally k-critically n-connected graphs Jianji Su Department of Mathematics,

Guang Xi Normal University, Guilin, Guangxi, 541004, China

Received 7 November 1989 Revised 16 January 1992

Abstract Su, J., On locally k-critically

n-connected

graphs,

Discrete

Mathematics

120 (1993) 183-190.

Let 0 # W’g V(G). The graph G is called a W-locally k-critically n-connected graph or simply a W-locally (n, k)-graph, if for all V’G W with 1V’I 6 k and each fragment F of G we have that K(GV’)=n-1 V’ and Fn W#@I. In this paper we prove that every non-complete W-locally (n, k)-graph has (2k+2) distinct fragments and 1WI>2k+2. From this result it follows that: (1) Let G be a non-complete (n. k)-graph. If all ends of G are proper, then G has (2k + 2) pairwise disjoint ends. (2) Slater’s conjecture on (n, k)-graphs holds, i.e., the complete graph Kn+ 1 is the unique (n, k)graph for 2k>n.

1. Introduction By a graph we shall mean a finite, undirected, F G V(G). We denote the set of all vertices of G-F The F is said to be a fragment of G if F= V(G)-(FuN(F, G)) and K(G) denotes the

simple graph. Let G be a graph and adjacent to a vertex of F by N(F, G). IN(F, G)I = K(G) and I@#@, where connectivity of G. An end of G is

a fragment of G that contains no other fragments as a proper subset. Let F be a fragment or end of G. F is called proper if 1F I< IFI. A fragment with minimum cardinality is called an atom of G. Let rc(G) = n and Ts V(G). T is said to be a n-cutset of G if ( TI = II and G - T has at least two components. To generalize the well-known concepts of k-critically n-connected graphs, we introduce the concepts of a W-locally k-critically n-connected graphs. Let @# WC V(G). The graph G is called a W-locally k-critically n-connected graph or simply a W-locally (n, k)-graph, if for all V’S W with / VI
Correspondence to: Jianji Guangxi, 541004, China. 0012-365X/93/$06.00

0

Su, Department

1993-Elsevier

of Mathematics,

Science Publishers

Guang

Xi Normal

B.V. All rights reserved

University,

Guilin,

184

J. Su

(n, k)-graphs complete

G (see [2]) is V(G)-locally

W-locally

of G which contains

I”. The following

Slater [2] and Mader

Conjecture

It is easy to see that if G is a non-

conjectures

on (n, k)-graph

are proposed

by

[3] respectively.

Conjecture A (Slater). The complete

joint

(n, k)-graphs.

and I” E W with 1V’I d k, then there exists a n-cutset

(n, k)-graphs

B (Mader).

Every

graph K,, 1 is the unique

non-complete

(n, k)-graphs

(n, k)-graph

has (2k+2)

for 2k > n.

pairwise

dis-

ends.

Recently,

Conjecture

A is verified

by the author

in [6]. Conjecture

B is stronger

than Conjecture A. Knowing Conjectue B to be true one could settle several other difficult conjectures on (n, k)-graphs (see [3]). Hence it is very interesting and important that Conjecture B can be verified. Conjecture B is verified for k = 1 by Mader [4] and author [S]. Recently, the author in [7] proved that this conjecture is true for k = 2. For k > 3, Conjecture B is open. In this paper, we explore the number of ends in W-locally (n, k)-graphs and give a proof of Conjecture A.

2. Some properties of fragments and ends Let G be a non-complete graph. We define 9 (G) = ( [F, T,F] (F is a proper fragment of G, T=N(F, G), F= V(G)-(FuN(F, G))}. Let VOG V(G). We denote the set of proper ends of G which are contained in VOby &‘c(VO). The maximum possible number of pairwise disjoint ends of G which are contained in VO is denoted by bG( V,). Suppose that C,, Cz,...,C, are r non-empty distinct subsets of V(G) and TL V(G).We say that Tn [Cl,C2, . . . . C,] #@, if TnCi#@, i=l, 2 ,..., r. Lemma 1. Let [F, T, F], (1)

IfFnp#&

then

(2)IfFnF’#@#FnF’, (3) If FnF’#Q),

[F’,

T’, F’] EF(G).

(FnT'(3(F'nT(,(pnT/3(FnT'(. then both FnF' and FnF’ are fragments

then both FnF’

and FnF’

are fragments

Proof. For the proofs of (1) and (2), see Mader FnF’#@. If FnF'=@, then

[4, Lemma

of G.

of G.

l(a),(b)].

Now assume

IF'I=IFnF'I+/TnF'I+IFnF'I>jFnF'I+IFnT')+IFnF'J >IFnT'I+I~nFfI=IF13(lGI--(G))/2>,/F'I, which is a contradiction.

Thus FnF'#@. Therefore

Lemma 2. Let [F, T, F],

[F’,

(1) Z’F

T’, F’] EF(G).

is an end, then FnF’=@

(2) If F is an end and T’nF#&

or FcF’. then

FnF'=@

(3) follows from (2).

0

On locally graphs

185

Proof. Suppose that F is an end. If FnF’#@, by Lemma l(3), we have that FnF’ a fragment, hence F = FnF’s F’, Therefore (1) and (2) are true. 0

is

Lemma 3. Let F be a fragment of G. Let B and B’ be two distinct ends of G which are contained in F. Then BnB’=O. Proof. Assume BnB’ #@ Since B E F and B’ & F, then FG Bnl?‘, and by Lemma we have that BnB’ is a fragment Therefore

BnB’=@

Lemma 4. Let [F,,

and hence B= BnB’ = B’, which contradicts

l(2),

B # B’.

0 TO, F,] ~9 (G), where Fo is a proper fragment of G with maximum

cardinality. Let dc(FO)= {S,, . . . . SIO}, b,(FO)=pO and S;, . . . . S;, are pO pairwise disjoint ends contained in FO. Set S = (ufz 1 Si)u(u,“” 1Si). Take any [F, T, F] E 9(G). Zf Tn[S1, . . ..S.,]#@ or Tn[S; , . . . . SLJ #8, then Fn(FOuFO)=O. Thus FnS=@ Proof. First assume

Tn[S;

, . . . , S;,] #0, we shall show that Fn(FOuFO)

= 0. Assume

not, then we have that FnFo # 8 or FnF,, # 0. We shall obtain a contradiction. Suppose FnFo #0. By Lemma l(3), we have that FnpO is a fragment of G. Let B be an end of G which is contained in FnFo. Since Tn[S;, . . . . S;,] #0 and TnB=@, by Lemma 3, we have that BnBi=@, j= 1,. . . , ,uo, which contradicts bo(FO)=pO. Hence FnF,,=@ Suppose FnFO #0. Then FnF, =8. For otherwise, since FnFO #0 and FnF,, #0. by Lemma l(2), we have that FnF, is a fragment of G. A similar argument as above obtains a contradiction. Hence FnFo = 8. Since FnFo # 8, from Lemma l(l), we have thatIFnT,I3)TnF,I,itfollowsthatIFI=jFnF,I+IFnT,I>IF,nTI=IF,I,i.e.,Fis a proper fragment of G with cardinality greater than IFJ, which contradicts the definition of FO. Therefore FnFO #0, and Fn(F,u~O)=O. 0 If Tn[SI, . . . . S,,]#& by the same argument, it follows that Fn(F,uF,)=@

3. On W-locally (n, k)-graph Let G be a non-complete W-locally (n, k)-graph. Let B1, . . . , BA, B; , . . . , B; be A+ ,D distinct ends of G and B”,, . .., iA, 2 1, . . . . tp be A+y subsets of V(G). The end set B1, . . . . BA, B;, . . . . B; and subset set B1, . . . , B1, B”;, . . . , BP of G are said to be satisfying conditions (P), if they satisfy. (i) B”i, . . ..B.,B’ 1, . . . , & (ii) @#EiGBinW, (I?

are pairwise

0#2jcB;nW,

(iii) for any [F, T,F]E~(G), :

Tn[@l,...,&J#O,

disjoint;

i=l,...,

A, j=l,...,

if Tn[l?,,...,l?J#@

then FnB=@,

,u; or

where B=(~~~lBi)u(~~=lBj).

186

J. Su

5. Let G be a non-complete W-locally (n, k)-graph. Suppose the end set (BI, . . ..Bn. B;, . . ..BI} and subset set {B,, . . . . Bn, PI, . . . . PI} of G satisfy the conditions (I’). Take a [F,, T1, F,] EY(G) such that

Lemma

IF,(=max{IFII[F, Let B=(u~=,~i)u(u~=,BIi).

T,F]EB(G),

Tn[B, ,..., L?,J#@ or Tn[B; ,..., 8;]#@}

Take any [F, T,F]EP(G) C,} and Tn[g,, . . . . fiA,C1, . . . . C,]#@

(1) Lf~o(F1)={C1,..., C 1, ..., C,] #@, then Fn(FluF,)=@ (2) Zf bo(F,-&=t>l and

Tn[L?, ,..., BA,D1 ,..., D,]#@

or Tn[B”;, . . . . BP, or

Tn[B;

,..., BP,

D 1, ..., D,]#& where D1, . . . . D, are t pairwise disjoint ends contained in F, -& Fn(F, uF,) = fJ. (3) Ifbo(F,-@=O and there exists an iE{l, . . ..A} such that vi=BinF,#Q)

then

Tn[B”,, . . ..Bi_l. Vi, Bi+l,..., B”n]#@ or there exists a jE{l,...,p} such V>=B”>nF,#@ and Tn[B; ,...) &1, Vi, I?>+, ,..., 2;]=& then Fn(F,uF,)=O.

that

and

Proof. Suppose that Fn(F,uF,)#O. We shall show this leads to a contradiction. Proof of (1). By the symmetry, we may suppose that Tn[fi,, . . . , DA, C,, . . . , C,] ~8. Suppose that FnF, #0. By Lemma l(3), we have that FnF, is a fragment. Note that Tn[C1,..., C,] #0 and FnF, is a proper fragment, by Lemma 2(2), we have that (FnF,)nCi =0, i= 1, . . ..r. It follows that FnF1 is a proper fragment contained in F1-(U~=,Ci)Y which contradicts -C40(F1)={C1,...,C,.). Hence FnF1=@ Suppose that Fd, ~0. Then FnF, = 0. For otherwise, by Lemma l(2), we have that F;\F, is a fragment. Using a similar argument as above it follows that FnFl is a proper fragment contained in F r -(u I= I Ci), which leads to a contradiction. Hence FnF, =8. Since FnF, =@=FnF, and FnF, #f#~,by Lemma l(l), we have that (F( > 1F1 /, which contradicts the maximality of 1F1 ( (note that Tn[B”, , . . . , l?,J #0). Thus (1) is proved. Proof of (2). By the symmetry, suppose that Tn[g,, . . ., gA,, D1,. .., D,] #0. Suppose that FnF, ~0. By Lemma l(3), FnF, is a fragment and N(FnF,,

G)=(~nT,)u(TnT,)u(~,nT)=N(FuF,,

G).

Set F’=FuF, and T’=N(FuF,, G). Then F’=FnF,. Since T,n[B,, . . ..8.]#@ or T1 n[BI;, . . . , &] #0, by the condition (P)(iii), we have that F,nB =8. From this it follows that TnB=(T-F,)nBcT’nB, hence T’n[f?,,...,~J#0 (note that Tn[L?,, . . ..l?J ~0). In this case, if (F’J < IF’I, then by the condition (P)(iii), F’nB=@, hence F’nB=@. Assume that D is an end contained in F’. Note that in T’n[D,, . . . . D,]#@ and by Lemma 3, it follows that D is an end contained (F, -g)-( U;=IDf), which contradicts b,(F, -B”)= t. Hence (F’I lF,I, which contradicts the maximality of (F1 I (note that T’n[B,, ., DA] #8). Thus FnF, =@.

Suppose that FnFl #8. Then FnFi =8. For otherwise, we have that FnF, is a fragment. Let D be a proper end contained in FnFl. Set F’= FlvF and T’ = N( F’, G), then F’ = FnF,

. If IF’I < 1F’ 1,by a similar argument

as above, it follows

that D is a proper end contained in (F, -B”)-(u;= 1D,), which contradicts b&i -B”)=t. If IF’1 < IF’I, by a similar argument as above, it follows IF’1 > IF, I which contradicts the maximality of IFI 1. Hence FnF, =8. Since FnF, =fj=FnF, and FnF, #8, by Lemma 1 (I), we have that IF I > IF1 1, which contradicts mality of IF1 I. Thus FnFl =8 and (2) is proved. Proof of (3). By the symmetry, Vi=BinFi#@

and Tn[l?,,

suppose

..*,Bi-l,

the maxi-

that there exists an in (1, . , i} such that

Vi, Bi+l)..,)

8,]+QI.

Suppose that FnF1 #8. Using the notation F’, T’, F’ in the proof of (2), we have that T’n [i?, , . . . , l?,J # 0. By a similar argument as in the proof of (2), it follows that if IF’I < IF’l, then P’ is a fragment contained in Pi - fi, which contradicts b&F, - @ = 0; if IF’I < IP’I, then IF’I > IF, 1, which contradicts the maximality of IFI I. Hence FnF1 = 0. Suppose that FnF, #0. By a similar argument as in the proof of (2), it follows that FnFl =8 and IFI > IF1 1,which contradicts the maximality of IF1 I. Hence FnF, =0. 0 Therefore (3) is proved.

Lemma 6. Let G be a non-complete W-locally (n, k)-graph. Suppose that end set {B 1,..., BA, B; ,... , B;} and subset set {B”,, . . . . BA, PI, . . . . 2fl} of G satisfy the conditions (P) that maximizes A+/A. Let B”= (u f= 1Bi) u( WY= 1B;) and bo( V(G)- B”)= h. If A+p>l,

then h+A+p>2k+2. Suppose that h + ,I + p d 2k + 1. We shall lead to a contradic-

Proof. By contradiction.

tion. First we assert that A
Tl,F,]~g(G)

lF,I=max{lFI

pLdk. Suppose that p>k+l, then h+;l
such that

I[F, T, F] E@-(G), Tn[l?,,

. . . . l?J #8

or

Tn[gl;,

...,2P] #8}.

A+~21 and ibk and y C,>. We shall consider the following cases. Case 1: bG(F1-g)=t>l. Suppose that D1,. .., D, are t pairwise disjoint ends contained in F, -g. Let

Since

Let

188

J.

Then Ia0 and r+t+1+3,+~L~++++~22k+l. p + t < k. For otherwise, suppose that a [F, T, F] E F(G) such that Tn[MlnW By Lemma

,..., M,nW,

ilci)U(

then

r+A
and

l+t+p
Take

and hence

. . . . M,] #0 so that, by the condition

(P)

2(2), it follows that

Fn(Bu( Thus F is a proper

which

we show that

,1, DJ))=@.

Note that Tn[B”;, . . . . $1 #$!I and Tn[M,, (iii) and Lemma

First r+;l>k+l,

DlnW ,..., D,nW,B”; ,..., &]#@.

5(2) we have that Fn(F,uF,)=~

Fn(

su

i,

Mm))=@.

end contained

is a contradiction.

Hence

in

r+ld

k. By the same argument,

it follows

that

t+pbk. Nextweshowthattheendset(B, ,..,, BA,C1 ,..., C,,B\ ,.,., B;,D1 ,..., D,)andthe subset set {B”,, . . . . BA,, C,n W, . . . . C,n W,B”;, . . . . B”;, D,n W, . . . . D,n W} of G satisfy the conditions (P). Clearly, they satisfy the condition (P)(i)(ii). In order to show that they also satisfy the condition (P)(iii), let

Takeany[F,T,p]Eg(G).IfTn[B, ,..., fiA,ClnW ,..., C,nWl#@orTn[B”; D,nW, . . . . D,n W] #0, then by Lemma 5(l)(2), we have that Fn(FluF,)=O.

,..., l$, Hence

Fn(( $l+(iID,))=@ Note that FnB =& so that FnB* =@. Thus the above mentioned end set and subset sef of G satisfy the conditions (P). But A+ r + p+ t > A + p, which contradicts the maximality of A+ p. Therefore b,(F, - i) = 0. Since b,(F, -l?) =O, then F,&#@, Thus we have the following. Case 2: bG(F1 -B”) = 0 and there exists an i E { 1, . . . , A> such that Vi = EinFl# @ or there exists an je (1, . . ..p} such that V>=@nF1 #8. Without loss of generality we may assume that there exists an i E { 1, . . . , A} such that Vi = tiiinFi #0. Let

On locally

graphs

189

then I> 0 and r + l+ 1,+ p d 2k + 1. Using a similar argument

as above, it follows that

r + p < k and Ad k. By Lemma 5(l) and (3), a similar argument gives that the end set {B 1, . . ..Bn. B;, . . . . BL, C1 ,..., C,} and the subset set {El )...) B”i-1, I’i,B”i+l,...,E~~ B;, . ..) B;, Cm w, . ..) C,n W} of G satisfy the conditions

(P). But A+,u+r

>]++/A,

which contradicts the maximality of A+ p. Since we were led to contradictions in Case 1 and Case 2, hence h+;1+~>2k+2 and the lemma

follows.

0

Now we show the main theorem Theorem 1. Every non-complete 1WI>2k+2.

in this paper.

W-locally

(n, k)-graph

has 2k+2

distinct ends and

Proof. Take a [F,, T,, FO] E F(G), where F0 is a proper fragment with maximum cardinality. Let -r40(F0)= {S,, . . . . S,,} and b,(F,,)=pO. Suppose that S;, . . . . Sh,, are p0 pairwise disjoint ends contained in FO. By Lemma 4, we have that the end set {S 1, . ..> Slo,S;, ..., S;,,} and the subset set {Srn W, . . . . Slon W, Sin W, . . . . SL,n W} of G satisfy the conditions (P). Choose an end set {B,, . . , BI, B;, . . . , B;} and a subset set {B”I)..., &,B”; )... , I$} of G that satisfy the conditions (P) and maximize /1+~. Let fi=( wise It is and joint.

u f= 1 Bi)u( uy= 1 ii) and b,( I’(G)- fi) = h. Suppose that M1 , . . . , Mh are h pairdisjoint ends contained in V(G) - g. By Lemma 6, we have that h + A+ ~12 2k + 2. easy to see that the ;1+p+ h ends B1, . . ., BA, B;, . . . . B;, MI, . . . . Mh are distinct the A+ /J + h subsets g,, . . . , fin, B”;, . . . , 2p, MI n W, . . . , M,n W are pairwise disNote that Bs W, 1WI >I.+p+ h>,2k+2. The theorem follows. 0

By the definition, we have that if G is an (n, k)-graph, then G is a W( = V(G))-locally (n, k)-graph. Thus we have the following. Corollary 1. Every non-complete Since any two distinct

proper

(n, k)-graph has 2k+2 ends are disjoint,

distinct ends.

then we have the following.

Corollary 2. Let G be a non-complete G has 2k + 2 pairwise disjoint ends.

(n, k)-graph in which every end is proper.

Corollary 3. Let G be a non-complete

(n, k)-graph.

Then

Then n>2k.

Proof. Take an atom A of G. Then we have that G-A is a non-complete (n k- 1)-graph and FnN(A, G)#& where F is a fragment of G-A (see [3. p. 3951). W= N(A, G). Clearly, Ws V(G - A); hence G--A is a non-complete W-locally (n k- 1)-graph. By Theorem 1, it follows that n= IN(A, G)I = I WI >2(k1) +2 =2k. corollary follows.

I A 1, Let 1A 1, The

190

J. su

From the Corollary is true. 0

3, it follows immediately

that Conjecture

A (Slater’s conjecture)

References [l] [2] [3] [4] [S]

[6] [7]

R.C. Entringer and P.J. Slater, A note on k-critically n-connected graphs, Proc. Amer. Math. Sot. 66 (1977) 3722375. S.B. Maurer and P.J. Slater, On k-critical n-connected graph, Discrete Math. 20 (1977) 255-262. W. Mader, On k-critically n-connected graphs, in: J.A. Bondy and U.S.R. Murty, eds., Progress in Graph Theory (Academic Press, Orlando, FL, 1984) 389-398. W. Mader, Disjunkte Fragment in kritisch n-fach zusammenhangenden Graphen, European J. Combin. 6 (1985) 353-359. J. Su, On some properties of critically h-connected graphs and k-critically h-connected graphs, Graph Theory and its Applications: East and West Proceedings of the First China-USA International Graph Theory Conference, Ann. NY. Acad. Sci., Vol. 576 (1989) 536-541. J. Su, Proof of Slater’s conjecture on k-critical n-connected graphs, Kexue Tongbao, Sci. Bull. 20 (1988) 1675-1678. J. Su, Fragments in 2-critically n-connected graphs, to appear.