Two local and one global properties of 3-connected graphs on compact 2-dimensional manifolds

http://www.elsevier.com/locate/jctb

Journal of Combinatorial Theory, Series B 93 (2005) 1–21

Two local and one global properties of3-connected graphs on compact

2-dimensional manifolds

S. Jendrol’a and H.-J. Vossb,w

a Institute of Mathematics, Faculty of Science, P.J. Safarik University, Jesenna 5, 041 54 Kosice, SlovakiabDepartment of Algebra, Technical University Dresden, Mommsenstrasse 13. D-01062 Dresden, Germany

Avaulable online 14 May 2004

Abstract

Let GðMÞ be the family of all 3-connected graphs which can be embedded in a compact

2-manifold M with Euler characteristic wðMÞo0: We have proved the following two results:

1. Each graph GAGðMÞ having a k-path, a path on k-vertices, kX4; contains ak-path Pk such that its maximum degree DGðPkÞ in G satisfies

DGðPkÞp2þ ð6k � 6� 2EÞ 1þ jwðMÞj3

� �� ;

where E ¼ 0 for even k and E ¼ 1 for odd k: This bound is best possible.2. Each graph GAGðMÞ of order at least kX5 contains a connected subgraph H of

order k such that its maximum degree DGðHÞ satisfies

DGðHÞp2þ ð4k � 2Þ 1þ jwðMÞj3

� �� :

This bound is best possible.The sharp values of DGðPkÞ and DGðHÞ are determined for kAf2; 3; 4g as well.

r 2004 Elsevier Inc. All rights reserved.

MSC: 05C10; 05C38; 52B10

Keywords: 3-connected graphs; Embeddings; Path; Light subgraphs; Spanning subgraphs

ARTICLE IN PRESS

E-mail address: [email protected] (S. Jendrol’).wDeceased.

0095-8956/$ - see front matter r 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.jctb.2004.03.004

1. Introduction

This paper continues the investigations of [FaJe1,FaJe2,JeVo1,JeVo2,JeVo3].Some of the definitions of [JeVo2] are repeated. All graphs are simple, andmultigraphs can have loops and multiple edges.

In this paper all manifolds are compact two-dimensional manifolds withoutboundary. If a multigraph G is embedded in a manifold M then the closure of theconnected components of M� G are called the faces of G: If each face is an opendisc then the embedding is called a 2-cell embedding. If each vertex has degree atleast three and each vertex of degree h is incident with h different faces then G iscalled a map in M: If, in addition, G is 3-connected and the embedding hasrepresentativity at least three, then G is called a polyhedral map in M; see e.g.[RV,M]. Let us recall that the representativity repðG;MÞ (or face width) of a (2-cell)embedded graph G into a compact 2-manifold M is equal to the smallest number t

such that M contains a noncontractible closed curve that intersects the graph G in t

points. If an embedding of a graph G in M triangulates M we say that G is atriangulation on M: If G is a multigraph triangulating M we say that G is asemitriangulation of M:

Let Sg (Nq) be an orientable (a non-orientable) compact two-dimensional

manifold (called also a surface [Rin2]) of genus g (q; respectively). Let us recall thatthe relationship between Euler characteristic wðMÞ and the genus of a surface M isthe following:

wðSgÞ ¼ 2� 2g or wðNqÞ ¼ 2� q:

We say that H is a subgraph of a map G if H is a subgraph of the underlying graphof the map G:

The degree degG ðaÞ of a face a of a map is the number of edges incident to a whereeach bridge of a is counted twice. Vertices and faces of degree i are called i-vertices

and i-faces, respectively. Let viðGÞ and pjðGÞ denote the number of i-vertices and

j-faces, respectively. For a map G let VðGÞ; EðGÞ; and FðGÞ be the vertex set, theedge set and the face set of G; respectively. The degree of a vertex A in G is denotedby degG ðAÞ or degðAÞ if G is known from the context. A path and a cycle on k

vertices is defined to be the k-path and the k-cycle; respectively. A k-path is denotedby Pk: A k-path passing through vertices A1;A2;y;Ak is denoted by ½A1;A2;y;Akprovided that AiAiþ1AEðGÞ for any i ¼ 1; 2;y; k � 1:

In Sections 1.1 and 1.2 local properties are investigated, namely, the existence ofsmall subgraphs with a required structure will be proved. From this a globalproperty is derived, namely, the existence of a spanning 2-connected subgraph of arequired shape.

1.1. Light paths of order k

It is an old classical consequence of the famous Euler’s formula that each planargraph contains a vertex of degree at most 5. A beautiful theorem of Kotzig[Ko1,Ko2] states that every 3-connected planar graph contains an edge with degree

ARTICLE IN PRESSS. Jendrol’, H.-J. Voss / Journal of Combinatorial Theory, Series B 93 (2005) 1–212

sum of its endvertices being at most 13. This result was further developed in variousdirections and served as a starting point for discovering many structural propertiesof embeddings of graphs, see e.g. [GrSh,Ivan,HJT,JeVo1,JeVo2,Zaks].

In 1997 Fabrici and Jendrol’ [FaJe1] posed the following problem: For a givenconnected graph H let GðH;MÞ be the family of all 3-connected graphs which can beembedded on a compact 2-manifold M with Euler characteristic wðMÞ having asubgraph isomorphic with H: What is the minimum integer jðH;MÞ such that everygraph GAG (H;M) contains a subgraph K isomorphic with H for which

degG ðAÞpjðH;MÞ for every vertex AAVðKÞ?

If such minimum does not exist we write jðH;MÞ ¼ N: If jðH;MÞoN then thegraph H is called light in GðH;MÞ:

The answer to this question for S0 and N1 is contained in

Theorem 1 (Fabrici and Jendrol’ [FaJe1]). Let k be an integer, kX1: Then for M ¼S0 and M ¼ N1 it holds

(i) jðPk;MÞ ¼ 5k; and

(ii) jðH;MÞ ¼ N for all HaPk:

For the sphere M ¼ S0 Fabrici and Jendrol’ proved the result in [FaJe1]. Fromthis proof a proof for the projective plane M ¼ N1 can easily be derived (see[JeVo4]).

For compact 2-manifolds M with wðMÞ ¼ 0 we have proved.

Theorem 2 (Jendrol’ and Voss [JeVo4]). Let k be an integer, kX1: Then

jðPk;S1Þ ¼ jðPk;N2Þ ¼6k for k ¼ 1 and even kX2;

6k � 2 for odd kX3:

�

Ivanco [Ivan] has proved that each graph on Sg of minimum degree at least three

contains an edge with degree sum of their end vertices being at most 2g þ 13 if0pgp3 and at most 4g þ 7; if gX4; and these bounds are sharp. This implies

jðP2;SgÞ ¼2g þ 10 for 0pgp3; and

4g þ 4 for gX4:

�

This result is generalized by our Theorem 3 for 3-connected graphs on compact 2-manifolds M:

Theorem 3. Let k be an integer, kX1; and M a compact 2-manifold of Euler

characteristic wðMÞo0: Let E ¼ 0 for even kX2 and E ¼ 1 for odd kX3: Then

(i) jðP1;MÞp 12ð5þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49þ 24jwðMÞj

pÞ

� ;

(ii) jðP2;N3Þ ¼ 12; jðP2;N4Þ ¼ jðP2;S2Þ ¼ 14; jðP2;N5Þ ¼ 16;

ARTICLE IN PRESSS. Jendrol’, H.-J. Voss / Journal of Combinatorial Theory, Series B 93 (2005) 1–21 3

(iii) jðP3;N3Þ ¼ 17; jðP3;N4Þ ¼ jðP3;S2Þ ¼ 19;(iv) jðPk;MÞ ¼ 2þ ð6k � 6� 2EÞ 1þ jwðMÞj

3

�j kin all other cases.

The corresponding bound for the family of all 3-connected loopless multigraphs

which can be embedded in M without contractible 2-cycles is 6k 1þ jwðMÞj3

�for even

kX2; and ð6k � 2Þ 1þ jwðMÞj3

�j kfor odd kX3; (see [JeVo4]).

The bound for the family of all polyhedral maps has another structure, namely,

the bound lies between 2 k2

� 5þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49�24wðMÞ

p2

� �and k

5þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49�24wðMÞ

p2

� �; for kX1; see

[JeVo1].More information can be found in our survey article [JeVo6].

1.2. Light subgraphs of order k

Fabrici and Jendrol’ [FaJe2] have proved that every 3-connected planar graph G

of order at least k contains a subgraph on k vertices such that each vertex of thissubgraph has, in G; a degree p4k þ 3; for kX3: More precisely, for the sphere thefollowing problem has been investigated, which is more generally formulatedhere.Let k be a positive integer. Let Gðk;MÞ be the family of all 3-connected graphsof order Xk which can be embedded on the compact 2-manifold M with Eulercharacteristic wðMÞ: What is the minimum integer Yðk;MÞ such that every graphGAGðk;MÞ contains a connected subgraph H of order k such that

degG ðAÞpYðk;MÞ

holds for every vertex AAVðHÞ?Fabrici and Jendrol’ [FaJe2], and Jendrol’ and Voss [JeVo3] have proved the

following result for the sphere M ¼ S0 and the projective plane M ¼ N1;respectively.

Theorem 4 (Fabrici and Jendrol’ [FaJe2], Jendrol’ and Voss [JeVo4]). Let k be a

positive integer. Then for M ¼ S0 and M ¼ N1 it holds

(i) Yð1;MÞ ¼ 5;(ii) Yð2;MÞ ¼ 10;(iii) Yðk;MÞ ¼ 4k þ 3 for any kX3:

From Theorems 12 and 14 of [JeVo4] it follows

Theorem 5 (Jendrol’ and Voss [JeVo4]). Let k be a positive integer. Then for M ¼ S1

and M ¼ N2

(i) Yð1;MÞ ¼ 6(ii) Yðk;MÞ ¼ 4k þ 4 for any kX2:


For a compact 2-manifold M with wðMÞo0 we will show:

Theorem 6. Let k be a positive integer, and M a compact 2-manifold of Euler

characteristic wðMÞo0: Then

(i) Yð1;MÞp 12ð5þ


pÞ

� ;

(ii) Yð2;N3Þ ¼ 12; Yð2;N4Þ ¼ Yð2;S2Þ ¼ 14; Yð2;N5Þ ¼ 16;(iii) Yð3;N3Þ ¼ 17; Yð3;N4Þ ¼ Yð3;S2Þ ¼ 19;(iv) Yð4;N3Þ ¼ 22;(v) Yðk;MÞ ¼ 2þ ð4k � 2Þ 1þ jwðMÞj

3

�j kin all other cases

The corresponding bound for the family of all 3-connected loopless multigraphs

which can be embedded in M without contractible 2-cycles is ð4k þ 4Þ 1þ jwðMÞj3

�j kfor any kX2; see [JeVo4]. The bound for the family of all polyhedral maps has

another structure, namely, the bound lies between 2kþ23

� 5þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49�24jwðMÞj

p3

� �� 3

2

� �and

ðk þ 1Þ5þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49�24jwðMÞj

p3

� �for kX4; see [JeVo5].

1.3. Spanning 2-connected subgraphs of small maximum degree

We will deal with global properties. We shall say that a subgraph H of a graph G

spans G if H contains every vertex of G: A 2-connected graph G is s-coverableprovided that it can be spanned by a 2-connected subgraph H of maximum degree atmost s:

In [Bar1] Barnette proved that every planar 3-connected graph has a spanning treeof maximum degree at most 3. In [Bar2] the same author showed that every planar 3-connected graph is 15-coverable. Gao [Gao] improved this second result by showingthat each 3-connected graph in the plane, projective plane, torus, or Klein bottle is 6-coverable. Sanders and Zhao [SaZh] have shown:

Theorem 7 (Sanders and Zhao [SaZh]). Every 3-connected graph G of the genus wðGÞis ð10� 2wðGÞÞ-coverable. If wðGÞp� 5; then G is ð8� 2wðGÞÞ-coverable and for

wðGÞp� 10 G is even ð6� 2wðGÞÞ-coverable. For wðGÞp� 10 the assertion is sharp.

Our next proposition does not provide the precise solution and is not as strong asthe result proved by Sanders and Zhao (Theorem 7). The purpose of our proof ofProposition 8 is to show that our local results imply such a global result, namely,that G is ð20� 6wðMÞÞ-coverable. The opposite direction is not true. Such a globalresult does not imply one of the local results.

Applying our Theorem 3 with k ¼ 4 we are able to prove, in a simple way, theglobal result:


Proposition 8. Every 3-connected graph G embeddable in a compact 2-manifold M

with Euler characteristic wðMÞ is ð20� 6wðMÞÞ-coverable.

2. Minimum degrees of graphs and multigraphs on M

In our paper we shall use

Lemma 1. Let G be an embedding of a graph of order n in a compact 2-manifold M of

Euler characteristic wðMÞ: Let n0 ¼ 12ð7þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49� 24wðMÞ

pÞ

� : Then the number of

edges of G is

eðGÞp1

2nðn � 1Þ for npn0;

3ðn � wðMÞÞ for nXn0

8<:

and the minimum degree of G is

dðGÞpn � 1 for npn0

6 1� wðMÞn

� �� for nXn0:

8<:

Proof. By Lemma 2 of [JeVo4] we have eðGÞp3ðn � wðMÞÞ: Obviously the graph

G has at most 12nðn � 1Þ edges. Both bounds are equal for

n ¼ n0 ¼ 12ð7þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49� 24wðMÞ

pÞ

� : &

3. Proof of Theorem 3—upper bounds

For k ¼ 1 Theorem 3 (assertion (i)) has been proved in [JeVo2]. In the followinglet kX2:

The proof follows the ideas of [FaJe1,JeVo4]. Suppose that there is a counter-example having n vertices. Let G be a counterexample with the maximum number ofedges among all counterexamples having n vertices. A vertex A of the graph G iscalled major (minor) if its degree is greater than (is at most) 12, 14, 14, 16, 17, 19, or19 for the pair ðPk;MÞ ¼ ðP2;N3Þ; ðP2;N4Þ; ðP2;S2Þ; ðP2;N5Þ; ðP3;N3Þ; ðP3;N4Þ; orðP3;S2Þ; respectively. In all other cases the vertex A is called major (minor) if its

degree is greater than (at most) 2þ ð6k � 6� 2EÞ 1þ jwðMÞj3

�j k: Let H ¼ HðGÞ be

the subgraph of G induced on all major vertices of G and let nðHÞ be the number ofvertices of H: Since G is also a 3-connected graph we can apply to G some resultsobtained in [JeVo4] for 3-connected multigraphs without loops and multiple edges.Assertions (5), (7) and (9) of Section 3 in [JeVo4] implyX

AAVðHÞdegG ðAÞp2eðHÞ þ ð6k � 6� 2EÞðnðHÞ þ jwðMÞjÞ; ð1Þ


if nðHÞX3; where E ¼ 0 for even kX2; and E ¼ 1 for odd kX3:XAAVðHÞ

degG ðAÞp2eðHÞ þ tðk � 1Þ; if nðHÞp2; ð2Þ

where 1ptp2ðjwðMÞj þ nðHÞ þ 1Þ:Since G and H are graphs the number eðHÞ of edges of H is

eðHÞpminn�ðn� � 1Þ

2; 3ðn� þ jwðMÞjÞ

� �; where n� ¼ nðHÞ: ð3Þ

We consider three cases.Case 1: Let nðHÞ ¼ n�

X3: Conditions (1) and (3) implyXAAVðHÞ

degG ðAÞpminfn�ðn� � 1Þ; 6ðn� þ jwðMÞjÞg

þ ð6k � 6� 2EÞðn� þ jwðMÞjÞ: ð4Þ

Hence there exists a major vertex B of degree

degG ðBÞpmin n� � 1; 6 1þ jwðMÞjn�

� �� þ ð6k � 6� 2EÞ 1þ jwðMÞj

n�

� �: ð5Þ

By Lemma 1 with n�0 :¼ 1

2ð7þ


pÞ the minimum in (5) is attained at

n� � 1 for n�pn�0 and at 6 1þ jwðMÞj

n�

�for n�

Xn�0; and

degG ðBÞpdðn�Þ :¼n� � 1þ ð6k � 6� 2EÞ 1þ jwðMÞj

n�

� �for n�pn�

0;

ð6k � 2EÞ 1þ jwðMÞjn�

� �for n�

Xn�0:

8>>><>>>: ð6Þ

Here dðn�Þ is a convex function of the real variable n� in ½3; n�0; and a

monotonously decreasing function in [n�0;N). (Note dðn�Þ is a monotonously

decreasing function in ½3;NÞ for large k:) Hence the maximum of dðn�Þ in ½3;NÞ isattained at n� ¼ 3 or n� ¼ n�

0; i.e., degG ðBÞpmaxfdð3Þ; dðn�0Þg; a bound independent

on n�:

dð3Þ ¼ 2þ ð6k � 6� 2EÞ 1þ jwðMÞj3

� �

X k � E3

�5þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49þ 24jwðMÞj

p2

¼ dðn�0Þ ð7Þ

if and only if k ¼ 2; jwðMÞjX5 and k ¼ 3; jwðMÞjX3; and kX4; jwðMÞjX2; and kX6;jwðMÞjX1:

Proof of (7). First inequality (7) is transformed into the equivalent inequality

7þ 4jwðMÞj � 24

3k � E� 12

3k � EjwðMÞjX


p: ð8Þ


This transformation can be obtained by multiplying 63k�E to both sides of (7) and

then adding �5 to them. Instead of checking (7) it is now sufficient to check thevalidity of (8). In each of the four cases k ¼ 2; wðMÞX5; and k ¼ 3; wðMÞX3; andkX4; wðMÞX2 and kX6; wðMÞX1 the left-hand side of (8) can be estimated frombelow (i.e., the smallest possible k is taken in each case) and it can be shown that the

obtained value is always not smaller thanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49þ j24wðMÞj

p: In order to complete the

proof it is sufficient to show that for the remaining eight pairs ðk; wðMÞÞ inequality(8) is not satisfied. &

Conditions (6) and (7) imply the contradiction: the major vertex B has the degree

degG ðBÞpIdð3Þm ¼ 2þ ð6k � 6� 2EÞ 1þ jwðMÞj3

� ��

for the pairs ðk;MÞ described in (7). A contradiction!In all other eight cases

jwðMÞj ¼ 1; 2pkp5; jwðMÞj ¼ 2; 2pkp3;

jwðMÞj ¼ 3; k ¼ 2; jwðMÞj ¼ 4; k ¼ 2;

the bound is degG ðBÞpIdðn�0Þm ¼ k � E

3

� �5þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49þ24jwðMÞj

p2

� �: In the following this

bound will be improved. Now Idðn�Þm is considered only for integers n�A½3;NÞ:The maximum of Idðn�Þm is attained at n� ¼ 3 or n� ¼ In�

0m or n� ¼ Jn�0n; i.e.

degG ðBÞpmaxfIdð3Þm;IdðIn�0mÞm;IdðJn�

0nÞmg: ð9Þ

In Table 1 the maximum is calculated for six of the eight cases of interest. For thetwo remaining cases jwðMÞj ¼ 1; k ¼ 2 and jwðMÞj ¼ 1; k ¼ 4 the stronger formula(10) is derived from the concluding remark of Section 3 in [JeVo4]. It follows from

assertion (15) of that remark (note that 6 1þ jwðMÞjn�0

�¼ n�

0 � 1Þ:

degGðBÞpd 0ðn�Þ :¼n� � 1þ ðk � 1Þ 6 1þ jwðMÞj

n�

� �� for n�pn�

0

k 6 1þ jwðMÞjn�

� �� for n�

Xn�0:

8>>><>>>: ð10Þ

Observe: putting E ¼ 0 in (6) results in a formula similar to (10). For even k

assertion (10) is stronger than assertion (6) because of the use of the integer function

1þ jwðMÞjn�

j k: We use this fact for the cases jwðMÞj ¼ 1; k ¼ 2; and jwðMÞj ¼ 1; k ¼ 4:

For odd k the assertion (10) is weaker than (6) because in (6) E is one and not zero.Hence we cannot replace (10) by (6) in the whole proof.

Similar reasoning that established (9) implies (11).

degG ðBÞpmaxfd 0ð3Þ; d 0ðIn�0mÞ; d 0ðJn�

0nÞg: ð11Þ

In Tables 1 and 2 the bounds so obtained are scheduled.


Besides the case no. 6 in all cases 1–8 the bound for degG ðBÞ leads to acontradiction!

The case no. 6, i.e., jwðMÞj ¼ 1 and k ¼ 3; will be investigated more precisely.Idð3Þm ¼ 15 and Idð7Þm ¼ 17 imply Idðn�Þmp17 for integers 3pn�p7: SinceIdðn�Þm is a monotonously decreasing function Idðn�ÞmpIdð9Þm ¼ 17 for allintegers n�

X9: Only Idð8Þm ¼ 18; i.e., the value 18 is obtained at the case when thesubgraph H of major vertices of G has precisely 8 vertices. With (1) we have

XAAVðHÞ

degG ðAÞp2eðHÞ þ ð6k � 6� 2EÞðnðHÞ þ jwðMÞjÞ:

If H is a triangulation of N3 then eðHÞ ¼ 3ðnðHÞ þ jwðMÞjÞ ¼ 27: Then HCK�8 :

By Ringel [Rin1] we know that K�8 has no triangular embedding in any surface.

Hence the subgraph H of major vertices is no triangulation of N3; and eðHÞp26:With (1) we obtain

XAAVðHÞ

degG ðAÞp52þ 10ð8þ 1Þ ¼ 142:

Thus H contains a major vertex B of degree degG ðBÞp 1428

� ¼ 17: Hence

degG ðBÞp17; a contradiction! &

Case 2: nðHÞ ¼ 2: By (2) we haveXAAVðHÞ

degG ðAÞp2þ 2ðk � 1ÞðjwðMÞj þ 3Þ

ARTICLE IN PRESS

Table 1

No. jwðMÞj k In�0m Jn�0n Idð3Þm IdðIn�0mÞm IdðJn�0nÞm max.

1 4 2 9 10 16 16 16 16

2 3 2 9 9 14 16 16 16

3 2 3 8 9 18 19 19 19

4 2 2 8 9 12 14 14 14

5 1 5 7 8 31 31 31 31

6 1 3 7 8 15 17 18 18

Table 2

No. jwðMÞj k In�0m Jn�0n d 0ð3Þ d 0ðIn�0mÞ d 0ðJn�0nÞ max.

7 1 4 7 8 26 24 26 26

8 1 2 7 8 10 12 12 12

S. Jendrol’, H.-J. Voss / Journal of Combinatorial Theory, Series B 93 (2005) 1–21 9

and there is a major vertex BAVðHÞ of degree

degG ðBÞp 1þ ðk � 1ÞðjwðMÞj þ 3Þ ¼ 1þ ð3k � 3Þ jwðMÞj3

þ 1

� �

o 2þ ð3k � 3Þ jwðMÞj3

þ 1

� �þ ð3k � 3� eÞ jwðMÞj

3þ 1

� �

¼ 2þ ð6k � 6� 2eÞ jwðMÞj3

þ 1

� �;

a contradiction. &

Case 3: nðHÞ ¼ 1: Let B denote the only major vertex of H: By (2) we have

degG ðBÞp 2ðk � 1ÞðjwðMÞj þ 2Þ ¼ 2þ ð6k � 6� 2EÞ 1þ jwðMÞj3

� �

� 2 k � E 1þ jwðMÞj3

� �� p2þ ð6k � 6� 2EÞ 1þ jwðMÞj

3

� �;

if kXE 1þ jwðMÞj3

�; i.e., if kX2 is even or kX 1þ jwðMÞj

3

�is odd. In order to obtain

the sharp upper bound for all k’s we use a different proof technique. Add B-edges(i.e., edges with end vertex B) so that each face incident with B is a triangle(new loops and multiple edges are allowed). The obtained result is denoted by G�; itis a 3-connected multigraph and each path of k vertices of G� contains the majorvertex B: The number of B-edges in G� is d :¼ degG� ðBÞ: Since G is 3-connected nocontractible 1- or 2-cycle can be generated. But there may be noncontractible 1- or 2-cycles. Each small topological neighborhood of B is an open 2-cell. Thus we canintroduce a direction around B and denote the B-edges by h1; h2;y; hd so that hj is

directly followed by hjþ1 (indices modulo d). Let Rj be the endvertex of hj so that

RjaB if hj is not a loop, and Pj ¼ B if hj is a loop. Since in G� the vertex B is incident

only with triangular faces the edges hj ; hjþ1 and RjRjþ1 bound a triangular face for all

j: This implies

If hm and hn are in a trivial cycle C so that hmþ1; hmþ2;y; hn�1 are in

its inner 2-cell parts then the endvertices of these edges form a path

½Rm;Rmþ1;y;Rn�1;Rn: ð12Þ

Let S denote the set of all B-edges joining B with a vertex of G\fBg: Two B-edgeshm; hnAS are said to be equivalent, if hm; hmþ1;y; hn�1; hnAS or hn; hnþ1;y; hm�1;

hmAS: The ‘‘equivalence’’ is an equivalence relation and S is partitioned into

equivalence classes. If T ¼ fhm;y; hng; nam� 1 is such a class then Rm�1 and Rnþ1

are not in G\fBg; i.e., Rm�1 ¼ Rnþ1 ¼ B:

We partition each equivalence class T ¼ fhm;y; hng of B-edges of S into s

pairwise disjoint nonempty classes of a most k � 1 consecutive B-edges so that each

of the classes DT1 ; DT

2 ;y;DTs�1 contains precisely k � 1 consecutive B-edges of T ;

and the class DTs contains the remaining consecutive B-edges of T : Obviously, s ¼

n�mþ1k�1

� �where n� m (modulo d) is to be chosen so that 0pn� mod:


Let ri :¼ mþ ði � 1Þðk � 1Þ; 1pips: Then

DT1 :¼ fhm; hmþ1;y; hmþðk�1Þ�1g;

Dii :¼ fhri

; hriþ1;y; hriþ1�1g; 2pips � 1;

DTs :¼ fhrs

; hrsþ1;y; hng:

Obviously, r1 ¼ m; and DT1 ¼ fhr1

; hr1þ1;y; hr2�1g: In each equivalence class T we

label the first edge of each DTi ; i.e., we label hri

; 1pips: Next, we consider the system

O of all subclasses of all equivalence classes T and denote the members so that

O ¼ fC1;C2;y;Ctg; i.e., for each subclass DTi of each equivalence class T there

exists precisely one index j with Cj ¼ DTi : In Section 3 of [JeVo4] we found a bound

for the total number t of such subclasses (the number t here is the same as in Section 3of [JeVo4]).

In assertion (9) of Section 3 in [JeVo4] we have proved thattp2ðjwðMÞj þ 2Þ: ð13Þ

Till here the proof for nðHÞ ¼ 1 is the same as in Case II of Section 3 fornðHÞAf1; 2g in [JeVo4]. The difference in the proofs is in assertion (14), which isonly true for graphs.

At most one class contains k � 1 B-edges of the graph G;

all other classes have only pk � 2 B-edges of G: ð14Þ

Proof of (14). Assume two classes, say C1 and C2; contain k � 1 B-edges of G: Thenthe endvertices of Ci form in G\fBg a path pj of k � 1 vertices, j ¼ 1; 2: (Note G is a

graph.) These paths are longest paths of G\fBg: Since G is 3-connected G\fBg isconnected, and it is well known that p1 and p2 have at least one common vertex V :Since at most one edge of G joins V and B; the path p1 or p2 is joined with B bypk � 2 edges. This contradiction proves (14). &

With (13) and (14) we obtain

degG ðBÞp ðk � 1Þ þ ðt � 1Þðk � 2Þ

¼ 1þ tðk � 2Þp1þ 2ðk � 2ÞðjwðMÞj þ 2Þ

¼ 1þ ð6k � 6� 6Þ jwðMÞj3

þ 2

3

� �

o 2þ ð6k � 6� 2EÞ jwðMÞj3

þ 1

� �:

4. Proof of Theorem 6—upper bounds

If kp3 then Theorem 6 coincides with Theorem 3. Hence, we have to prove Theorem6 only for kX4: The proof follows the ideas of the proof of Theorem 3 in Section 3.


Suppose there is a counterexample having n vertices. Let G be a counterexamplewith the maximum number of edges among all counterexamples having n vertices. Avertex A of the graph G is major (minor) if its degree is greater than (at most) 22, if

k ¼ 4 and M ¼ N3; and greater than (at most) 2þ ð4k � 2Þ 1þ jwðMÞj3

�j kotherwise.

Again we can apply to G some results obtained in [JeVo4]. Let H be a subgraph of G

induced by major vertices of H: Let nðHÞ be the number of vertices in H: Assertion(1) of Section 9 in [JeVo4] impliesX

AAVðHÞdegG ðAÞp2eðHÞ þ ð2k � 1Þ2ðnðHÞ þ jwðMÞjÞ: ð1Þ

With Lemma 1 we have

eðHÞpminnðHÞðnðHÞ � 1Þ

2; 3ðnðHÞ þ jwðMÞjÞ

� �: ð2Þ

Case 1: Let n� :¼ nðHÞX3: Conditions (1) and (2) implyXAAVðHÞ

degG ðAÞpminfn�ðn� � 1Þ; 6ðn� þ jwðMÞjÞg þ ð4k � 2Þðn� þ jwðMÞjÞ:

ð3Þ

Hence, there exists a vertex B of degree

degG ðBÞp 2eðHÞn� þ ð4k � 2Þ 1þ jwðMÞj

n�

� �

pmin n� � 1; 6 1þ jwðMÞjn�

� �� þ ð4k � 2Þ 1þ jwðMÞj

n�

� �: ð4Þ

By Lemma 1 with n�0 :¼ 1

2ð7þ


pÞ the minimum is attained at n� � 1

for n�pn�0 and at 6 1þ jwðMÞj

n�

�for n�

Xn�0; and

degG ðBÞpdðn�Þ :¼n� � 1þ ð4k � 2Þ 1þ jwðMÞj

n�

� �for n�pn�

0;

ð4k þ 4Þ 1þ jwðMÞjn�

� �for n�

Xn�0:

8>>><>>>: ð5Þ

The function dðn�Þ is a convex one of the real variable n� in ½3; n�0; and a

monotonously decreasing function in ½n�0;NÞ: Hence, the maximum of dðn�Þ in

½3;NÞ is attained at n� ¼ 3 or n� ¼ n�0; i.e., degG ðBÞpmaxfdð3Þ; dðn�

0Þg; a bound

independent on n�:

dð3Þ ¼ 2þ ð4k � 2Þ 1þ jwðMÞj3

� �X

2k þ 2

3

5þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49þ 24jwðMÞj

p2

¼ dðn�0Þ ð6Þ

if and only if kX4; jwðMÞjX2; and kX7; jwðMÞjX1:


Proof of (6). Assertion (6) can be proved by studying the equivalent inequality

7þ 4jwðMÞj � 12

k þ 1� 6

k þ 1jwðMÞjX


p: &

Inequalities (5) and (6) imply the contradiction

degG ðBÞpIdð3Þm ¼ 2þ ð4k � 2Þ 1þ jwðMÞj3

� ��

for the pairs ðk;MÞ described in (6).

In all other three cases jwðMÞj ¼ 1; 4pkp6; the bound is degG ðBÞpIdðn�0Þm ¼

2kþ23

5þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi49þ24jwðMÞj

p2

� �: Thus for k ¼ 6 the degree degG ðBÞp31; and for k ¼ 4 the

degree degG ðBÞp22; a contradiction!For k ¼ 5; i.e., for M ¼ N3 and k ¼ 5 the degree degG ðBÞp27: In the following

this bound will be improved. If the integer n� is in the interval 3pn�p7 ¼ In�0m then

the convex function Idðn�Þm ¼ n� � 1þ ð4k � 2Þ 1þ jwðMÞjn�

�j khas its maximum at

n� ¼ 3 or n� ¼ In�0m ¼ 7; i.e., Idðn�ÞmpmaxfIdð3Þm;Idð7Þmg ¼ 26: If the integer

n�XJn�

0nþ 1 then the monotonously decreasing function Idðn�Þm ¼

ð4k þ 4Þ 1þ jwðMÞjn�

�j khas its maximum at n� ¼ Jn�

0nþ 1 ¼ 9; i.e.,

Idðn�ÞmpIdð9Þm ¼ 26 for all integers n�X9:

Hence by (5) the degree degG ðBÞp26 for all n�a8: Next let n� ¼ Jn�0n ¼ 8; i.e.,

the subgraph H of major vertices of G has precisely 8 vertices.If H is a triangulation of N3 then eðHÞ ¼ 3ðnðHÞ þ jwðMÞjÞ ¼ 27: Then HCK�

8 :By Ringel [Rin1] we know that K�

8 has no triangular embedding in any surface.

Hence the subgraph H of major vertices is no triangulation of N3 and eðHÞp26:With (4) we obtain

degG ðBÞp 2eðHÞJn�

0nþ ð4k � 2Þ 1þ jwðMÞj

Jn�0n

� �� ¼ 26:

Hence for k ¼ 5; and jwðMÞj ¼ 1 the graph G contains a vertex of degreedegG ðBÞp26 a contradiction. Thus in Case 1 the proof is complete.

Case 2: Let 1pn�p2: Since G is 3-connected the subgraph H 0 induced by theminor vertices of G has precisely one component K of order pk � 1: Hence eachvertex of H is joined by at most k � 1 edges with vertices of H 0: Consequently, eachvertex B of H has a degree

degG ðBÞpko2þ ð4k � 2Þ 1þ jwðMÞj3

� �� :

This contradiction completes the proof.


5. Proofs of Theorems 3(iv) and 6(v)—lower bounds

In order to prove the lower bound we use a construction of Zaks [Zaks] for thecase of the path P2 on orientable compact 2-manifolds and generalize it to all Pk;kX2; on an arbitrary 2-manifold M of Euler characteristic wðMÞp0: Theconstruction starts with the triangulation of the rectangle presented in Fig. 1.Identifying opposite sides results in a semitriangulation T1 of the torus S1 by amultigraph on three vertices, each pair of vertices is joined by three edges.

Reversing one side of the rectangle R of Fig. 1 and then identifying opposite sidesof R results in a semitriangulation Q2 of the Klein bottle N2 by a multigraph on threevertices each pair of vertices is joined by three edges. Introducing a crosscap into T1

and adding a 3-cycle (i.e. adding three further edges forming a cycle) via a crosscapas indicated in Fig. 2 results in a semitriangulation of N3 by a multigraph on threevertices each pair of vertices is joined by four edges. We define Tg and Qq inductively.

Suppose Tg�1; gX2; or Qq�2; qX4; have already been constructed. Delete a

triangular face in each of the two triangulations Tg�1 and T1 or Qq�2 and T1;

respectively, and identify the boundaries properly along the three edges. Theobtained semitriangulation of Sg or Nq is denoted by Tg and Qq; respectively.

ARTICLE IN PRESS

Fig. 1.

Fig. 2.

S. Jendrol’, H.-J. Voss / Journal of Combinatorial Theory, Series B 93 (2005) 1–2114

Tg and Qq are triangular embeddings of the multigraph on three vertices into

M ¼ Sg or M ¼ Nq; respectively, each pair of vertices is joined by jwðMÞj þ 3 edges,

and consequently Tg and Qq have 3 vertices, 3ðjwðMÞj þ 3Þ edges and f ¼ 2ðjwðMÞj þ3Þ triangular faces. Each of the three vertices has degree 2jwðMÞj þ 6:

Next let H ¼ Tg if M ¼ Sg and H ¼ Qq if M ¼ Nq: Let A1; A2; A3 be the vertices

of H: Into each triangle we insert a generalized 3-star S3 consisting of a central vertexZ and three paths, say p1; p2 and p3; starting in Z and including Z; the path p3 has

length Jk2n and the other two paths p1 and p2 have length Ik

2m:

We partition the set of the triangle faces of H into three classes C1; C2; and C3 sothat jC1jXjC2jXjC3j and jjCij � jCjjjp1 for all 1pi; jp3: In Ci we join Ai with all

2Ik2m� 1 vertices of p1,p2; and Aiþ1 and Aiþ2 with all Jk

2nþ Ik

2m� 1 ¼ k � 1

vertices of p2,p3 or p3,p1; respectively (indices modulo 3). Finally we delete allAiAj-edges, 1pi; jp3; and add the 3-cycle ½A1;A2;A3;A1: The obtained graph is G:

It is obviously not polyhedral. The degree of the vertex Ai in the new subgraph H isnow 2.

If kX2 is an even integer then

degG ðAiÞ ¼ degHðAiÞ þ ðk � 1ÞðjC1j þ jC2j þ jC3jÞ

¼ degHðAiÞ þ ðk � 1Þf ¼ 2þ ðk � 1Þð6þ 2jwðMÞjÞ:

If kX3 is an odd integer then

degG ðA3ÞX degG ðA2ÞXdegG ðA1Þ

¼ degHðA1Þ þ jC1jðk � 2Þ þ jC2jðk � 1Þ þ jC3jðk � 1Þ

¼ 2þ ðk � 1ÞðjC1j þ jC2j þ jC3jÞ � jC1j ¼ 2þ ðk � 1Þf � jC1j

¼ 2þ ðk � 1Þð6þ 2jwðMÞjÞ � 6þ 2jwðMÞj3

� �:

If k ¼ 1 or kX2 is even then each Pk of G contains a vertex of degree 2þ ðk �1Þð6þ 2jwðMÞjÞ: If kX3 is odd then each Pk of G contains a vertex of degree 2þ

ðk � 1Þð6þ 2jwðMÞjÞ � 6þ2jwðMÞj3

l m¼ 2þ k � 1� 1

3

� �ð6þ 2jwðMÞjÞ

� :

The lower bound of Theorem 6(v) is obtained by replacing the generalized 3-starS3 by the generalized 3-star S�

3 of order k � 1 in each face of G: The generalized

3-star S�3 consists of a central vertex Z and three paths, say p1; p2 and p3; starting in

Z and including Z; one of length kþ13

� ; the second of length kþ2

3

� ; and the third of

length kþ33

� :

6. Proof of Theorems 3(i)—(iii) and 6(i)–(iv)—lower bounds

The bound jðP2;MÞX2 12ð5þ


pÞ

� for MAfN3;N4;N5g,fS2g is

attained at the kleetope of the embedding Hn of the complete graph Kn into M with


n ¼ 12ð5þ


pÞ

� þ 1; here kleetope means that in each region of Hn a

new vertex is placed, and this vertex is joined with each vertex on the boundary ofthat region by precisely one edge.

In order to obtain the bounds jðP3;N3ÞX17; jðP3;N4ÞX19 and jðP3;S2ÞX19we start with embeddings of K7 into N3; and of K8 into N4 and S2; respectively.

Fig. 3 represents a 2-cell embedding of a 3-regular graph H into N3 with 16vertices and 7 faces (see [Rin2]). At each 3-valent vertex an arrow points to one of its

neighbouring faces. The dual eHH is a triangular embedding of a multigraph with 7vertices and 16 triangular faces containing an embedding U of K7:

Let F be any vertex of H; and A1; A2; A3 the adjacent faces of F ; where the arrow

at F points to A3: Let eFF denote the corresponding triangular face of eHH and ½fA1A1; fA2A2;fA3A3; fA1A1 its bounding cycle, where eAiAi corresponds to Ai: Then we introduce into eFF aK2 and join one vertex with A1; A2; A3 and the second vertex only with A1 and A2

(see Fig. 4).

We apply this construction to all triangular faces of eHH; and finally we delete all

edges of Eð eHHÞ � EðUÞ: The obtained graph G is embedded in N3 and each vertex A

of eHH has a degree degG ðAÞX17: Hence jðP3;N3ÞX17:

ARTICLE IN PRESS

Fig. 3.


The embedding of H can also be described by Scheme 1.In Scheme 1 the line i : i1; i2; i3;y describes the face i and its neighbouring faces

i1; i2; i3;y in this cyclic order (at this moment forget the sign �). From Scheme 1 theembedding H in N3 can be reconstructed as follows: represent each face i by apolygon with sides i1; i2; i3;y; then identify side i1 of the polygon i with the side i ofthe polygon i1 so that side i2 of the polygon i is adjacent to the side i2 of the polygoni1 (see Fig. 5).

Repeating this procedure for all sides of all polygons results in an embedding of H

into N3 which is homeomorphic to the embedding depicted in Fig. 3. In order toobtain the embedding G from H we have to interprete the sign �: The sign � in line i

between it and itþ1 means that K2 introduced in the triangle ½i; it; itþ1; i of the dual

embedding eHH is joined with vertex i by only one edge, otherwise K2 is joined withvertex i by precisely two edges.

Next Schemes 2, 3 and 4 are used to represent embeddings of graphs into compact2-manifolds. Schemes 5.6 and 5.7 of Chapter 5 in [Rin2] represent 2-cell embedding

ARTICLE IN PRESS

Fig. 4.

Scheme 1.


of 3-regular graphs into N4 and S2; respectively, with 24 vertices and 10 faces.Identifying two pairs of faces in each of them results in 2-cell embeddings of3-regular graphs H1 and H2 into N4 and S2; respectively, with 20 vertices and 8 faces

0; 1;y; 7 represented by Schemes 2 and 3. The duals fH1H1 and fH2H2 are triangular

ARTICLE IN PRESS

Fig. 5.

Scheme 2.

Scheme 3.


embeddings of a multigraph, into N4 or S2 with 8 vertices and 20 triangular facescontaining an embedding U1 or U2 of K8; respectively.

The construction introduced above is applied to fHjHj; j ¼ 1; 2: The obtained graphs

G1 and G2 are embedded in N4 or S2; respectively. In Gj each vertex of fHjHj has a

degree at least 19. Hence each path with three vertices contains a vertex of fHjHj of

degree least 19.Next we turn to the embedding of the graph H of Fig. 1, also described by Scheme

1. The dual eHH contains an embedding U of K7: The embedding of H is againpresented in Scheme 4 with a new placement of the signs �:

Let F be any vertex of H and eFF the corresponding triangular face of the dual

embedding eHH: Then we introduce in eFF a path P3 with three vertices and join P3 with

one vertex of eFF by three edges and with the other two vertices of eFF by precisely two

edges. We apply this construction to all triangle faces of eHH: The sign � in line i

between it and itþ1 means that the path P3 introduced in the triangle ½i; it; itþ1; i ofthe dual embedding eHH is joined with vertex i by three edges, otherwise P3 is joinedwith vertex i by precisely two edges.

Finally we delete all edges of EðHÞ � EðUÞ: The obtained graph G0 is embedded in

N3 and each vertex A of eHH has a degree degG0 ðAÞX22: Hence Yð4;N3ÞX2:

7. Spanning 2-connected graphs in embedded 3-connected graphs

For the sake of completeness we repeat the most important definitions. A graph isk-connected provided that between any two of its vertices there are k paths meetingonly at their endpoints. An edge of a 3-connected graphs G is contractible ifshrinking it to a vertex produces a new 3-connected graph (when we shrink an edge,resulting multiple edges are coalesced). An edge is removable if removing it producesa new 3-connected graph. (When we remove an edge two edges at any resultingvertex of degree 2 are coalesced into a single edge.) We shall say that a subgraph H

of a graph G spans G if H contains every vertex of G: A 2-connected graph G iss-coverable provided that it can be spanned by a 2-connected subgraph H ofmaximum degree at most s:

ARTICLE IN PRESS

Scheme 4.


We shall use the following lemma of Thomassen [Thom]:

Lemma 2. If e is any edge in a 3-connected graph of order 44; then either e is

contractible or removable.

Proof of Proposition 8. Our proof is by induction on the number of edges of G: If G

has 4 or 5 edges then the theorem clearly holds. We assume the theorem to be truefor all graphs with at most n edges and suppose G has n þ 1 edges.

By Theorem 3(iv), G has a 4-path P4 ¼ ½VXYZ having all vertices of degreepr ¼ ð20� 6wðMÞÞ: Let e be the edge XY of P4: By Lemma 2 the edge e is eithercontractible or removable. We treat two cases.

Case 1: e is contractible. We contract e to a vertex W producing a graph G0 withfewer edges which is coverable by a subgraph H 0: Now we split the vertex W toproduce the original graph G: This induces a splitting in H 0: Let K be the graphproduced from H 0 by this splitting. All vertices of K remain of degree at most r:

Subcase 1.1: No edge of K meets X. Therefore, in K ; degðVÞpr � 1 anddegðYÞpr � 1 because both V and Y are of degree pr in G: If we add the edgesVX and XY to K we obtain a 2-connected spanning subgraph H of G whosemaximum degree is at most r.

Subcase 1.2: Edges of K meet both X and Y. In this case, in K ; degðXÞpr � 1 anddegðYÞpr � 1: If we add the edge XY to K we obtain the required spanningsubgraph H of G:

Case 2: e is removable. We remove e producing a graph G0 with fewer edges, whichby induction, is r-coverable by a subgraph H 0: We return the edge e and consider H 0

as a subgraph of G:Subcase 2.1: X (or Y or both) is of degree 3 in G. In this case H 0 does not meet X

(or Y or both). We add the edges VX and XY (XY and YZ or VX ;XY and ZY ;respectively) to H 0: Since the vertices V ;X ;Y ;Z have a degree pr in G they havealso a degree pr in the subgraph K : The result is a 2-connected spanning subgraphH of G having required properties.

Subcase 2.2: Neither X or Y is of degree 3 in G. In this case H 0 is the requiredspanning subgraph of G: &

Acknowledgments

Support of Slovak VEGA Grant 1/0424/03 is acknowledged. The authors thanktwo anonymous referees for detailed and constructive comments.

References

[Bar1] D. Barnette, Trees in polyhedral graphs, Canad. J. Math. 18 (1966) 731–736.

[Bar2] D. Barnette, 2-connected spanning subgraphs of planar 3-connected graphs, J. Combin. Theory

B 61 (1994) 210–216.


[FaJe1] I. Fabrici, S. Jendrol’, Subgraphs with restricted degrees of their vertices in planar 3-connected

graphs, Graphs Combin 13 (1997) 245–250.

[FaJe2] I. Fabrici, S. Jendrol’, Subgraphs with restricted degrees of their vertices in planar graphs,

Discrete Math. 191 (1998) 83–90.

[Gao] Z. Gao, 2-connected coverings of bounded degree in 3-connected graphs, J. Graph Theory 20

(1995) 327–338.

[GrSh] B. Grunbaum, G.C. Shephard, Analogues for tiling of Kotzig’s theorem on minimal weights of

edges, Ann. Discrete Math. 12 (1982) 129–140.

[HJT] J. Harant, S. Jendrol’, M. Tkac, On 3-connected plane graphs without triangular faces,

J. Combin. Theory B 77 (1999) 150–161.

[Ivan] J. Ivanco, The weight of a graph, Ann. Discrete Math. 51 (1992) 113–116.

[JeVo1] S. Jendrol’, H.-J. Voss, Subgraph with restricted degrees of their vertices in large polyhedral

maps on compact 2-manifolds, European J. Combin. 20 (1999) 821–832.

[JeVo2] S. Jendrol’, H.-J. Voss, A local property of polyhedral maps on compact 2-dimensional

manifolds, Discrete Math. 212 (2000) 111–120.

[JeVo3] S. Jendrol’, H.-J. Voss, Light paths with an odd number of vertices in polyhedral maps,

Czechoslovak Math. J. 50 (125) (2000) 555–564.

[JeVo4] S. Jendrol’, H.-J. Voss, Light subgraphs of multigraphs on compact 2-dimensional manifolds,

Discrete Math. 33 (2001) 329–351.

[JeVo5] S. Jendrol’, H.-J. Voss, Subgraph with restricted degrees of their vertices in polyhedral maps on

compact 2-manifolds, Ann. Combin. 5 (2001) 211–226.

[JeVo6] S. Jendrol’, H.-J. Voss, Light subgraphs of graphs embedded in 2-dimensional manifolds of

Euler characteristic p0—a survey, in: G. Halasz, L. Lovasz, M. Simonorits, V.T. Sos (Eds.), P.

Erdos and His Mathematics, Bolyai Society Mathematical Studies, Vol. 11, Springer, Budapest,

2002, pp.375–411.

[Ko1] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Math. Cas. SAV(Math. Slovaca)

5 (1955) 111–113.

[Ko2] A. Kotzig, Extremal polyhedral graphs, Ann. New York Acad. Sci. 319 (1979) 569–570.

[M] B. Mohar, Face-width of embedded graphs, Math. Slovaca 47 (1997) 35–63.

[Rin1] G. Ringel, Wie man die geschlossenen nichtorientierbaren Flachen in moglichst wenig Dreicke

zerlegen kann, Math. Ann. 130 (1955) 317–326.

[Rin2] G. Ringel, Map Color Theorem, Springer, Berlin, 1974.

[RV] N. Robertson, R.P. Vitray, Representativity of surface embeddings, in: B. Kozte, L. Lovasz,

H.J. Promel, A. Schrijver (Eds.), Paths, Flows and VLSI—Layout, Springer, Leipzig, 1972.

[SaZh] D.P. Sanders, Y. Zhao, On 2-connected spanning subgraphs with low maximum degree,

J. Combin. Theory B 74 (1998) 64–86.

[Thom] C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press,

Canada, 1984, pp. 43–69.

[Zaks] J. Zaks, Extending Kotzig’s theorem, Israel J. Math. 45 (1983) 281–296.


Documents

Two local and one global properties of 3-connected graphs on compact 2-dimensional manifolds