Degree sequence conditions for equal edge-connectivity and minimum degree, depending on the clique number

Degree SequenceConditions for EqualEdge-Connectivity andMinimum Degree,Depending on theClique Number

Lutz Volkmann*LEHRSTUHL II FUR MATHEMATIK

RWTH AACHEN, 52056 AACHEN, GERMANY

E-mail: [email protected]

Received December 15, 2000; Revised August 26, 2002

DOI 10.1002/jgt.10087

Abstract: Using the well-known Theorem of Turan, we present in thispaper degree sequence conditions for the equality of edge-connectivity andminimum degree, depending on the clique number of a graph. Differentexamples will show that these conditions are best possible and inde-pendent of all the known results in this area. � 2003 Wiley Periodicals, Inc. J Graph

Theory 42: 234–245, 2003

Keywords: edge-connectivity; degree sequence; clique number

——————————————————

*Correspondence to: Lutz Volkmann, Lehrstuhl II fur Mathematik, RWTH Aachen,Templergraben 55, Aachen D-52062 Germany.E-mail: [email protected]

� 2003 Wiley Periodicals, Inc.

1. INTRODUCTION

Sufficient conditions for the equality of the edge-connectivity �ðGÞ ¼ � and the

minimum degree �ðGÞ ¼ � of a graph G were given by several authors. For other

graph theory terminology we follow Chatrand and Lesniak [3]. The most general

results for arbitrary graphs are the following statements.

As a generalization of results of Chartrand [2], Lesniak [7], Plesnık [8], and

Plesnık, Znam [9], Dankelmann and Volkmann [4] (see also the textbook of

Volkmann [13, p. 318ff]) proved the following theorem.

Theorem 1.1 [4]. Let G be a connected graph. If for all 3-distance maximal

pairs of sets X; Y � VðGÞ the condition �ðG½X [ Y �Þ ¼ 0 is fulfilled, then � ¼ �.

For a connected graph G, we define the distance dðu; vÞ between two vertices u

and v as the minimum of the lengths of the u� v paths of G. For two subsets X

and Y of the vertex set VðGÞ in a connected graph G, let dðX;YÞ ¼ minfdðx; yÞ jx 2 X; y 2 Yg. A pair of sets X; Y � VðGÞ with distance dðX;YÞ ¼ 3 is called

3-distance maximal, if there exist no sets X1 6¼ X or Y1 6¼ Y with X � X1 and

Y � Y1 such that dðX1;Y1Þ ¼ 3.

The next result is an improvement of a well-known degree sequence condition

of Bollobas [1].

Theorem 1.2 [5]. Let G be a graph of order n with edge-connectivity � and

degree sequence d1 � d2 � � � � � dn ¼ �. If � � bn=2c or if � � bn=2c � 1 and

Xki¼1

ðdi þ dnþi��1Þ � kðn� 2Þ þ 2� � 1

for some k with 1 � k � �, then � ¼ �.

A further degree sequence condition can also be found in [5].

Theorem 1.3 [5]. Let G be a graph of order n with edge-connectivity � and

degree sequence d1 � d2 � � � � � dn ¼ �. If � � bn=2c or if � � bn=2c � 1 and

X2k

i¼1

dnþ1�i � kn� 3

for some k with 2 � k � �, then � ¼ �.

Theorem 1.4 [6]. Let G be a graph of order n with edge-connectivity �,

clique number ! � p, and degree sequence d1 � d2 � � � � � dn ¼ �. If n �2b�p=ðp� 1Þc � 1 or if n � 2b�p=ðp� 1Þc � 2p and

Xki¼1

di þXð2p�1Þk

i¼1

dnþ1�i � kðp� 1Þnþ 2� � 1

for some k with 1 � k � b�=ðp� 1Þc, then � ¼ �.

DEGREE SEQUENCE CONDITIONS 235

Let G be a graph with vertex set VðGÞ, edge set EðGÞ, and clique number

!ðGÞ ¼ ! � p. Then, the well-known Theorem of Turan [10] leads to

2jEðGÞj� p� 1

pjVðGÞj2:

Using this bound, we present in this paper degree sequence conditions for the

equality of edge-connectivity and minimum degree, depending on the clique

number, which are independent of Theorems 1.1–1.4. Examples will show that

these conditions are best possible, and one of our theorems implies a result of

Dankelmann and Volkmann [4].

2. MAIN RESULTS

Theorem 2.1. Let G be a graph of order n � 6 with edge-connectivity �, clique

number ! � p, and degree sequence d1 � d2 � � � � � dn ¼ �. Furthermore, let

� ¼ 1 when n is even and � ¼ 0 when n is odd. If � � bn=2c or if � � bn=2c � 1

and

X�þ1

i¼1

dnþ1�i � ð� þ 1Þ p� 1

p

nþ 1 þ �

2� 2� þ 2

pðn� 3 þ �Þ ;

then � ¼ �.

Proof. Suppose to the contrary that � < �. Then, there exist two disjoint sets

X;Y � VðGÞ with X [ Y ¼ VðGÞ such that the number of edges jðX; YÞj from X

to Y satisfy the inequality jðX; YÞj < �.We first show that the sets X and Y contain at least � þ 1 vertices, and therefore

� � bn=2c � 1.

Suppose that X contains at most � vertices. Then, we obtain the contradiction

jXj� �Xx2X

degðxÞ � jXjðjXj � 1Þ þ jðX;YÞj � �ðjXj � 1Þ þ jðX;YÞj:

Similarly one can show that jY j � � þ 1.

Now assume without loss of generality that jXj � n2

and let S � X be a ð� þ 1Þ-set that contains the � þ 1 vertices of smallest degree in X. Then, it follows from

our hypothesis

Xv2S

degðvÞ �X�þ1

i¼1

dnþ1�i � ð� þ 1Þ p� 1

p

nþ 1 þ �

2� 2� þ 2

pðn� 3 þ �Þ :

236 JOURNAL OF GRAPH THEORY

This implies

degðuÞ � p� 1

p

nþ 1 þ �

2� 2

pðn� 3 þ �Þ

for every vertex u 2 X � S and hence,

Xv2X

degðvÞ � jXj p� 1

p

nþ 1 þ �

2� 2jXjpðn� 3 þ �Þ : ð1Þ

On the other hand, our assumption � < � and the theorem of Turan lead to

Xv2X

degðvÞ � p� 1

pjXj2 þ � � 1: ð2Þ

Now, we shall investigate three cases:

Case 1. Let 2jXj � n� 3 þ �. It follows from (1)

Xv2X


pðjXj þ 2Þ � 2jXj

pðn� 3 þ �Þ ;

and hence together with (2)

� � 2jXj p� 1

pþ 1 � 2jXj

pðn� 3 þ �Þ � jXj;

a contradiction to jXj � � þ 1.

Case 2. Let 2jXj ¼ n� 1 þ � and let ð� þ 1Þ p�1p

nþ1þ�2

be an integer. Since2�þ2

pðn�3þ�Þ < 1, the condition of our theorem is equivalent with the condition

X�þ1

i¼1

dnþ1�i � ð� þ 1Þ p� 1

p

nþ 1 þ �

2:

As above, this inequality yields

Xv2X


p

nþ 1 þ �

2¼ jXj p� 1

pðjXj þ 1Þ:

Together with (2) we obtain

� � p� 1

pjXj þ 1;


and therefore

Xv2X

degðvÞ � �jXj � p� 1

pjXj2 þ jXj:

We deduce from (2) the contradiction � � jXj þ 1.

Case 3. Let 2jXj ¼ n� 1 þ � and let ð� þ 1Þ p� 1p

nþ 1þ �2

be no integer. Then,

it follows that

jXj ¼ psþ r with 0 � r � p� 2:

From (1) we obtain

Xv2X


pðjXj þ 1Þ � 2jXj

pðn� 3 þ �Þ ;

and hence together with (2)

� � jXj þ 1 � jXjp

� 2jXjpðn� 3 þ �Þ ¼ jXj þ 1 � s� r

p� psþ r

pðpsþ r � 1Þ :

Since

r

pþ psþ r

pðpsþ r � 1Þ < 1;

we conclude that even

� � jXj þ 1 � s ¼ jXj þ 1 � jXjp

þ r

p� jXj þ 1 � jXj

p:

This leads to

Xv2X

degðvÞ � �jXj � jXj2 þ jXj � jXj2

p:

Hence, we deduce from (2) the contradiction � � jXj þ 1, and the proof is

complete. &

Theorem 2.1 generalizes results of Volkmann [11], [12] as well as a theorem of

Dankelmann and Volkmann [4].


Corollary 2.1 [4]. Let G be a graph of order n. If ! � p and

n � 2p

p� 1�

� �� 1;

then � ¼ �.

Proof. If � � bn=2c, then in view of Theorem 2.1, � ¼ �. In the other case,

it follows from the hypothesis that n � 2b pp�1

�c � 1 for n odd and n � 2b pp�1

�c�2 for n even. This implies

X�þ1

i¼1

dnþ1�i � ð� þ 1Þ� � ð� þ 1Þ p� 1

p

nþ 1

2

� �:

According to Theorem 2.1, we conclude that � ¼ �. &

Example 2.1. Let G1 ¼ G2 ¼ K2, H1 ¼ K2;2;...;2, the complete t-partite graph

with t � 3 and the partite sets fx2i�1; x2ig for i ¼ 1; 2; . . . ; t, and H2 ¼ K1;2;2;...;2,

the complete t-partite graph with the partite sets fy2i�1; y2ig for i ¼ 1; 2; . . . ;t � 1, and the partite set fy2t�1g, consisting of one vertex.

We define the graph G as the union of G1;G2;H1, and H2 together with the

following edges. We join all vertices of G1 with all vertices of H1 and we join all

vertices of G2 with all vertices of H2. In addition, G contains the edges x1y1;x2y2; . . . ; x2t�2y2t�2 and x2t�3y2t�4; x2t�2y2t�3; x2t�1y2t�2, as well as x2ty2t�1. The

resulting graph G is of order 4t þ 3 with clique number ! ¼ t þ 2 and minimum

degree � ¼ 2t. Furthermore, G has 2t � 3 vertices of degree � ¼ 2t, two vertices,

namely x2t�2 and x2t�3, of degree 2t þ 2, and 2t þ 4 vertices of degree 2t þ 1.

It follows that

X�þ1

i¼1

dnþ1�i ¼X2tþ1

i¼1

dnþ1�i ¼ 4t2 þ 2t þ 4

and

ð� þ 1Þ p� 1

p

nþ 1

2¼ ð2t þ 1Þ t þ 1

t þ 2ð2t þ 2Þ:

Now, it is easy to see that

4t2 þ 2t þ 4 � ð2t þ 1Þ t þ 1

t þ 2ð2t þ 2Þ;

and consequently, �ðGÞ ¼ �ðGÞ, by Theorem 2.1.


Finally, it is straightforward to verify that the graph G fulfills neither the

conditions of Corollary 2.1 nor these of Theorems 1.1–1.4.

The next example will show that Theorem 2.1 is best possible, when n is odd.

Example 2.2. Let H1 ¼ K2;2;...;2 and H2 ¼ K3;2;2;...;2, be two complete t-partite

graphs with t � 3, VðH1Þ ¼ fx1; x2; . . . ; x2tg and VðH2Þ ¼ fy1; y2; . . . ; y2tþ1gsuch that fy1; y2; y3g is the partite set with 3 vertices. We define the graph G as the

union of H1 and H2 together with the new edges x1y1; x2y2; . . . ; x2t�2y2t�2. Then,

G is of order 4t þ 1 with clique number ! ¼ t and minimum degree � ¼ 2t � 2.

It follows

X�þ1

i¼1

dnþ1�i ¼X2t�1

i¼1

dnþ1�i ¼ 4t2 � 4t � 1

¼ ð2t � 1Þ t � 1

tð2t þ 1Þ � 4t � 2

tð4t � 2Þ

¼ ð� þ 1Þ p� 1

p

nþ 1

2� 2� þ 2

pðn� 3Þ :

The following example, which shows again that Theorem 2.1 is independent of

Theorems 1.1–1.3, will also be used at the end of this paper.

Example 2.3. Let H ¼ K2, H1 ¼ K2;2;...;2, the complete t-partite graph, and

H2 ¼ K3;3;2;2;...;2, the complete ðt � 1Þ-partite graph with t � 3, and denote by V1

and V2, the two partite sets of H2 with 3 vertices. Let a; b, and u be three further

vertices. The vertex set of the graph G is the union of the vertex sets of H;H1;H2

together with a; b, and u. In addition, we join all vertices of H with all vertices of

H2 and the vertices a; b; u with all vertices of H1. Furthermore, we join b with all

vertices of H2 and a with the vertices of V1 [ V2. The resulting graph G is of order

4t þ 5 with clique number ! ¼ t þ 1 and minimum degree � ¼ degðuÞ ¼ 2t. It

follows that

X�þ1

i¼1

dnþ1�i ¼X2tþ1

i¼1

dnþ1�i ¼ 4t2 þ 4t

and

ð� þ 1Þ p� 1

p

nþ 1

2¼ ð2t þ 1Þ t

t þ 1ð2t þ 3Þ:

Because of

4t2 þ 4t � ð2t þ 1Þ t

t þ 1ð2t þ 3Þ;

Theorem 2.1 implies �ðGÞ ¼ �ðGÞ.


Our second result is an improvement of Theorem 2.1, when the order of the

graph is even.

Theorem 2.2. Let G be a graph of order n � 6 with edge-connectivity �, clique

number ! � p, and degree sequence d1 � d2 � � � � � dn ¼ �. If � � bn=2c or if

� � bn=2c � 1 and

X2�þ2

i¼1

dnþ1�i � ð� þ 1Þ p� 1

pðnþ 2Þ � 4� þ 4

pðn� 2Þ ;

then � ¼ �.

Proof. Suppose to the contrary that � < �. Then, there exist two disjoint sets

X; Y � VðGÞ with X [ Y ¼ VðGÞ and jðX; YÞj < �. As shown in Theorem 2.1,

jXj; jYj � � þ 1 and � � bn=2c � 1.

Now, let S � X and T � Y be two ð� þ 1Þ-sets that contain the � þ 1 vertices

of smallest degree in X and in Y , respectively. The theorem of Turan yields

Xv2X

degðvÞ � p� 1

pjXj2 þ � � 1; ð3Þ

as well as

Xv2Y

degðvÞ � p� 1

pjY j2 þ � � 1: ð4Þ

First of all, we will show that

Xv2S

degðvÞ < ð� þ 1Þ ðp� 1ÞðjXj þ 1Þp

� � þ 1

pðjXj � 1Þ : ð5Þ

If, we suppose to the contrary that

Xv2S

degðvÞ � ð� þ 1Þ ðp� 1ÞðjXj þ 1Þp

� � þ 1

pðjXj � 1Þ ;

then it follows

Xv2X


pðjXj þ 1Þ � jXj

pðjXj � 1Þ : ð6Þ

Now, we shall investigate two cases:


Case 1. LetjXj þ 1

pbe an integer. Since

jXjpðjXj�1Þ < 1, it follows from (6)

Xv2X


pðjXj þ 1Þ;

and hence together with (3),

� � jXj p� 1

pþ 1:

This yields

Xv2X

degðvÞ � �jXj � p� 1

pjXj2 þ jXj;

a contradiction to (3) and jXj � � þ 1.

Case 2. LetjXj þ 1

pbe no integer. Then, it follows that

jXj ¼ psþ r with 0 � r � p� 2:

Then, the inequalities (3) and (6) imply

� � jXj þ 1 � jXjp

� jXjpðjXj � 1Þ ¼ jXj þ 1 � s� r

p� psþ r

pðpsþ r � 1Þ :

Since

r

pþ psþ r

pðpsþ r � 1Þ < 1;

we conclude that even

� � jXj þ 1 � s ¼ jXj þ 1 � jXjp

þ r

p� jXj þ 1 � jXj

p:

This yields

Xv2X

degðvÞ � �jXj � jXj2 þ jXj � jXj2

p;

and hence, we deduce from (3)

� � jXj þ 1;


a contradiction, and the proof of (5) is complete. Using (4), we can show

analogously

Xv2T

degðvÞ < ð� þ 1Þ ðp� 1ÞðjY j þ 1Þp

� � þ 1

pðjYj � 1Þ : ð7Þ

The inequalities (5) and (7) lead to

Xv2S

degðvÞ þXv2T

degðvÞ < ð� þ 1Þ p� 1

pðnþ 2Þ � � þ 1

pðjXj � 1Þ

� � þ 1

pðjYj � 1Þ :ð8Þ

Now, it is easy to show that

� þ 1

pðjXj � 1Þ þ� þ 1

pðjY j � 1Þ �4� þ 4

pðn� 2Þ :

As a consequence of (8) we see that

X2�þ2

i¼1

dnþ1�i < ð� þ 1Þ p� 1

pðnþ 2Þ � 4� þ 4

pðn� 2Þ ;

a contradiction to our hypothesis. This completes the proof of the theorem. &

Example 2.4. Let L1 and L2 two copies of the complete graph K2 and H1 and H2

two copies of the complete t-partite graph K2;2;...;2 with t � 2, VðH1Þ ¼fx1; x2; . . . ; x2tg, and VðH2Þ ¼ fy1; y2; . . . ; y2tg. We define the vertex set of the

graph G as the union of the vertex sets of L1; L2;H1, and H2. In addition, we join

all vertices of L1 with all vertices of H1 and all vertices of L2 with all vertices of

H2. Furthermore, we join xi with yi for i ¼ 1; 2; . . . ; 2t and x1 with y2 and x2 with

y1. The resulting graph G is of order 4t þ 4 with clique number !ðGÞ ¼ t þ 2 and

minimum degree �ðGÞ ¼ 2t þ 1. It follows that

X2�þ2

i¼1

dnþ1�i ¼X4tþ4

i¼1

dnþ1�i ¼ 8t2 þ 12t þ 8;

and

ð� þ 1Þ p� 1

pðnþ 2Þ ¼ ð2t þ 2Þ t þ 1

t þ 2ð4t þ 6Þ:


Now, it is easy to see that

8t2 þ 12t þ 8 � ð2t þ 2Þ t þ 1

t þ 2ð4t þ 6Þ;

and consequently, �ðGÞ ¼ �ðGÞ, by Theorem 2.2.

It is straightforward to verify that the graph G fulfills neither the conditions

of Theorems 1.1–1.4 nor of Theorem 2.1. In addition, Example 2.3 doesn’t fulfill

the conditions of Theorem 2.2. Consequently, Theorem 2.2 is independent of

Theorem 2.1.

The next example will show that Theorem 2.2 is best possible when n is even.

Example 2.5. Let H1 ¼ H2 ¼ K2;2;...;2, be the complete t-partite graphs with

t � 3, VðH1Þ ¼ fx1; x2; . . . ; x2tg, and VðH2Þ ¼ fy1; y2; . . . ; y2tg. We define the

graph G as the union of H1 and H2 together with the new edges x1y1; x2y2; . . . ;x2t�2y2t�2. Then, G is of order 4t with clique number !ðGÞ ¼ t and minimum

degree �ðGÞ ¼ 2t � 2. It follows

X2�þ2

i¼1

dnþ1�i ¼X4t�2

i¼1

dnþ1�i ¼ 8t2 � 8t � 2

¼ ð2t � 1Þ t � 1

tð4t þ 2Þ � 8t � 4

tð4t � 2Þ

¼ ð� þ 1Þ p� 1

pðnþ 2Þ � 4� þ 4

pðn� 2Þ :

REFERENCES

[1] B. Bollobas, On graphs with equal edge-connectivity and minimum degree,

Discrete Math 28 (1979), 321–323.

[2] G. Chartrand, A graph theoretic approach to a communications problem,

SIAM J Appl Math 14 (1966), 778–781.

[3] G. Chartrand and L. Lesniak, Graphs and Digraphs, 3rd Edition, Wadsworth,

Belmont, CA, 1996.

[4] P. Dankelmann and L. Volkmann, New sufficient conditions for equality of

minimum degree and edge-connectivity, Ars Combin 40 (1995), 270–278.

[5] P. Dankelmann and L. Volkmann, Degree sequence conditions for maximally

edge-connected graphs and digraphs, J Graph Theory 26 (1997), 27–34.

[6] P. Dankelmann and L. Volkmann, Degree sequence conditions for maximally

edge-connected graphs depending on the clique number, Discrete Math 211

(2000), 217–223.

[7] L. Lesniak, Results on the edge-connectivity of graphs, Discrete Math 8

(1974), 351–354.


[8] J. Plesnık, Critical graphs of given diameter, Acta Fac Rerum Natur Univ

Commenian Math 30 (1975), 71–93.

[9] J. Plesnık and S. Znam, On equality of edge-connectivity and minimum

degree of a graph, Arch Math (Brno) 25 (1989), 19–25.

[10] P. Turan, An extremal problem in graph theory, Mat-Fiz Lapok 48 (1941),

436–452 (in Hungarian).

[11] L. Volkmann, Remarks on the p-fold edge connectivity of graphs. An Univ

Buccuresti Mat 37 (1988), 75–79 (in German).

[12] L. Volkmann, Edge-connectivity in p-partite graphs, J Graph Theory 13

(1989), 1–6.

[13] L. Volkmann, Foundations of graph theory, Springer-Verlag, Wien New York

(1996) (in German).


Documents

Degree sequence conditions for equal edge-connectivity and minimum degree, depending on the clique number