Long cycles in 3-connected graphs in orientable surfaces

Long Cycles in3-Connected Graphs inOrientable Surfacesy

Laura Sheppardson* and Xingxing YuSCHOOL OF MATHEMATICS

GEORGIA INSTITUTE OF TECHNOLOGY

ATLANTA, GEORGIA 30332

E-mail: [email protected]

Received August 21, 2000

DOI 10.1002/jgt.10051

Abstract: In this article, we apply a cutting theorem of Thomassen toshow that there is a function f : N ! N such that if G is a 3-connected graphon n vertices which can be embedded in the orientable surface of genusg with face-width at least f (g), then G contains a cycle of length at leastcnlog32, where c is a constant not dependent on g. � 2002 Wiley Periodicals, Inc.

J Graph Theory 41: 69–84, 2002

Keywords: cycle; circuit graph; 3-connected

——————————————————

yMSC Primary 05C38 and 05C50 Secondary 57M15.Contract grant sponsor: NSF grant (to X.U.); Contract grant number; DMS-9970527.*Correspondence to: Laura Sheppardson, School of Mathematics, Georgia Tech,Atlanta, GA 30332. E-mail: [email protected]

� 2002 Wiley Periodicals, Inc.

1. INTRODUCTION

Whitney [11] proved in 1931 that every 4-connected triangulation of the planecontains a Hamilton cycle. In 1956, Tutte [9] showed that in fact all 4-connectedplanar graphs are Hamiltonian. Thomas and Yu [7] have extended this to showthat 4-connected projective planar graphs are also Hamiltonian. Archdeacon et al.[1] showed that this result does not extend to graphs embedded in other surfaces.They proved that for each n, there is a triangulation of an orientable surface whichis n-connected and in which every spanning tree contains a vertex of degree atleast n. (Note that a Hamilton path is a spanning tree with maximum degree 2.)However, in any 5-connected triangulation of any surface with large face-width(see definition below), Yu [12] proved that there is a Hamiltonian path. (This isstill open for non-triangulations.)

We also lose Hamiltonicity in planar graphs if we relax the connectivitycondition. Moon and Moser [6] showed that there exist 3-connected planartriangulations on n vertices whose longest cycles are of length Oðnlog32Þ. Theyconjectured, however, that any 3-connected planar graph on n vertices must con-tain a cycle of length at least �ðnlog32Þ. Chen and Yu [3] recently proved thisconjecture, not only for planar graphs, but also for graphs embeddable in theprojective plane, or the torus, or the Klein Bottle.

This result can be extended to graphs on other surfaces under certain condi-tions. The face-width (or representativity) of a graph G embedded in a surface �is defined to be the number minfj� \ VðGÞj : � is a homotopically non-trivialclosed curve in �, and � \ G � VðGÞg. We will show that if a graph on n verticesis embeddable in some fixed orientable surface with sufficiently large face-width,then it contains a cycle of length at least cnlog32, where c ¼ 1

2ð1

4Þlog32

.The techniques used here are similar to those in [3], combined with a cutting

technique of Thomassen in [8]. In this article, we first review the definitions tobe used in graph cutting and the necessary lemmas. Our main theorem is statedfollowing these definitions. We then define circuit graphs and annulus graphs,special classes of 2-connected planar graphs. We show the existence of ‘‘heavy’’paths in vertex-weighted circuit or annulus graphs. Finally, the previous resultsare combined to prove the main theorem. The theorem addresses ‘‘heavy’’ cyclesin vertex-weighted graphs, which easily extends to long cycles by applying auniform weighting.

Graph Cutting. If H is a graph embedded in an orientable surface �, and Cis a cycle in H, we talk about cutting H and � along C. To cut along C,where C ¼ x1x2 � � � xkx1 is an orientation preserving cycle, we add cycleC0 ¼ x01x

02 � � � x0kx01 immediately to the left of C. Where edge vxi is incident with

C on its left, we remove vxi and replace it with edge vx0i. We then remove theopen region of � between C and C0, and attach a disc to each of C and C0.This yields a new graph H0 embedded in a new surface �0, or new graphs H0

1 andH0

2 embedded in new surfaces �01 and �0

2, respectively.

70 JOURNAL OF GRAPH THEORY

Let H be a 2-connected graph embedded in some surface �, and let C0 and C1

be two disjoint cycles in H which are homotopic as curves in �. Let G be the2-connected subgraph of H contained in the annulus region of � between C0 andC1, including C0 and C1. Note that G can be viewed as a plane graph such that C0

and C1 are its facial cycles and C1 bounds its infinite face. Hence, we call G acylinder with outer cycle C1 and inner cycle C0. The cylinder-width of G is thelargest integer q such that G contains q pairwise disjoint cycles R0; . . . ;Rq�1, allhomotopic to C0 and C1 in �. (We distinguish between inner and outer cycles forconvenience of description. Note that interchanging these labels on C0 and C1

does not affect the cylinder-width.)Suppose that H is embedded in the orientable surface � of genus g, and that G

is an induced subgraph of H which is a cylinder with inner cycle C0 and outercycle C1. Cutting along both C0 and C1 produces graphs H0 (or H0

1;H02 if C0 and

C1 separate the surface) and G on some set of surfaces. Here, we say that thegraph H0 is produced from H by cutting and deleting the cylinder G. Note that ifC0 and C1 are homotopically non-trivial, then H0 (or each of H0

1 and H02) is

embedded in a surface �0 of genus at most g� 1. We use Sg to identify theorientable surface of genus g, which has Euler genus 2g.

The following result is proven in [8] for triangulations, and is given in [2] as asimple extension to all graphs.

Lemma 1.1. For any natural numbers g; r there exists a natural number f ðg; rÞsuch that any 2-connected graph H in Sg having face-width at least f ðg; rÞcontains g pairwise disjoint cylinders Q1; . . . ;Qg of cylinder-width at least rwhose cutting and deletion results in a connected plane graph.

Let Ci be the outer cycle of Qi and let Di be the inner cycle of Qi. Let Q0i � Qi

be a cylinder of H with outer cycle C0i and inner cycle D0

i such that C0i and D0

i

are homotopic to Ci and Di as curves in Sg. We observe that, after cuttingand deletion of Q0

1; . . . ;Q0g from H, the resulting graph is also a connected plane

graph. To see this, let Pi denote the subgraph of G contained in the annulus regionbetween Ci and C0

i, and let Ri denote the subgraph of G contained in the annulusregion between D0

i and Di. Assume that C0i and D0

i are chosen such thatPi \ Ri ¼ ;. Let G be the resulting graph after cutting and deletion of Q1; . . . ;Qg

from H, and let G0 be the resulting graph after cutting and deletion of Q01; . . . ;Q

0g

from H. Clearly, G0 ¼ G [ ðSg

i¼1ðPi [ RiÞÞ. Hence, G0 is also a connected planegraph.

Note that f ðg; rÞ is �ðr4gÞ and f ðg; rÞ � 4g. Using Lemma 1.1, and applying theresult of Chen and Yu [3], B�ohme et al. [2] proved that there is a function cðgÞsuch that every 3-connected graph G with n vertices embeddable in Sg contains acycle of length at least cðgÞnlog32. No face-width requirement is placed on thegraph G, but cðgÞ depends on the genus of the surface. In this article, we showthat by imposing a lower bound on the face-width of G, cðgÞ can be replaced by aconstant which is not dependent on g. To do this, we prove a stronger result forweighted graphs.

LONG CYCLES IN GRAPHS IN SURFACES 71

We will work with graphs which have non-negative vertex weighting. Wherethere is no danger of confusion, we use v 2 G to mean v 2 VðGÞ. Let Rþ denotethe set of non-negative real numbers. Let G be a graph, and w : VðGÞ ! Rþ.If H is a subgraph of G, we define wðHÞ ¼

Pv2H wðvÞ. Define wð;Þ ¼ 0.

We now have the definitions required to precisely state the main result ofthis article.

Theorem 1.1. Let G be a 3-connected graph embedded in Sg and havingface-width � f ðg; 6Þ, and let w : VðGÞ ! Rþ. Then, G contains a cycle R such

thatP

v2R wðvÞlog32 � 1

214wðGÞ

� �log32.

2. CIRCUIT GRAPHS—DEFINITIONS AND BACKGROUND

Let us begin with some notation.Let S be a subgraph of a graph G. Denote by G� S, the subgraph of G induced

by VðGÞ n VðSÞ. Let H be a component of G� S, and let B be the subgraphinduced by EðHÞ [ fxy 2 EðGÞ : x 2 VðSÞ; y 2 VðHÞg. Then, B is called anS-bridge of G. We also call a subgraph B an S-bridge of G if B is induced by asingle edge xy 2 EðGÞ � EðSÞ, where x; y 2 VðSÞ. In both cases, the vertices inVðBÞ \ VðSÞ are called the attachments of B.

Let x; y 2 VðGÞ for a graph G, and let P be a path in G with x; y 2 VðPÞ. ByxPy, we mean the subpath of P between x and y, inclusive. Where, C is a cycle ina plane graph G and x; y 2 VðCÞ, we use xCy to indicate the subpath of C fromx to y, inclusive, which follows the clockwise orientation of C. Where x and y

are the endpoints of a path R in G, we call R an x–y path. Let Y � VðGÞ; x =2 Y . IfR is an x–y path with R \ Y ¼ fyg, we say that R is an x–Y path. A set of x–Ypaths is called an x–Y fan if each pair of the paths has only x in common. If thenumber of paths in an x–Y fan is n, we call it an x–Y n-fan. Given a 2-connectedplane graph G, we call the cycle boundary of the infinite face of G the outer cycle

of G.A circuit graph is a pair ðG;CÞ, where G is a 2-connected plane graph and C is

a facial cycle of G, such that, for every 2-cut S of G, every component of G� Scontains a vertex of C. Given a circuit graph ðG;CÞ and distinct verticesx; y 2 VðCÞ, we say that ðG; xCyÞ is a strong circuit graph if, for every 2-cut S ofG, S \ ðyCx� fx; ygÞ 6¼ ;.

An annulus graph is a triple ðG;C1;C2Þ, where G is a 2-connected plane graph,and C1 and C2 are facial cycles of G, such that, for any 2-cut S of G, everycomponent of G� S contains a vertex of C1 [ C2.

Where G is a connected graph which is not 2-connected, we describe thestructure of G using Tutte’s block graph construction [10]. A block of G is amaximal 2-connected subgraph. For this purpose, we define any subgraphinduced by a single edge to be 2-connected. We construct a graph BlkðGÞ suchthat VðBlkðGÞÞ ¼ fcutvertices ui of Gg [ fblocks Bj of Gg, and ui is adjacent toBj in BlkðGÞ if ui 2 Bj in G. We call a block B an extremal block of G if B


contains exactly one cutvertex. If the graph BlkðGÞ is a simple path, we say that Gis a chain of blocks. Note that for a connected graph G this is equivalent to thecondition that each block of G contains at most two cutvertices, and eachcutvertex of G is contained in at most two blocks. Suppose that G is a chain ofblocks, and that each block of G is induced by a single edge or is a circuit graphðB;CÞ with all cutvertices of G in B lying on the cycle C. Then, G is called achain of circuit graphs.

Let G be a chain of circuit graphs such that, for each block B of G, one of thefollowing holds:

(i) B ¼ G; either B is induced by an edge xy, or there is a facial cycle C of Band there are distinct x; y 2 VðCÞ such that ðB; xCyÞ is a strong circuitgraph.

(ii) B 6¼ G is an extremal block containing cutvertex y of G; either B isinduced by an edge xy, or there is a facial cycle C of G containing y andthere is some x 2 VðCÞ � fyg such that one of ðB; xCyÞ or ðB; yCxÞ is astrong circuit graph.

(iii) B 6¼ G is a non-extremal block containing distinct cutvertices x; y of G;either VðBÞ ¼ fx; yg or there is some facial cycle C in B such thatfx; yg � VðCÞ and one of ðB; xCyÞ or ðB; yCxÞ is a strong circuit graph.

We call G a chain of strong circuit graphs.We will repeatedly make use of the following fact.

Lemma 2.1. Let ni 2 Rþ; i ¼ 1; . . . ;m, and let 0 < r � 1. Then,Pm

i¼1 nri �

ðPm

i¼1 niÞr.

Lemma 2.2. Let ðG; xCyÞ be a strong circuit graph, and let w : VðGÞ ! Rþ.Then G contains an x-y path P such that

Pv2P�y wðvÞ

log32 � wðG� yÞlog32.

Lemma 2.2 is given in [3]. The vertex y is excluded in the above inequality fortechnical reasons, (counting becomes straightforward when piecing togetherpaths), which will be adopted in this article. As a warm-up, we prove a similarresult for the larger class of circuit graphs.

Lemma 2.3. Let ðG;CÞ be a circuit graph, let x; y 2 VðCÞ be distinct, and letw : VðGÞ ! Rþ. Then, G contains an x–y path P such that

Pv2P�y wðvÞ

log32 �½12wðG� yÞ�log32

.

Proof. We assume without loss of generality that G is embedded in theplane with outer cycle C. If either of ðG; xCyÞ or ðG; yCxÞ is a strong circuitgraph, we simply apply Lemma 2.2. Let us assume then that neither is a strongcircuit graph. Hence, G has a 2-cut contained in VðxCyÞ and a 2-cut containedin VðyCxÞ. Let fSk : k ¼ 1; 2; . . . ;mg be the collection of 2-cuts of G withSk � VðxCyÞ, and let fTl : l ¼ 1; 2; . . . ; ng be the collection of 2-cuts of Gwith Tl � VðyCxÞ. For each k ¼ 1; 2; . . . ;m let Bk be the union of all Sk-bridges


of G which do not contain yCx. Define H ¼ ðS

k BkÞ � fx; yg. Similarly, for eachl ¼ 1; 2; . . . ; n define Al as the union of Tl-bridges of G which do not contain xCy,and let L ¼ ð

Sl AlÞ � fx; yg. Since ðG;CÞ is a circuit graph, every component of

G� Sk (respectively, G� Tl) contains a vertex of C. Hence, H \ L ¼ ;.By symmetry, assume that wðHÞ � wðLÞ. A 2-cut Sk � VðxCyÞ is called

maximal if there is no other 2-cut Sj � VðxCyÞ for which Bk � Bj, Bk 6¼ Bj.For each maximal 2-cut Sk of G, remove the subgraph Bk � Sk from G and replaceit with a single edge ek between the vertices of Sk so that the resulting graph is aplane graph. Call the resulting graph G0, and its outer cycle C0.

Now ðG0; xC0yÞ is a strong circuit graph. To see this, note that any 2-cut S0 in G0

would also be a 2-cut in the original graph G. Hence, S0 � C0. Note also that sincewe replaced the bridges of all maximal 2-cuts contained in VðxCyÞ, S0 6� xC0y.Hence, S0 \ ðyC0x� fx; ygÞ 6¼ ;, and so, ðG0; xC0yÞ is a strong circuit graph.

Since wðHÞ � wðLÞ and H \ L ¼ ;, we have wðG0 � yÞ � 12wðG� yÞ.

Applying Lemma 2.2 to ðG0; xC0yÞ, we find an x–y path P0 in G0 such that

Xv2P0�y

wðvÞlog32 � ½wðG0 � yÞ�log32 � 1

2wðG� yÞ

� �log32

:

We then expand P0 to an x–y path P in G as follows: if ek 2 EðP0Þ andSk ¼ fu; vg, simply replace ek with any u–v path in Bk. Now,

Xv2P�y

wðvÞlog32 � 1

2wðG� yÞ

� �log32

: &

Let G be a chain of circuit graphs, and let B be a block of G, where ðB;CÞ is acircuit graph or B is induced by a single edge. Call x 2 VðBÞ an outer vertex of Bif x is not a cutvertex of G, and either x 2 VðCÞ or x has degree 1 in B.

Lemma 2.4. Let G be a chain of circuit graphs, and let w : VðGÞ ! Rþ. If Ghas only one block, let x; y be distinct outer vertices of G. If G has at least twoblocks, let x 2 VðGÞ be an outer vertex of one extremal block of G, and let

y 2 VðGÞ be an outer vertex of the other extremal block of G. Then, G contains anx–y path P such that

Pv2P�y wðvÞ

log32 � ½12wðG� yÞ�log32

.

Proof. Let the blocks of G be labeled by B1; . . . ;Bk, let the cutverticesbe labeled x1; . . . ; xk�1, and let x0 ¼ x and xk ¼ y, so that xi�1; xi 2 Bi fori ¼ 1; . . . ; k. To construct the desired long path in G, we apply Lemma 2.3 toeach of Bi, finding an xi�1 � xi path Pi such that

Pv2Pi�xi

wðvÞlog32 �½12wðBi � xiÞ�log32

. Let P ¼Sk

i¼1 Pi. By applying Lemma 2.1, we have

Xv2P�y

wðvÞlog32 ¼Xi

Xv2Pi�xi

wðvÞlog32 �Xi

1

2wðBi � xiÞ

� �log32

� 1

2wðG� yÞ

� �log32

: &


Let us make a few simple observations about the structure of circuit andannulus graphs. These are stated in a fairly weak form, but they will be helpful tosimplify the extension of our results to annulus graphs. Where G is a plane graphand C is any cycle in G, we define intðCÞ to be the subgraph of G contained in theclosed disc in the plane which is bounded by C.

Lemma 2.5. Let G be a 2-connected plane graph with outer cycle C andanother facial cycle D. Let F be any cycle in G.

(i) If ðG;CÞ is a circuit graph, then ðintðFÞ;FÞ is a circuit graph.(ii) If ðG;C;DÞ is an annulus graph and D � intðFÞ, then ðintðFÞ;F;DÞ is an

annulus graph.

(iii) If ðG;C;DÞ is an annulus graph and D 6� intðFÞ, then ðintðFÞ;FÞ is acircuit graph.

Proof. If intðFÞ is 3-connected, then it is clearly a circuit graph, so weassume that intðFÞ is not 3-connected. Let S be an arbitrary 2-cut in intðFÞ, andlet T be any component of intðFÞ � S.

(i) Assume ðG;CÞ is a circuit graph. Suppose that T \ F ¼ ;, and henceT \ C ¼ ;. Since G is planar, we see that there can be no edges from T toG� intðFÞ. So, S is a 2-cut in G, and T is a component of G� S with T \ C ¼ ;.But, this is a contradiction to the hypothesis that ðG;CÞ is a circuit graph. Hence,T \ F 6¼ ;. Since S and T are arbitrary, ðintðFÞ;FÞ is a circuit graph.

(ii) Suppose ðG;C;DÞ is an annulus graph and D � intðFÞ. IfT \ ðF [ DÞ ¼ ;, then S is a 2-cut of G and T is a component of G� S, andT \ ðC [ DÞ ¼ ;, contradicting the assumption that ðG;C;DÞ is an annulusgraph. Hence, T \ ðF [ DÞ 6¼ ;. Since S and T are arbitrary, ðintðFÞ;F;DÞ is anannulus graph.

(iii) Now suppose ðG;C;DÞ is an annulus graph and D 6� intðFÞ. If T \ F ¼ ;,then S is a 2-cut of G and T is a component of G� S, and T \ ðC [ DÞ ¼ ;,contradicting the assumption that ðG;C;DÞ is an annulus graph. Since S and T arearbitrarily chosen, ðintðFÞ;FÞ must be a circuit graph. &

3. LONG PATHS IN ANNULUS GRAPHS

Lemma 3.1. Let ðA;C;DÞ be an annulus graph. Let w : VðAÞ ! Rþ, and letx; y 2 VðC [ DÞ be distinct. Then, there is an x–y path P in A such that

Xv2P�y

wðvÞlog32 � 1

4wðA� yÞ

� �log32

:

Proof. Throughout the proof, we assume without loss of generality that A isembedded in the plane with outer cycle C.


Case 1. jC \ Dj � 3.

We will define two subgraphs of A, each of which is a circuit graph or chain ofcircuit graphs. We begin by defining U ¼ VðC \ DÞ. Label the vertices of U byu1; . . . ; um in clockwise order along C, and let umþ1 ¼ u1. Let Mi denote the unionof fui; uiþ1g-bridges of A not containing uiþ1Cui [ uiþ1Dui. (see Fig. 1).

We claim that there are Mk and Ml (not necessarily distinct) such that either(1) x 2 Mk � ukþ1, y 2 Ml � ul, and uk 6¼ ulþ1 or (2) x 2 Mk � uk, y 2 Ml � ulþ1,and ul 6¼ ukþ1 (see Fig. 1). Assume without loss of generality that x 2 Mk � ukþ1

and y 2 Ml � ul. If uk 6¼ ulþ1, then (1) holds. So assume uk ¼ ulþ1. Then,ukþ1 6¼ ul because m � 3. If x 6¼ uk and y 6¼ uk, then we have x 2 Mk � uk,y 2 Ml � ulþ1, and ul 6¼ ukþ1, and so, (2) holds. So either x ¼ uk or y ¼ uk. Ifx ¼ uk, then y 6¼ uk, and so, x 2 Ml � ul, y 2 Ml � ulþ1, and ulþ1 6¼ ul, that is, (2)holds with Mk ¼ Ml. So, y ¼ uk and x 6¼ uk. Then, x 2 Mk � uk, y 2 Mk � ukþ1,and uk 6¼ ukþ1, and so, (2) holds with Ml ¼ Mk.

By symmetry, assume (1) holds. Without loss of generality, assume that1 � k � l � m. Let B1 ¼ [l

i¼kMi, and let B2 ¼ A� ðB1 � fuk; ulþ1gÞ. Notethat each Mi is either a subgraph of A induced by a single edge or (by (iii) ofLemma 2.5) a circuit graph with its outer cycle as the special cycle. Hence,B1 and B2 are chains of circuit graphs. Note that when k < l, x and y lie in distinctextremal blocks of B1. If k ¼ l, then B1 is a circuit graph. Also, note that eitherwðB1 � yÞ � 1

2wðA� yÞ or wðB2 � yÞ � 1

2wðA� yÞ; otherwise, we have the

contradiction that wðB1 � yÞ þ wðB2 � yÞ < wðA� yÞ.(1A) Suppose wðB1 � yÞ � 1

2wðA� yÞ. We apply Lemma 2.4 to find an x–y

path P in B1 withXv2P�y

wðvÞlog32 � 1

2wðB1 � yÞ

� �log32

� 1

4wðA� yÞ

� �log32

:

(1B) Now, suppose wðB1 � yÞ < 12wðA� yÞ, so wðB2 � yÞ � 1

2wðA� yÞ. By

Lemma 2.4, there is an uk � ulþ1 path P2 in B2 withP

v2P2�ulþ1wðvÞlog32 �

FIGURE 1. Case 1, jC \ Dj � 3:


½12wðB2 � ulþ1Þ�log32

. Note also that B1 contains two disjoint paths Px;Py from x; yto uk; ulþ1, respectively, which are internally disjoint from B2.

We define an x–y path by P ¼ Px [ P2 [ Py. Then, with an application ofLemma 2.1, we find

Xv2P�y

wðvÞlog32 ¼X

v2Px�uk

wðvÞlog32 þX

v2P2�ulþ1

wðvÞlog32 þX

v2Py�y

wðvÞlog32

�X

v2P2�ulþ1

wðvÞlog32 þ wðulþ1Þlog32

� 1

2wðB2 � ulþ1Þ

� �log32

þwðulþ1Þlog32

� 1

2wðB2Þ

� �log32

� 1

4wðA� yÞ

� �log32

:

Case 2. jC \ Dj � 2.

First, we define subgraphs B1 and B2. Their definitions depend upon the size ofC \ D and are illustrated in Fig. 2.

When jC \ Dj ¼ 2: Let U ¼ fu; vg ¼ C \ D. If uCv ¼ uDv, then letB1 ¼ uCv, which is induced by uv 2 EðGÞ. Otherwise, let B1 ¼ intðF1Þ, whereF1 ¼ uCv [ uDv. If vCu ¼ vDu, let B2 ¼ vCu, which is induced by uv 2 EðGÞ.Otherwise, let B2 ¼ intðF2Þ, where F2 ¼ vCu [ vDu.

When jC \ Dj ¼ 0: Since A is 2-connected, there must be disjoint paths Q andR from VðCÞ to VðDÞ. We may assume that each path intersects C and D only atits endpoints. Let qc 2 VðCÞ and qd 2 VðDÞ be the endpoints of Q, and rc 2 VðCÞand rd 2 VðDÞ the endpoints of R. Let U ¼ fqc; qd; rc; rdg. Define B1 ¼ intðF1Þ,where F1 ¼ qcCrc [ Q [ qdDrd [ R. Define B2 ¼ intðF2Þ, where F2 ¼ rcCqc[Q [ rdDqd [ R.

FIGURE 2. Case 2, jC \ Dj � 2:


When jC \ Dj ¼ 1: Let fug ¼ C \ D. Since A is 2-connected, A� u isconnected. We may find a path Q from VðCÞ to VðDÞ in A� u, such that Q

intersects C and D only at its endpoints. Let fqcg ¼ Q \ C and fqdg ¼ Q \ D.Let, U ¼ fqc; qd; ug. Define B1 ¼ intðF1Þ, where F1 ¼ uCqc [ Q [ uDqd. DefineB2 ¼ intðF2Þ, where F2 ¼ qcCu [ Q [ qdDu.

We proceed in the same manner for all three of the above situations.By the above definitions, D 6� B1 and D 6� B2. From (iii) of Lemma 2.5 both

ðB1;F1Þ and ðB2;F2Þ are circuit graphs, except when B1 or B2 is induced by asingle edge.

By symmetry, assume that wðB1 � yÞ � wðB2 � yÞ. Then, wðB1 � yÞ �12wðA� yÞ. In this case, B1 cannot beinduced by a single edge, and so, ðB1;F1Þ

is a circuit graph.

(2A) First, we consider the case x; y 2 B1. Then, by Lemma 2.3, there is an x–ypath P in B1 with

Xv2P�y

wðvÞlog32 � 1

2wðB1 � yÞ

� �log32

� 1

4wðA� yÞ

� �log32

:

(2B) Now consider the case x; y =2 B1, noting that B2 may not be induced by asingle edge in this case. We recall that x; y 2 VðC [ DÞ. Hence B2 containsdisjoint paths Px, Py from x; y to vertices ux; uy 2 U, respectively, withðPx [ PyÞ \ U ¼ ðPx [ PyÞ \ B1 ¼ fux; uyg.

By Lemma 2.3, there is a ux-uy path P1 in B1 with

Xv2P1�uy

wðvÞlog32 � 1

2wðB1 � uyÞ

� �log32

:

Now, we define an x–y path by P ¼ Px [ P1 [ Py. Then, with an application ofLemma 2.1, we have

Xv2P�y

wðvÞlog32 ¼X

v2Px�ux

wðvÞlog32 þX

v2P1�uy

wðvÞlog32 þX

v2Py�y

wðvÞlog32

�X

v2P1�uy

wðvÞlog32 þ wðuyÞlog32

� 1

2wðB1 � uyÞ

� �log32

þwðuyÞlog32

� 1

2wðB1Þ

� �log32

� 1

4wðA� yÞ

� �log32

:


(2C) Finally, suppose x =2B1; y 2 B1. The case x 2 B1; y =2B1 is symmetric.Since x 2 VðC [ DÞ, B2 contains a path P2 from x to some vertex ux 2 U, suchthat P2 \ U ¼ P2 \ B1 ¼ fuxg.

By Lemma 2.3, there is a ux � y path P1 in B1 withP

v2P1�y wðvÞlog32 �

½12wðB1 � yÞ�log32

.Now, we define an x-y path by P ¼ P1 [ P2. Then

Xv2P�y

wðvÞlog32 �X

v2P1�y

wðvÞlog32

� 1

2wðB1 � yÞ

� �log32

� 1

4wðA� yÞ

� �log32

: &

4. PROOF OF THEOREM 1.1

Given a 3-connected graph G embedded in Sg with sufficiently large face-width,as defined by f ðg; rÞ of Lemma 1.1, we apply Lemma 1.1 to create a connectedplane graph G0. We do not use the graph G0 directly. Instead, we use observationsabout the cutting cylinders used to create G0 from G, to construct from G a3-connected plane graph H. With the results of [3], we find that H has a heavycycle. A modification of this cycle is then used to produce a heavy cycle in theoriginal graph G.

Where S and T are disjoint subgraphs of a graph G, we define ½S;T � ¼fxy 2 EðGÞ : x 2 VðSÞ; y 2 VðTÞg.

(1) Selecting the Cutting Cylinders. Let G be a 3-connected graph embeddedin Sg with face-width at least f ðg; 6Þ. Then, by Lemma 1.1, G contains g pairwisedisjoint cylinders Q1; . . . ;Qg of cylinder-width at least six whose cutting anddeletion results in a connected plane graph G0. Let C0

i and C5i be the cycles in

Qi; i ¼ 1; . . . ; g, along which we cut, with C0i the inner cycle of Qi and C5

i itsouter cycle. Note that each of these cutting cycles is homotopically non-trivial inSg (because otherwise G0 would not be connected). By the observation followingLemma 1.1, we may choose Q1; . . . ;Qg such that any cylinder of G properlycontained in Qi with outer cycle and inner cycle homotopic to C0

i and C5i has

cylinder-width <6.In the following discussion, where C is a cycle in G, we will also use C to refer

to the corresponding cycle in G0 or in Qi. Let us view each of the deletedcylinders, Qi, as a plane graph with C5

i as its outer cycle and C0i as a facial cycle.

Since Qi has cylinder-width at least six, there exist disjoint cycles C1i ;C

2i ;C

3i ;C

4i

in Qi, also disjoint from C0i and C5

i , with C0i � intðC1

i Þ � � � � � intðC4i Þ � intðC5

i Þ.Furthermore, we choose C1

i ;C2i ;C

3i ;C

4i such that the following graphs are

minimal: intðC1i Þ, intðC3

i Þ � ðintðC2i Þ � C2

i Þ, and intðC5i Þ � ðintðC4

i Þ � C4i Þ.


Claim 1. There are no vertices between C0i and C1

i , between C2i and C3

i , andbetween C4

i and C5i . More precisely, VðintðC1

i Þ � intðC0i ÞÞ ¼ VðC1

i Þ, VðintðC3i Þ�

intðC2i ÞÞ ¼ VðC3

i Þ, and VðintðC5i Þ � intðC4

i ÞÞ ¼ VðC5i Þ.

Proof. Suppose there is some vertex v 2 ðintðC1i Þ � intðC0

i ÞÞ with v =2 VðC1i Þ.

Since G is 3-connected, there is a v � VðC0i [ C1

i Þ 3-fan in intðC1i Þ. That is, there

are three distinct paths from v to the set VðC0i [ C1

i Þ which are disjoint exceptfor v and which intersect VðC0

i [ C1i Þ only at their endpoints. Two of these paths,

say P and Q, must end on the same cycle. Let p; q be the endpoints of P;Q,respectively, other than v.

Case 1. p; q 2 C1i .

Then either C0i � intðP [ Q [ pC1

i qÞ or C0i � intðP [ Q [ qC1

i pÞ. Assume by sym-metry that C0

i � intðP [ Q [ pC1i qÞ. Since EðqC1

i pÞ is outside intðP [ Q [ pC1i qÞ,

intðC1i Þ is properly contained in intðP [ Q [ pC1

i qÞ. Note that P [ Q [ pC1i q is

disjoint from C0i ;C

2i ;C

3i ;C

4i ;C

5i . Hence, P [ Q [ pC1

i q contradicts the choice ofC1i such that intðC1

i Þ is minimal.

Case 2. p; q 2 C0i .

Then either C0i � intðP [ Q [ pC0

i qÞ or C0i � intðP [ Q [ qC0

i pÞ. By symmetry,assume that C0

i � intðP [ Q [ pC0i qÞ, and let C0

i ¼ P [ Q [ pC0i q. Let Q0

i be thecylinder of G with outer cycle C5

i and inner cycle C0i. Since EðqC0

i pÞ is notcontained in Q0

i, Q0i is properly contained in Qi. Clearly, C0

i is homotopic to C0i in

Sg, and Q0i has cylinder width 6, contradicting the choice of Qi.

Hence, we proved that VðintðC1i Þ � intðC0

i ÞÞ ¼ VðC1i Þ. By a similar argument,

we can prove VðintðC5i Þ � intðC4

i ÞÞ ¼ VðC5i Þ. By an argument similar to Case 1,

we can show that VðintðC3i Þ � intðC2

i ÞÞ ¼ VðC3i Þ. This proves the claim.

Let Ai ¼ intðC2i Þ � C0

i . Then, ðAi;C2i ;C

1i Þ can be viewed as an annulus graph

(i.e., it has a natural planar embedding with outer cycle C2i which is an annulus

graph). Let A0i ¼ intðC4

i Þ � intðC2i Þ, with ðA0

i;C4i ;C

3i Þ also seen as an annulus

graph (see Fig. 3).

(2) Constructing the New Graph. Recall that G0 is the connected plane graphobtained from H by cutting and deleting Q1; . . . ;Qg.

Claim 2. G0 is 2-connected, and for any 2-cut S of G0, ð1Þ S � VðC0i Þ or

S � VðC5i Þ for some 1 � i � q, and ð2Þ G0 � S has exactly two components,

each containing a vertex of C0i if S � VðC0

i Þ, or C5i if S � VðC5

i Þ.

FIGURE 3. Cutting cylinder Qi .


Proof. Let S � VðG0Þ be a cutset in G0 with jSj � 2. Let T1; . . . ; Tk be thecomponents of G0 � S. For each j 2 f1; . . . ; kg, there is a simple closed curve �jin the plane such that �j \ G0 ¼ S, and Tj is contained in one open region of theplane bounded by �j, while

Si6¼j Ti is containedin the other open region bounded

by �j. Let �j denote the simple closed curve in Sg corresponding to �j. Then�j \ G ¼ S. Since G has face-width greater than three, �j must be homotopicallytrivial in Sg. Since G is 3-connected, we see that Tj \ Dj 6¼ ; for some cycleDj 2 fC0

i ;C5i : 1 � i � qg. But, since Dj is homotopically non-trivial, no Dj is

contained in the closed region in the plane bounded by �j and containing Tj.Hence, S � Dj for all j 2 f1; . . . ; kg. This forces k ¼ 2 and jSj ¼ 2 (because thecycles C0

i ;C5i are disjoint). This proves the claim.

Now, we turn to the construction of the graph H. Beginning from G, remove all½C2

i ;C3i �, i ¼ 1; . . . ; g. Contract each subgraph Ai to the vertex ai and contract

each A0i to the vertex a0i, deleting multiple edges, and letting ~wðaiÞ ¼ wðAiÞ;

~wða0iÞ ¼ wðA0iÞ, for i ¼ 1; . . . ; g. Let ~wðxÞ ¼ wðxÞ for all x 2 VðG� ðAi [ A0

iÞÞ.Call the resulting graph H. Then, ~wðHÞ ¼ wðGÞ. Note that G�

Sgi¼1ðAi [ A0

iÞ ¼G0. By Claim 2 and since the face-width of G is at least three, we see that H is a3-connected plane graph.

(3) Constructing a Long Cycle. Let C be a facial cycle of H, and let e ¼xy 2 EðCÞ. Since H is 3-connected, ðH; xCyÞ is a strong circuit graph. ApplyingLemma 2.2, we find that there is some x–y path P in H with

Xv2P�y

~wðvÞlog32 � ~wðH � yÞlog32:

Note that we may choose P so that VðPÞ 6¼ fx; yg. Now, let T be the cycle in H

obtained from P by adding the edge e. By Lemma 2.1, we find that

Xv2T

~wðvÞlog32 � ~wðH � yÞlog32 þ ~wðyÞlog32 � ~wðHÞlog32:

However, the cycle T may pass through contracted vertices ai or a0i.Note that any cycle T in H which passes through the contracted vertex ai can

be easily extended to a cycle in G which passes through the annulus subgraph Ai.To reach ai in H, T must pass through two distinct vertices ui; vi on the cyclesurrounding ai, which corresponds to the cutting cycle C0

i in G. We will callvertex ai good in cycle T if there exist edges uixi; viyi 2 EðGÞ with xi; yi distinctvertices on the cycle C1

i . If no such edges exist, we call ai bad in T . Similardefinitions are used for a0i, with respect to C4

i and C5i . (Note that if ai =2VðTÞ, then

ai is considered neither good nor bad in T .)The notation of the following discussion will be simplified by a re-indexing

of the vertices a0i. We let aiþg ¼ a0i for i ¼ 1; . . . ; g, so that faig2gi¼1 ¼ faiggi¼1 [

fa0iggi¼1. Similarly, let Aiþg ¼ A0

i for i ¼ 1; . . . ; g. Define W ¼ T � faig2gi¼1. Let

I ¼ fi 2 Z : ai is good in Tg, J ¼ fj 2 Z : aj is bad in Tg.


We construct a cycle R in G based on the cycle T in H as follows. Foreach i 2 I, we may apply Lemma 3.1 to find an xi–yi path Ri in Ai withP

v2Ri�yiwðvÞlog32 � ½1

4wðAi � yiÞ�log32

. For each j 2 J, let xj 2 Aj withujxj; vjxj 2 EðGÞ. Define the cycle R in G as W [ ð

Si2I uixiRiyiviÞ [ ð

Sj2J ujxjvjÞ.

Note that from our choice of the original cycle T in H, we have

Xv2W

~wðvÞlog32 þXi2I

~wðaiÞlog32 þXj2J

~wðajÞlog32

¼Xv2T

~wðvÞlog32 � ~wðHÞlog32

¼ wðGÞlog32:

So considering the weight of the new cycle R, with an application ofLemma 2.1, we haveX

v2RwðvÞlog32 ¼

Xv2W

wðvÞlog32 þXi2I

Xv2Ri�yi

wðvÞlog32

þXi2I

wðyiÞlog32 þXj2J

wðxjÞlog32

�Xv2W

wðvÞlog32 þXi2I

1

4wðAi � yiÞ

� �log32

þXi2I

1

4wðyiÞ

� �log32

� 1

4

� �log32 Xv2W

~wðvÞlog32 þXi2I

~wðaiÞlog32

" #:

If we haveP

v2W ~wðvÞlog32 þP

i2I ~wðaiÞlog32 � 1

2

Pv2T ~wðvÞlog32

then, we find

Xv2R

wðvÞlog32 � 1

2

1

4wðGÞ

� �log32

:

We may assume then that

Xv2W

~wðvÞlog32 þXi2I

~wðaiÞlog32 <1

2

Xv2T

~wðvÞlog32:

So we must have

Xj2J

~wðajÞlog32 >1

2

Xv2T

~wðvÞlog32:

In this case, we will show that H contains a cycle T 0 through all aj with j 2 J

such that aj is good with respect to T 0. That is, if ujajvj � T 0, then9 xj; yj 2 VðC1

j Þ (or VðC4j�gÞ if j > g) such that xj 6¼ yj and ujxj; yjvj 2 EðGÞ.


For each j 2 J, consider the bipartite graph Gj induced by ½C0j ;C

1j � (or by

½C4j�g;C

5j�g� if j > g). Let Vj be a vertex cover for Gj. Then, we can find a

homotopically non-trivial closed curve �j lying between C0j and C1

j (or C4j�g and

C5j�g) in Sg with �j \ G ¼ Vj. The face-width of G is at least f ðg; 6Þ, so

jVjj � f ðg; 6Þ. Applying Konig’s theorem, we find that there must be a matchingMj in Gj with jMjj � f ðg; 6Þ. Let H0 be obtained from H by deleting edges in½C0

j ;C1j � �Mj (or ½C4

j�g;C5j�g� �Mj if j > g).

Note that jJj � 2g and f ðg; 6Þ as specified in Lemma 1.1 is greater than 4g.Note also that H0 has no set S with jSj < f ðg; 6Þ separating two vertices offaj : j 2 Jg, otherwise the face-width of G is < f ðg; 6Þ. Hence, H0 contains acycle T 0 through all aj; j 2 J. By the construction of H0, the edges of T 0 at aj,j 2 J, correspond to edges in Mj. Since Mj is a matching in G, every aj is goodwith respect to T 0.

Now, let uj; vj be the neighbors of aj in T 0, and let xj 6¼ yj 2 VðC1j Þ (or VðC4

j�gÞif j > g) such that xjuj; yjvj 2 EðGÞ. For j 2 J, applying Lemma 3.1 to Aj, we findan xj–yj path Rj in Aj such that

Xv2Rj�yj

wðvÞlog32 � 1

4wðAj � yjÞ

� �log32

:

Define the cycle R0 in G as ðT 0 � faj : j 2 JgÞ [ ðS

j2J ujxjRjyjvjÞ.We then have, again with an application of Lemma 2.1,

Xv2R0

wðvÞlog32 �Xj2J

Xv2Rj�yj

wðvÞlog32 þXj2J

wðyjÞlog32

�Xj2J

1

4wðAj � yjÞ

� �log32

þXj2J

1

4wðyjÞ

� �log32

�Xj2J

1

4~wðajÞ

� �log32

� 1

2

1

4wðGÞ

� �log32

: &

REFERENCES

[1] D. Archdeacon, N. Hartsfield, and C. H. C. Little, Nonhamiltonian triangula-tions with large connectivity and representativity, J Combin Theory Ser B 68(1996), 45–55.

[2] T. Bohme, B. Mohar, and C. Thomassen, Long cycles in graphs on a fixedsurface, University of Ljubljana, Institute of Mathematics, Physics, andMechanics, Dept. of Mathematics, Preprint Series, 37 no 649 (1999).


[3] G. Chen and X. Yu, Long cycles in 3-connected graphs, J Combin TheorySer B, to be published.

[4] Z. Gao and X. Yu, Convex programming and circumference of 3-connectedgraphs of low genus, J Combin Theory Ser B 69 (1997), 39–51.

[5] B. Jackson and N. Wormwald, Longest cycles in 3-connected planar graphs,J Combin Theory Ser B 54 (1992), 291–321.

[6] J. W. Moon and L. Moser, Simple paths on polyhedra, Pacific J Math 13(1963), 629–631.

[7] R. Thomas and X. Yu, 4-connected projective planar graphs are hamiltonian,J Combin Theory Ser B 62 (1994), 114–132.

[8] C. Thomassen, Color-critical graphs on a fixed surface, J Combin Theory SerB 70 (1997), 67–100.

[9] W. T. Tutte, A theorem on planar graphs, Trans Amer Math Soc 82 (1956),99–116.

[10] W. T. Tutte, Graph Theory, Addison-Wesley Publishing Co., 1984.

[11] H. Whitney, A theorem on graphs, Ann Math 32 (1931), 378–390.

[12] X. Yu, Disjoint paths, planarizing cycles, and spanning walks, Trans AmerMath Soc Vol 349 no 4 (1997), 1333–1358.


Documents

Long cycles in 3-connected graphs in orientable surfaces