Hamilton paths in toroidal graphs

Journal of Combinatorial Theory, Series B 94 (2005) 214–236www.elsevier.com/locate/jctb

Hamilton paths in toroidal graphs

Robin Thomasa,1, Xingxing Yua,b,2, Wenan Zangc,3aSchool of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USAbCenter for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR China

cDepartment of Mathematics, University of Hong Kong, Hong Kong, China

Received 21 March 2002Available online 5 March 2005

Abstract

Tutte has shown that every 4-connected planar graph contains a Hamilton cycle. Grünbaum andNash-Williams independently conjectured that the same is true for toroidal graphs. In this paper, weprove that every 4-connected toroidal graph contains a Hamilton path.© 2005 Elsevier Inc. All rights reserved.

Keywords:Hamilton path; Tutte subgraph; Bridge; Toroidal graph; Face width

1. Introduction and notation

In 1931, Whitney [9] proved that every 4-connected triangulation of the sphere containsa Hamilton cycle (and hence, is 4-face-colorable). Tutte [8] generalized this result to 4-connected planar graphs. Extending the technique ofTutte,Thomassen [7] proved that in any4-connected planar graph there is a Hamilton path between any given pair of distinct vertices.Grünbaum [3] conjectured that every 4-connected graph embeddable in the projective planecontains a Hamilton cycle. This conjecture was proved by Thomas and Yu [5]. For graphsembeddable in the torus (toroidal graphs, for short), Grünbaum [3] and Nash-Williams [4]independently made the following

E-mail address:[email protected](X. Yu).1 Partially supported by NSF Grant DMS-9970514.2 Partially supported by NSF Grants DMS-9970527 and DMS-0245530, and by RGC Grant HKU7056/04P.3 Partially supported by RGC Grant HKU7056/04P.

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R. Thomas et al. / Journal of Combinatorial Theory, Series B 94 (2005) 214–236 215

Conjecture 1.1. Every4-connected toroidal graph contains a Hamilton cycle.

Conjecture 1.1 is established in[1] for 6-connected toroidal graphs. Brunet and Richter[2] proved Conjecture 1.1 for 5-connected triangulations of the torus. Later, Thomas andYu [6] proved Conjecture 1.1 for all 5-connected toroidal graphs. In this paper, we offerfurther evidence to Conjecture 1.1 by proving the following

Theorem 1.2. Every4-connected toroidal graph contains a Hamilton path.

Only simple graphs will be considered. LetG be a graph. We useV (G) andE(G) todenote the vertex set and edge set ofG, respectively. Ife is an edge ofG with endsu andv, then we also denotee by uv. ForX ⊆ E(G) orX ⊆ V (G), G − X denotes the graphobtained fromG by deletingX and (ifX ⊆ V (G)) by deleting all edges ofG incident withvertices inX. WhenX = {x}, we writeG−x instead ofG−{x}. LetSbe a set of 2-elementsubsets ofV (G); thenG+S denotes the graph with vertex setV (G) and edge setE(G)∪S.If S = {{xi, yi} : i = 1, . . . , k} then we sometimes useG + {xiyi : i = 1, . . . , k} insteadof G+ S, and ifS = {{x, y}} we useG+ xy instead ofG+ S.

Let G andH be subgraphs of a graph; thenG ∩ H (respectively,G ∪ H ) denotes theintersection(respectively,union) of G andH. We writeG−H instead ofG− V (G ∩H).We useP ⊆ G to mean thatP is a subgraph ofG. ForS ⊆ V (G), we also viewSas thesubgraph ofG with vertex setSand no edges. HenceP ∪ S makes sense forP ⊆ G andS ⊆ V (G). A block in a graph is a maximal 2-connected subgraph or is induced by a cutedge of the graph.

A graphG is embeddedin a surface� if it is drawn in� with no pair of edges crossing.Thefacesof Gare the connected components (in topological sense) of�−G. The boundaryof a face is called afacial walk. Theface widthor representativityof G in � is defined to bethe minimum number|� ∩G| taken over all non-null homotopic simple closed curves� in�. When� is the plane (or equivalently, the sphere),G is aplane graph. For convenience,an embedding of a graphG in the plane is called aplane representationof G. The boundaryof the infinite face of a plane graphG is called theouter walkof G, or outer cycleif it is acycle.

For a pathP and two verticesx, y ∈ V (P ), we usexPy to denote the subpath ofP withendsx andy. For a cycleC and distinct verticesx, y onC, anxy-segmentof C is a path inCwith endsx andy. If C is a cycle in a graph embedded in an orientable surface� such thatC bounds a closed disc in�, then we can speak of clockwise and counter clockwise ordersalongC. Given two verticesx andy on a cycleC bounding a closed disc, letxCy = {x} ifx = y, and otherwise, letxCydenote thexy-segment ofC which is clockwise fromx to y.

Let G be a graph andP ⊆ G. A P-bridgeof G is a subgraph ofG which either (1) isinduced by an edge ofG − E(P ) with both ends onP or (2) is induced by the edges in acomponent ofG − V (P ) and all edges from that component toP. For aP-bridgeB of G,the vertices ofB ∩ P are theattachmentsof B (on P). We say thatP is aTutte subgraphof G if everyP-bridge ofG has at most three attachments onP. ForC ⊆ G, P is aC-Tuttesubgraphof G if P is a Tutte subgraph ofG and everyP-bridge ofG containing an edge ofC has at most two attachments onP. A Tutte path(respectively,Tutte cycle) in a graph is apath (respectively, cycle) which is a Tutte subgraph.

216 R. Thomas et al. / Journal of Combinatorial Theory, Series B 94 (2005) 214–236

Note that ifP is a Tutte path in a 4-connected graph and|V (P )|�4, thenP is in facta Hamilton path. Hence, in order to prove Theorem 1.2, it suffices to find Tutte paths in2-connected toroidal graphs. This will be done by applying induction on the face-widthof a graph embedded in the torus. In Section2, we state and prove a few lemmas aboutTutte paths in plane graphs. In Section 3, we prove a result which will be used to treat theinduction basis: the face width of a graph in the torus is at most two. We then complete theproof of Theorem 1.2 in Section 4.

For convenience, we useA := B to renameB asA.

2. C-flaps and Tutte subgraphs

We begin this section with several known results on Tutte paths in plane graphs. The firstresult is the main theorem in [7], where aP-bridge is called a “P-component”.

Lemma 2.1. Let G be a2-connected plane graph with a facial cycle C. Assume thatx ∈V (C), e ∈ E(C), andy ∈ V (G − x). Then G contains a C-Tutte path P between x and ysuch thate ∈ E(P ).

Lemma 2.1 can easily be generalized as follows. LetG be a connected plane graph witha facial walkC. Assume thatx ∈ V (C), e ∈ E(C), y ∈ V (G − x), andG has a pathbetweenx andy and containinge. ThenG contains aC-Tutte pathP betweenx andy suchthate ∈ E(P ). Hence, when we apply Lemma 2.1, we actually apply this general version.

In order to state the next result, we need the following concept first introduced in[5]. LetCbe a cycle or a path in a graphG.A C-flapin G is either the null graph or an{a, b, c}-bridgeH of G such that

(i) a, b ∈ V (C) ∩ V (H), a �= b, andc ∈ V (H)− V (C);(ii) H contains anab-segmentSof C; and

(iii) H has a plane representation withSandc on its outer walk.

When aC-flap is an{a, b, c}-bridgeH, we say thata, b, c are its attachments and defineI (H) = V (H)− {a, b, c}. When aC-flap is null, we say thata, b, c are its attachments ifa = b = c ∈ V (C) and defineI (H) = ∅.

See Fig.1 for an illustration. Note that we do not specify the order ofa, b on C, andtherefore we do not need the condition in [5] thatH contains the clockwise segment ofCbetweena andb.

Lemmas 2.2 and 2.3 below are the first half and second half, respectively, of Lemma 2.5in [5], where aC-Tutte path is called anE(C)-snake. Lemma 2.2 is illustrated in Fig. 1.

Lemma 2.2. Let G be a2-connected plane graph with outer cycle C. Letx, y ∈ V (C) bedistinct, let e, f ∈ E(C), and assume thatx, y, e, f occur on C in this clockwise order.Then there exist a C-flap H in G with attachmentsa, b, c (a = b = c = y if H is null) anda (C − I (H))-Tutte path P between b and x inG − I (H) such thatx, a, y, b, e, f occuron C in this clockwise order, y ∈ (V (H)− {a}) ∪ {b}, {e, f } ⊆ E(P ), anda, c ∈ V (P ).


xy

b

cP

a

H

e f

Fig. 1.C-flap and Lemma 2.2.

Lemma 2.3. Let G be a2-connected plane graph with outer cycle C. Letx, y ∈ V (C) bedistinct, let e, f ∈ E(C), and assume thatx, y, e, f occur on C in this clockwise order.Then there exists a yCx-Tutte path P between x and y in G such that{e, f } ⊆ E(P ).

Note that the above three lemmas hold wheneor f or both are vertices ofC. This can beseen by choosing edges inE(C) incident with these vertices. Hence, when these lemmasare applied, we will alloweor f or both to be vertices.

The next lemma shows that if in Lemma 2.3 we do not insist thaty be an end ofP, thenwe can requireP be aC-Tutte path.

Lemma 2.4. Let G be a2-connected plane graph with outer cycle C. Letx, y ∈ V (C) bedistinct, let e, f ∈ E(C), and assume thatx, y, e, f occur on C in this clockwise order.Then there exists a C-Tutte path Q from x in G such thaty ∈ V (Q) and{e, f } ⊆ E(Q).

Proof. By Lemma 2.2, there exist aC-flapH in G with attachmentsa, b, c (a = b = c = y

if H is null) and a(C − I (H))-Tutte pathP betweenb and x in G − I (H) such thatx, a, y, b, e, f occur onC in this clockwise order,y ∈ (V (H)−{a})∪{b}, {e, f } ⊆ E(P ),anda, c ∈ V (P ). See Fig.1.

If H is null, thenQ := P gives the desired path. So assume thatH is non-null. SinceG is2-connected andc /∈ V (C),H ′ := H +{ac, bc} is 2-connected. Without loss of generality,assume thatac, bc are added so thatH ′ is a plane graph with outer cycleC′ := (aCb ∪{c}) + {ac, bc}. By Lemma 2.3 (withH ′, C′, b, c, ac, y asG,C, x, y, e, f , respectively),there exists acC′b-Tutte pathP ′ betweenbandc inH ′ such thaty ∈ V (P ′)andac ∈ E(P ′).Clearly,bc /∈ E(P ′).

Let Q := (P ′ − {a, c}) ∪ P . Then everyQ-bridge ofG is one of the following: aP-bridge ofG − I (H), or aP ′-bridge ofH ′, or a subgraph ofG induced by an edge ofP ′incident witha. Hence,Q is aC-Tutte path fromx in G such thaty ∈ V (Q) and{e, f } ⊆E(Q). �

Again, when Lemma 2.4 is applied,eor f or both may be vertices. Next result is a technicallemma, and it will be used many times in later proofs. In order to cover all situations whenthis lemma is applied, we need to state it in a fairly general setting. See Fig. 2 for anillustration.


tisi

Bi

q = sm+1p = t0

Q

L

Q ′

Tu

Fig. 2. Illustration of Lemma 2.5.

Lemma 2.5. Let K be a connected graph, let Q be a path in K with ends p and q, let L be asubgraph ofK −Q, letQ′ be a cycle in L, and letu ∈ V (Q′). Suppose the following threeconditions are satisfied.

(1) If B is a (L ∪Q)-bridge of G, then|V (B ∩ L)|�1 andV (B ∩ L) ⊆ V (Q′).(2) Let H be the union of Q,Q′, and those(L∪Q)-bridges of G with an attachment on Q;

then H has a plane representation such that Q and u are on a facial walk andQ′ is itsouter cycle.

(3) L contains aQ′-Tutte subgraph T such that(i) u ∈ V (T ) and|V (Q′)∩V (T )|�2,and(ii) every T-bridge X of L containing an edge ofQ′ has a plane representation such thatX ∩Q′ is a path on its outer walk.

ThenK − T contains a path S between p and q such thatS ∪ T is a Q-Tutte subgraphof K, and every T-bridge of L containing no edge ofQ′ is also an(S ∪ T )-bridge of K.

Remark. In most cases when Lemma 2.5 is applied,K is a plane graph,L is a block ofK−Q, Qanduare on a facial walk ofK, andQ′ is a facial cycle ofL which bounds the faceof L containingQ. Hence, conditions (1) and (2) of Lemma 2.5 hold. Moreover, condition(3) holds ifL has aQ′-Tutte subgraphT such thatu ∈ V (T ) and|V (T ) ∩ V (Q′)|�2.

Proof. Let W denote the set of attachments onQ′ of (L ∪Q)-bridges ofK. Note that foreachw ∈ W , eitherw ∈ V (T ) or there is aT-bridgeX of L such thatw ∈ V (X − T ).For w,w′ ∈ W , we definew ∼ w′ if w = w′ or there is aT-bridgeX of L such that{w,w′} ⊆ V (X − T ). Clearly,∼ is an equivalence relation onW. LetW1,W2, . . . ,Wmdenote the equivalence classes ofW with respect to∼. Then for i ∈ {1, . . . , m}, either|Wi | = 1 andWi ⊆ V (T ) (in this case,Bi := Wi ⊆ V (Q′)) or Wi ⊆ V (Bi − T ) forsomeT-bridgeBi of L. SinceT is aQ′-Tutte subgraph ofL andW ⊆ V (Q′) (by (1)),|V (Bi ∩ T )|�2. Hence, by (i) of (3),V (Bi ∩ T ) ⊆ V (Q′).

For eachi ∈ {1, . . . , m}, let si, ti ∈ V (Q) such that (I)p, si, ti , q occur onQ in thisorder, (II) there arews,wt ∈ Wi such that{si, ws} is contained in a(L ∪Q)-bridge ofKand{ti , wt } is contained in a(L ∪ Q)-bridge ofK, and (III) subject to (I) and (II),siQti


is maximal. By (2),siQti , i = 1, . . . , m, are edge disjoint. We may therefore assume thatp, s1, t1, s2, t2, . . . , sm, tm, q occur onQ in this order. Lett0 := p andsm+1 := q.

For eachi ∈ {0, . . . , m}, let Ti denote the union oftiQsi+1 and those(L ∪Q)-bridgesof K whose attachments are all contained inV (tiQsi+1). For eachi ∈ {1, . . . , m}, letUidenote the union ofsiQti , Bi , and those(L ∪Q)-bridges ofK whose attachments are allcontained inV (siQti) ∪Wi .

(a) By the definition ofsiQti , we conclude that fori�j , Ui ∩ Tj (and for i < j ,(Ui − T ) ∩ (Uj − T )) is one of the following:∅, or {ti}, or the union of those(L ∪Q)-bridges ofK with ti as their only attachment onL ∪Q. Similarly, for i < j , Ti ∩ Tj (andalsoTi ∩Uj ) is one of the following:∅, or {si+1}, or the union of those(L∪Q)-bridges ofK with si+1 as their only attachment onL ∪Q.

(b) We claim that for eachi ∈ {0, . . . , m}, Ti contains atiQsi+1-Tutte pathRi betweenti andsi+1.

If |V (tiQsi+1)|�2, thenRi := tiQsi+1 gives the desired path for (b). Now assume that|V (tiQsi+1)|�3. By (2),Ti has a plane representation such thattiQsi+1 is on its outer walk.Let Ci denote the outer walk ofTi , and choose an edgee from E(tiQsi+1). By applyingLemma 2.1 (withTi, Ci, ti , si+1 asG,C, x, y, respectively),Ti has aCi-Tutte pathRibetweenti andsi+1 such thate ∈ E(Ri). Clearly,Ri is atiQsi+1-Tutte path inTi .

(c) We claim that for eachi ∈ {1, . . . , m}, Ui − T contains a pathSi betweensi andtisuch thatSi ∪ (Ui ∩ T ) is ansiQti-Tutte subgraph ofUi .

Note that for alli ∈ {1, . . . , m}, |V (Ui ∩ T )| = |V (Bi ∩ T )|�2. By (ii) of (3), Bi hasa plane representation such thatBi ∩Q′ is a path on its outer walk. Hence, by (2),Ui hasa plane representation such thatsiQti andBi ∩ T are on its outer walk. We will work onsuch a plane representation ofUi .

If si = ti , then letSi := siQti , and clearly,Si ∪ (Ui ∩ T ) is ansiQti-Tutte subgraph ofUi (because|V (Ui ∩ T )|�2). So assume thatsi �= ti . We distinguish two cases.

First assume thatWi ⊆ V (T ). Then |Wi | = 1. So letw be the only vertex inWi .Without loss of generality, we can assume that(siQti ∪ {w}) + tiw is contained in theouter walkDi of Ui + tiw. By Lemma 2.1 (withUi + tiw,Di, si, w, tiw asG,C, x, y, e,respectively),Ui+ tiw contains aDi-Tutte pathS′

i betweensi andw such thattiw ∈ E(S′i ).

Let Si := S′i −w. ThenSi ⊆ Ui − T , and it is easy to see thatSi ∪ (Ui ∩ T ) = Si ∪ {w} is

ansiQti-Tutte subgraph ofUi .Now assume thatWi �⊆ V (T ). ThenBi �= Wi andBi is aT-bridge ofL containing an

edge ofQ′. Hence,V (Bi ∩ T ) consists of two verticesw andw′. Assume thatw,w′, ti , sioccur on the outer walk ofUi in this clockwise order. Note that(siQti ∪ {w,w′}) +{wsi, tiw′} is contained in a cycle ofUi + {wsi, tiw′}, and so, letU ′

i denote the blockin Ui + {wsi, tiw′} containing one such cycle. Without loss of generality, we may assumethat(siQti ∪{w,w′})+{wsi, tiw′} is contained in the outer cycleD′

i ofU ′i . By Lemma 2.3

(withU ′i , D

′i , w,w

′, tiw′, wsi asG,C, x, y, e, f , respectively),U ′i contains aw′D′

iw-TuttepathS′

i betweenw andw′ such that{wsi, tiw′} ⊆ E(S′i ). Clearly,S′

i is also ansiQti-Tuttepath inUi + {wsi, tiw′}. Let Si := S′

i − {w,w′}. ThenSi ⊆ Ui − T , and it is easy to seethatSi ∪ (Ui ∩ T ) = Si ∪ {w,w′} is ansiQti-Tutte subgraph ofUi .

By (a), (b) and (c),S := (⋃mi=0 Ri) ∪ (⋃m

i=1 Si) is a path betweenp andq in K − T .It is easy to see that every(S ∪ T )-bridge ofK is one of the following: aT-bridge ofL notcontained in anyUi , or aRi-bridge ofTi , or an(Si ∪ (Ui ∩ T ))-bridge ofUi . Thus,S ∪ T


is a Q-Tutte subgraph ofK, and everyT-bridge ofL containing no edge ofQ′ is also an(S ∪ T )-bridge ofK. �

3. Planar graphs

In this section, we prove Theorem 3.3 which will be used to take care of the base case inthe inductive proof of Theorem 1.2. First, we need two lemmas about Tutte paths in planargraphs, and for the sake of induction we will prove them simultaneously.

Lemma 3.1. Let G be a2-connected plane graph with outer cycle C and a facial cycle D,let y ∈ V (C), and letx ∈ V (D). Then G contains a(C ∪D)-Tutte path P from y such thatx ∈ V (P ) and no P-bridge of G contains vertices of bothC − P andD − P .

A separation(G1,G2) in a graphG is a pair of edge disjoint subgraphs ofG such thatG = G1 ∪G2 andE(G1) �= ∅ �= E(G2). A separation(G1,G2) in G is ak-separationif|V (G1 ∩G2)| = k.

Lemma 3.2. Let G be a2-connected plane graph with outer cycle C and another facialcycle D. Lety ∈ V (C), x ∈ V (D), ande ∈ E(C). Assume that there do not exist distinctverticesp, q ∈ V (C)and a2-separation(G′

1,G′2) in G such thatV (G′

1)∩V (G′2) = {p, q},

p, y, e, q occur on C in the clockwise order listed, pCq ⊆ G′1, andqCp ∪D ⊆ G′

2. Thenone of the following holds:

(A) there exist a C-flap H in G with attachmentsa, b, c (a = b = c = y if H is null) anda ((C − I (H)) ∪ D)-Tutte path P from b inG − I (H) such thatD ⊆ G − I (H),e, b, y, a occur on C in this clockwise order, y ∈ (V (H) − {a}) ∪ {b}, e ∈ E(P ),x, a, c ∈ V (P ), and no P-bridge of G contains vertices of bothC − P andD − P ; or

(B) there exista, b ∈ V (C) ∩ V (D), a separation(H,H ∗) in G withV (H) ∩ V (H ∗) ={a, b}, and a(C ∪D)-Tutte path P from b in G such thate, b, y, a occur on C in thisclockwise order,bCa∪bDa ⊆ H ,aCb∪aDb ⊆ H ∗,P ⊆ H ∗,y ∈ (V (H)−{a})∪{b},a, x ∈ V (P ), e ∈ E(P ), and no P-bridge of G distinct from H contains vertices ofbothC − P andD − P (Fig. 3).

Proof of Lemmas 3.1 and 3.2.We proceed by induction on|V (G)|. We may assume thatC �= D; for otherwise, choosef ∈ E(C) − {e} such thatf is incident withy, thenP := C−f satisfies both the conclusion of Lemma 3.1 and (A) of Lemma 3.2 withH null.Thus|V (G)|�4. Because we are proving Lemma 3.1 and Lemma 3.2 simultaneously, wecan inductively assume that both Lemma 3.1 and Lemma 3.2 hold for all graphs on strictlyless than|V (G)| vertices. Let us remark thate is not defined in Lemma 3.1; hence in theproof of Lemma 3.1 we are free to specifyewhen inductively applying Lemma 3.2.

LetG2 be a block ofG− y chosen as follows: for Lemma 3.1,G2 is a block ofG− y

containing an edge ofC− y; for Lemma 3.2, ife is not incident withy letG2 be the uniqueblock ofG− y containinge, and otherwise letG2 be the unique block ofG− y containingthe unique edge ofC adjacent toe but not incident withy. If G2 has only one edge let


y

b

e e

xDb

y

HH

c

a

a

DxP

P

(A) (B)

H*

Fig. 3. Illustration of Lemma 3.2.

G2

vi

v1

JkJi

vk

J1

y

Fig. 4.G2 andJ1, . . . , Jk .

C2 = G2; otherwise letC2 be the outer cycle ofG2. It follows that every(G2 ∪{y})-bridgeof Ghas exactly one attachment inG2, and this attachment belongs toC2. Letv1, v2, . . . , vkbe those attachments of(G2 ∪ {y})-bridges ofG onC2 listed in clockwise order such thatvkCv1 = vkC2v1. For i = 1,2, . . . , k, letJi be the union of those(G2 ∪ {y})-bridges ofGcontainingvi , and letDi denote the outer walk ofJi (Fig. 4).

SinceC �= D, we have the following.(a) Either there is somei ∈ {2, . . . , k} such thatD is the union ofvi−1C2vi , a subpath

of Ji−1, and a subpath ofJi ; or there is somei ∈ {1,2, . . . , k} such thatD is a subgraph ofJi ; or D is a subgraph ofG2.

Let us make three observations for the proof of Lemma 3.2 before we handle these casesseparately.

(b) If e is incident withy, and if we denote the other end ofe by z, then we may assumethaty, z, e occur onC in the clockwise order listed.

To prove (b) assume thaty, e, z occur onC in this clockwise order. Letp := y, q := z, letG′

1 be the subgraph ofG induced bye, and letG′2 := G− e. It follows from the hypothesis

of Lemma 3.2 thatD is not a subgraph ofG′2, and hencee ∈ E(D). Applying Lemma

2.1 to the graphG′2 (with G′

2, zCy ∪ zDy, y, z, x asG,C, y, x, e, respectively), we find a


(zCy∪zDy)-Tutte pathP ′ betweeny andz inG′2 such thatx ∈ V (P ′). Hence,T := P ′ +e

is a(C ∪D)-Tutte cycle inG with e ∈ E(T ) andx ∈ V (T ). By planarity, noT-bridge ofGcontains edges of bothC andD. Let P be the path obtained fromT by deleting the uniqueedgef of T − e incident withy. Note that anyP-bridge ofG either is induced byf or is aT-bridge ofG. Hence, noP-bridge ofG contains vertices of bothC − P andD − P , andso,P satisfies (A) of Lemma 3.2 withH null. Thus we may assume that (b) holds.

We deduce from (b) that(c) e ∈ E(J1) ∪ E(G2).Moreover, the choice ofG2 implies that(d) if e /∈ E(G2), thenehas endsy andv1.Now we distinguish three cases according to (a).Case1: There exists somei ∈ {2, . . . , k} such thatD is the union ofvi−1C2vi , a subpath

Qi−1 of Ji−1, and a subpathQi of Ji .In this case,y ∈ V (D),Qi−1 has endsvi−1 andy, andQi has endsvi andy.Supposei = 2 and assume bothe ∈ E(J1) andx ∈ V (J1). In this case, we only need

to prove Lemma 3.2 (because for Lemma 3.1, we specifyebelow so thate is inG2). Sincee ∈ E(J1) and by (d),E(v1Cy) = {e}. LetH ∗ := J1, H := G − (V (J1) − {y, v1}), andlet a := v1 andb := y. Then(H,H ∗) is a separation inG with V (H) ∩ V (H ∗) = {a, b},bCa ∪ bDa ⊆ H , aCb ∪ aDb ⊆ H ∗, andy ∈ (V (H) − {a}) ∪ {b}. If |V (J1)| = 2 thenx = v1 andP := J1 gives aD1-Tutte path fromb = y in H ∗ such thata, x ∈ V (P )

ande ∈ E(P ), and we have (B) of Lemma 3.2. Now assume thatJ1 is 2-connected. ByLemma 2.1 (withH ∗ asG), we find aD1-Tutte pathP betweenb = y andx in H ∗ suchthate ∈ E(P ). Therefore, we have (B) of Lemma 3.2.

So assume that eitheri �= 2 or one of{e, x} does not belong toJ1.LetG′ be obtained fromG by deletingV (Jj )− {vj , y} for all j = i, i + 1, . . . , k. Then

C′ := vkCy ∪ Qi−1 ∪ vi−1C2vk is the outer walk ofG′. The following will serve as aproof of both Lemma 3.1 and Lemma 3.2 with the proviso that when inductively applyingLemma 3.2 in the proof of Lemma 3.1 the edgee is chosen to be the unique edge ofC ∩G′incident withvk.

Next, we find a(C′ ∪ Di)-Tutte pathP ∗ from y in G′ ∪ Ji such thatx, vi ∈ V (P ∗)and e ∈ E(P ∗). Assume first thatx ∈ V (G′). If G′ is 2-connected, thenP ∗ can befound by applying Lemma 2.4 toG′ (with G′, C′, y, x, vi, e, P ∗ asG,C, x, y, e, f, P ,respectively). So assume thatG′ is not 2-connected. Theni = 2, and so, eithere /∈ E(J1)

or x /∈ V (J1). By Lemma 2.1 (withJ1,D1, y, v1, e or x asG,C, y, x, e, respectively),we find aD1-Tutte pathP1 betweeny andv1 in J1 such thate ∈ E(P1) if e ∈ E(J1)

andx ∈ V (P1) if x ∈ V (J1). If x /∈ V (G2) then by Lemma 2.1 (withG2, C2, v1, v2, e asG,C, x, y, e, respectively), and ifx ∈ V (G2) then by Lemma 2.4 (withG2, C2, v1, x, v2, e

asG,C, x, y, e, f , respectively), we find aC2-Tutte pathP2 from v1 in G2 such thatv2 ∈V (P2), e ∈ E(P2), andx ∈ V (P2). It is easy to check thatP ∗ := P1 ∪ P2 is the desiredpath. Now we may assume thatx /∈ V (G′). Thusx ∈ V (Ji)−{y, vi}. By Lemma 2.1 (withG′, C′, y, vi, e asG,C, x, y, e, respectively), we find aC′-Tutte pathP ′ in G′ with endsy andvi such thate ∈ E(P ′). Again by Lemma 2.1 (withJi,Di, vi, y, x asG,C, x, y, e,respectively), we find aDi-Tutte pathP ′′ with endsyandvi such thatx ∈ V (P ′′). It is easy tocheck thatP ∗ := P ′∪(P ′′−y)gives the desired path. This completes the construction of thepathP ∗.


Supposevk ∈ V (P ∗). If there exists noP ∗-bridge ofG containing edges of bothC andD, thenP := P ∗ satisfies both the conclusion of Lemma 3.1 and (A) of Lemma 3.2 withHnull. So assume that there exists aP ∗-bridge ofG containing edges of bothC andD, thenthis bridge isJk andi = k. In this case,y is an end ofP ∗ andx /∈ V (Jk)−{y, vk}. Thereforefor Lemma 3.2, the pathP := P ∗, graphsH := Jk andH ∗ = G − (V (H) − {y, vk}),and verticesb := y, a := vk satisfy (B). To prove Lemma 3.1, we apply Lemma 2.1(with G′, C′, y, vk, x asG,C, x, y, e, respectively) to find aC′-Tutte pathQ betweenyandvk in G′ such thatx ∈ V (Q). We also apply Lemma 2.1 toJk (with Jk,Dk, y, vk asG,C, x, y, respectively) to find aDk-Tutte pathR betweeny andvk. Then it is easy to seethatP := Q ∪ (R − y) satisfies the conclusion of Lemma 3.1.

Now assume thatvk /∈ V (P ∗). Theni �= k (sincevi ∈ V (P ∗)) ande is not incident withvk, and so, we only need to prove Lemma 3.2 (because for Lemma 3.1,e is incident withvk). LetH be theP ∗-bridge ofG containingvk, and leta, b, c be its attachments labeled sothatb = y, a ∈ V (C) andc ∈ V (C2) − V (C). Sincei �= k, D ⊆ G − I (H). It followsfrom the construction ofP ∗ (in particular,vi ∈ V (P ∗)) thatP ∗ is a ((C − I (H)) ∪ D)-Tutte path inG − I (H) and noP ∗-bridge ofG contains edges of bothC andD. Hence,P := P ∗, H, a, b, c satisfy (A) of Lemma 3.2.

Case2: There is somei ∈ {1,2, . . . , k} such thatD is a subgraph ofJi .First, we dispose of the casei = 1. To this end assume thati = 1, letp := y, q := v1,

let K be obtained fromG by deleting vertices and edges ofJ1 exceptp and q, and letus consider the separation(K, J1) in G. To prove Lemma 3.1, letJ ′

1 := J1 + yv1 andC′ := v1Cy + yv1 (so thatC′ is the outer cycle ofJ ′

1). Inductively applying Lemma3.1 to J ′

1 (with J ′1, C

′, x, y asG,C, x, y, respectively) we find a(C′ ∪ D)-Tutte pathP ′ from y in J ′

1 such thatx ∈ V (P ′) and noP ′-bridge ofJ ′1 contains vertices of both

C′ − P ′ andD − P ′. If yv1 /∈ E(P ′), thenP := P ′ satisfies the conclusion of Lemma3.1 (andK is aP-bridge ofG with two attachments and contains no vertex ofD). If yv1 ∈E(P ′), then we apply Lemma 2.1 toK (with K, y, v1, vk asG, x, y, e, respectively) tofind a yCv1-Tutte pathP ′′ betweeny andv1 such thatvk ∈ V (P ′′), and clearly,P :=(P ′ − yv1) ∪ P ′′ satisfies the conclusion of Lemma 3.1. Now let us prove Lemma 3.2.The hypothesis of Lemma 3.2 implies thate ∈ E(J1). By (d), the edgee has endsy andv1, and so,J1 is 2-connected. By inductively applying Lemma 3.2 toJ1 (with J1,D1, y, x

asG,C, y, x, respectively), we finda1, b1, c1, H1, P (asa, b, c,H, P , respectively) sat-isfying (A) of Lemma 3.2 or we finda1, b1, (H1, H

∗1 ), P , (asa, b, (H,H ∗), P , respec-

tively) satisfying (B) of Lemma 3.2. Notice thatK becomes aP-bridge ofG with attach-mentsy andv1. Also note that if the graphH1 obtained by induction is non-null thenH1contains no vertex ofC andy is an attachment ofH1. Hence, (A) of Lemma 3.2 holdswith H null.

So we may assume thati > 1.LetG′ be obtained fromG by deletingV (Jj )− {vj , y} for all j = i, i + 1, . . . , k. Note

thatG′ + yvi is 2-connected, and letC′ := (viC2vk ∪ vkCy) + yvi (so thatC′ is theouter cycle ofG′ + yvi). By Lemma 2.3 (withG′ + yvi, C′, y, vi, vk, e asG,C, x, y, e, f ,respectively), there exists aC′-Tutte pathP ′ betweeny andvi in G′ such thate ∈ E(P ′)andvk ∈ V (P ′). LetG3 = Ji + yvi and lete3 be the edge ofG3 with endsy andvi . Ifi = k, then letC3 be the cycle ofG3 consisting of the edgee3 and the pathyCvk. If i < klet C3 be an arbitrary facial cycle ofG3 that includes the edgee3. In either case we may


assume that a plane representation ofG3 is chosen so thatC3 is its outer cycle andy, vi, e3appear onC3 in the clockwise order listed.

To prove Lemma 3.1, we inductively apply Lemma 3.1 toG3 (with G3, C3, y, x asG,C, y, x, respectively), and we find a(C3 ∪ D)-Tutte pathP3 from y in G3 such thatx ∈ V (P3) and noP3-bridge ofG3 contains vertices of bothC3 − P3 andD − P3. Ife3 /∈ E(P3), thenP := P3 satisfies the conclusion of Lemma 3.1 (andG′ ∪ Ji+1 ∪ . . .∪ Jkis contained in aP3-bridge ofGwith two attachments). Ife ∈ E(P3), thenP := P ′∪(P3−y)satisfies the conclusion of Lemma 3.1.

To prove Lemma 3.2, we wish to inductively apply Lemma 3.2 toG3 (withG3, C3, y, x, e3asG,C, y, x, e, respectively). Sincee3 is incident withy and becausey, vi, e3 occur onC3 in this clockwise order, we see that the hypothesis of Lemma 3.2 is satisfied. Hence,there are two possibilities. (A) There exist aC3-flapH3 in G3 with attachmentsa3, b3, c3(a3 = b3 = c3 if H3 is null) and a((C3 − I (H3)) ∪ D)-Tutte pathP3 from b3 inG3 − I (H3) such thatD ⊆ G − I (H3), e3, b3, y, a3 occur onC3 in this clockwise order,y ∈ (V (H3) − {a3}) ∪ {b3}, e3 ∈ E(P3), x, a3, c3 ∈ V (P3), noP3-bridge ofG3 containsvertices of bothC3 −P3 andD−P3. (B) There exista3, b3 ∈ V (C3)∩V (D), a separation(H3, H

∗3 ) in G3 with V (H3) ∩ V (H ∗

3 ) = {a3, b3}, and a(C3 ∪ D)-Tutte pathP3 from b3in G3 such thate3, b3, y, a3 occur onC3 in this clockwise order,b3C3a3 ∪ b3Da3 ⊆ H3,a3C3b3∪a3Db3 ⊆ H ∗

3 ,P3 ⊆ H ∗3 , y ∈ (V (H3)−{a3})∪{b3}, e3 ∈ E(P3), a3, x ∈ V (P3),

and noP3-bridge ofG3 distinct fromH3 contains vertices of bothC3 − P3 andD − P3.Sincee3 is incident withy we see thatb3 = y. If i �= k, thenP := P ′ ∪ (P3 − y)

satisfies (A) of Lemma 3.2 withH null. If i = k, thenP := P ′ ∪ (P3 − y), H3 or(H3, H

∗3 ∪G′) (asH or (H,H ∗)), and the verticesa3, b3, c3 (asa, b, c) satisfy (A) or (B)

of Lemma 3.2.Case3: D is a subgraph ofG2.We will construct the desired pathP as the union of two pathsP1, P2 in graphsG1 (to

be defined),G2, respectively. This will be done simultaneously for Lemma 3.1 and Lemma3.2, with the understanding that when inductively applying Lemma 3.2 toG2 in the proofof Lemma 3.1 we letebe the unique edge ofC2 ∩ C incident withvk.

SinceD ⊂ G2 andG2 is a block,G2 is 2-connected. To constructP2 we wish toinductively apply Lemma 3.2 toG2, C2,D, v1, x, e (asG,C,D, y, x, e, respectively) ife ∈E(G2), or we wish to inductively apply Lemma 3.1 toG2, C2,D, v1, x (asG,C,D, y, x,respectively) ife /∈ E(G2). In order to apply Lemma 3.2 toG2, we must verify the absencein G2 of verticesp, q and a 2-separation as in the hypothesis of Lemma 3.2. Suppose fora contradiction that there exist distinct verticesp, q ∈ V (C2) and a 2-separation(G′′

2,G′2)

in G2 such thatV (G′′2) ∩ V (G′

2) = {p, q}, p, v1, e, q occur onC2 in the clockwise orderlisted,pC2q ⊆ G′′

2, andqC2p ∪ D ⊆ G′2. Sincev1 ande both belong topC2q ∩ C, we

deduce thatp, q ∈ V (C). LetG′1 := G′′

2∪J1∪J2∪ . . .∪Jk; then(G′1,G

′2) is a 2-separation

in G that contradicts the hypothesis of Lemma 3.2. This verifies the absence of verticesp, q

and a corresponding 2-separation inG2, and hence we may inductively apply Lemma 3.2toG2, C2,D, v1, x, e whene ∈ E(G2).

First, we constructP2. If e /∈ E(G2) (this only applies to the proof of Lemma 3.2), thenby inductively applying Lemma 3.1 toG2, we find a(C2 ∪D)-Tutte pathP2 from v1 inG2such thatx ∈ V (P2) and noP2-bridge ofG2 contains vertices of bothC2 − P andD− P ,and we defineH2 to be null. Now assumee ∈ E(G2). By inductively applying Lemma


3.2 toG2, we have two possibilities. (A) There exist aC2-flapH2 in G2 with attachmentsa2, b2, c2 (a2 = b2 = c2 = v1 if H2 is null), and a((C2−I (H2))∪D)-Tutte pathP2 fromb2inG2−I (H2) such thatD ⊆ G2−I (H2), e, b2, v1, a2 occur onC2 in this clockwise order,v1 ∈ (V (H2)−{a2})∪{b2}, e ∈ E(P2),x, a2, c2 ∈ V (P2), and noP2-bridge ofG2 containsvertices of bothC2 −P2 andD−P2. (B) There exista2, b2 ∈ V (C2)∩V (D), a separation(H2, H

∗2 ) in G2 with V (H2) ∩ V (H ∗

2 ) = {a2, b2}, and a(C2 ∪D)-Tutte pathP2 from b2in G2 such thate, b2, v1, a2 occur onC2 in this clockwise order,b2C2a2 ∪ b2Da2 ⊆ H2,a2C2b2∪a2Db2 ⊆ H ∗

2 ,P2 ⊆ H ∗2 , v1 ∈ (V (H2)−{a2})∪{b2},a2, x ∈ V (P3), e2 ∈ E(P2),

and noP2-bridge ofG2 distinct fromH2 contains vertices of bothC2 − P2 andD − P2.We now define the graphG1 and construct the pathP1 inG1. Assume first thatH2 is null.

LetG1 := J1, and apply Lemma 2.1 toJ1 to find aD1-Tutte pathP1 betweenyandv1 inG1such thate ∈ E(P1) if e ∈ E(J1). Now assume thatH2 is non-null. Thene ∈ E(G2). Sincee, b2, v1, a2 occur onC2 in the clockwise order listed, it follows thatb2 ∈ V (C). We mayassume thata2 ∈ V (v1C2vk); for otherwise, we only need to prove Lemma 3.2 (becausee ∈ E(P ) is not incident withvk), and the verticesa2, b2, c2 (asa, b, c) or a2, b2 (asa, b),the graphH2 ∪ J1 ∪ . . .∪ Jk (asH), and the pathP2 (asP) satisfy (A) or (B) of Lemma 3.2.LetG1 be the union ofH2 and thoseJi with vi ∈ (V (H2)− {a2})∪ {b2}. Assume first thatwe have (A) above forP2. Note thatG′

1 := G1 + {a2c2, c2b2, ya2} is 2-connected, and wemay assume thatG′

1 has a plane representation such thata2c2, b2c2, ya2 andC ∩ G1 areon its outer cycleC′

1. By applying Lemma 2.3 to the graphG′1 we find aC′

1-Tutte pathP ′1

betweeny anda2 such thata2c2, c2b2 ∈ E(P ′1), and letP1 := P ′

1 − {a2, c2}. Now assumethat we have (B) above forP2. Then we apply Lemma 2.1 to the graphG′′

1 := G1 + a2y

to get a((C ∪D) ∩G′′1)-Tutte pathP ′

1 betweena2 andb2 such thata2y ∈ E(P ′1), and let

P1 = P ′1 − a2.

LetP := P1 ∪ P2. If vk ∈ V (P ) we leta = b = c = y and letH be null. If vk /∈ V (P ),thenvk belongs to aP2-bridgeH ′ ofG2 with two attachments. Leta, c be the attachmentsof H ′ such thatc ∈ V (C2) − V (C) anda ∈ V (C) ∩ V (C2), let b = y, and letH be theunion ofH ′ and thoseJi with vi ∈ V (H ′) − {c}. ThenH is a C-flap with attachmentsa, b, c, andP is a path fromb in G − I (H) such thata, c ∈ V (P ) ande ∈ E(P ). Notewhenvk /∈ V (P ), e /∈ E(G2) and, therefore, we only need to show Lemma 3.2.

To prove thatP satisfies the conclusions of Lemma 3.1 and Lemma 3.2, we first noticethatP is a((C−I (H))∪D)-Tutte path inG1∪G2 = G1∪(G2−I (H2)), and by planarity,noP-bridge ofG1 ∪G2 contains vertices of bothC−P andD−P . LetJ be aP-bridge ofG distinct fromH. ThenJ is either aP-bridge ofG1 ∪G2, or J = Ji for somei (in whichcaseJ has at most two attachments), orJ is the union of aP2-bridgeJ ′ of G2 − I (H2)

and some of theJi ’s (in which case,J ′ includes an edge ofC2, and hence has exactly twoattachments onP2, and so we deduce thatJ has exactly three attachments onP). Hence,Pis a((C− I (H))∪D)-Tutte path inG− I (H). Note thatD ⊆ G− I (H), and so, we have(A) of Lemma 3.2. For Lemma 3.1, becausee is incident withvk, we havevk ∈ V (P2).ThereforeH is null, and henceP is a(C ∪D)-Tutte path inG satisfying Lemma 3.1. �

Now we show that Lemma 3.2 also holds when the vertexx is replaced by an edgef notincident withe. See Fig.3 for an illustration (withf replacingx).

Theorem 3.3. Let G be a2-connected plane graph with outer cycle C and another facialcycle D. Lety ∈ V (C), f ∈ E(D), and e ∈ E(C). Assume that{e, f } is a matching in


G, and assume that there do not exist distinct verticesp, q ∈ V (C) and a2-separation(G′

1,G′2) in G such thatV (G′

1) ∩ V (G′2) = {p, q}, p, y, e, q occur on C in the clockwise

order listed, pCq ⊆ G′1, andqCp ∪D ⊆ G′

2. Then one of the following holds:

(A) there exist a C-flap H in G with attachmentsa, b, c (a = b = c = y if H is null) anda ((C − I (H)) ∪ D)-Tutte path P from b inG − I (H) such thatD ⊆ G − I (H),e, b, y, a occur on C in this clockwise order, y ∈ (V (H) − {a}) ∪ {b}, e, f ∈ E(P ),a, c ∈ V (P ), and no P-bridge of G contains vertices of bothC − P andD − P ; or

(B) there exista, b ∈ V (C) ∩ V (D), a separation(H,H ∗) in G withV (H) ∩ V (H ∗) ={a, b}, and a(C ∪D)-Tutte path P from b in G such thate, b, y, a occur on C in thisclockwise order,bCa∪bDa ⊆ H ,aCb∪aDb ⊆ H ∗,P ⊆ H ∗,y ∈ (V (H)−{a})∪{b},a ∈ V (P ), e, f ∈ E(P ), and no P-bridge of G distinct from H contains vertices ofbothC − P andD − P .

Proof. LetG′,D′ denote the graphs obtained fromG,D, respectively, by subdividing theedgef with a vertexx. It is clear thatG′, C,D′, y, x, e (asG,C,D, y, x, e, respectively)satisfy the hypothesis of Lemma 3.2. By applying Lemma 3.2 toG′, C,D′, y, x, e, we findP ′ (asP in 3.2) andH (or, (H,H ∗)) satisfying (A) (or, (B)) of Lemma 3.2.

First assume thatx is not an end ofP ′. Then it is easy to check thatP := (P ′ − x)+ f

andH (or, (H, (H ∗ − x)+ f )) satisfy (A) (or, (B)) of Lemma 3.3.So assume thatx is an end ofP ′. Let x1, x2 denote the neighbors ofx, and assume

thatxx1 ∈ E(P ′). If x2 /∈ V (P ′) thenx2 is contained in someP ′-bridge ofG′ with twoattachments (one of these isx), and so, it is easy to check thatP := ((P ′−x)∪{x2})+f andH (or,(H, (H ∗−x)+f )) satisfy (A) (or, (B)) of Lemma 3.3. Therefore, we may assume thatx2 ∈ V (P ′). Choose the edgegofP ′ incident withx2 such thatP := ((P ′−x)+f )−g is apath. Because{e, f } is a matching inG, we haveg �= e. NowPandH (or,(H, (H ∗−x)+f ))satisfy (A) (or, (B)) of Lemma 3.3. �

4. Toroidal graphs

Let G be a graph embedded in the torus and letR be a face ofG. We define theR-widthof G to be the minimum number|� ∩G| taken over all non-null homotopic simple closedcurves� in the torus passing throughR. Note that we can homotopically shift curves in thetorus so that curves that we will deal with meetG only at vertices.

To prove Theorem 1.2, we will choose a faceRof G and apply induction on theR-widthof G. Lemma 4.1 below deals with the base case. For the sake of induction, we introduce(4,C )-connected graphs. LetG be a connected graph andC be a subgraph ofG; then wesay thatG is (4,C )-connected if, for anyT ⊆ V (G) such that|T |�3 andG − T is notconnected, every component ofG− T contains a vertex ofC.

Lemma 4.1. Let G be a2-connected graph embedded in the torus with face width at least2, let R be a face of G bounded by a cycle C in G, and lety ∈ V (C). Assume that the R-widthof G is2 and G is(4,C)-connected. Then there exist a C-flap H in G with attachmentsa, b, c

(a = b = c = y if H is null) and a(C − I (H))-Tutte path P from b inG− I (H) such thatb, y, a occur on C in this clockwise order, y ∈ (V (H)−{a})∪{b}, a, c ∈ V (P ), and every


c

H

y

a

b

D

v ′u ′

c

H

D

v ′

P ′v ′′

F ′′F ′′

v ′′u ′′ u ′′

a

(B)(A)

v*u*v*u*

ff

D ′

L

D ′

L

P ′

b = y = u ′

Fig. 5. Subcase 2.1.

P-bridge B of G containing an edge of C has a plane representation withB ∩ (C ∪ P) onits outer walk. Moreover, |V (P ) ∩ V (C)|�2 and|V (P )|�4.

Proof. Let u, v ∈ V (C) be distinct such that there is a non-null homotopic simple closedcurve� passing throughRand meetingG only in u andv. We may choose the triple(�, u, v)such thaty ∈ V (uCv)− {v}, and subject to this,uCy is minimal.

We cut the torus open along� and, as a result, we obtain a plane graphG′ and verticesu′, v′, u′′, v′′ of G′ such that (by choosing appropriate notation) (1)u′ andv′ belong to theouter walkC′ of G′, andE(uCv) induces a pathF ′ onC′ in the clockwise order fromu′to v′, (2)u′′ andv′′ belong to a facial walkC′′ ofG′, andE(vCu) induces a pathF ′′ onC′′in the clockwise order fromu′′ to v′′, and (3) identifyingu′ with u′′ asu and identifyingv′andv′′ asv, we obtainG fromG′.

Clearly,u′, v′ /∈ V (C′′)andu′′, v′′ /∈ V (C′). For otherwise, there is a non-null homotopicsimple closed curve in the torus intersectingGonly atuorv. This contradicts the assumptionthat the face-width ofG is at least 2.

Note that ify �= u theny ∈ V (F ′) − {u′, v′}. Let y := u′ if y = u. Also note thatif |V (F ′)|�3 thenF ′ + u′v′ is a cycle, and if|V (F ′)| = 2 thenF ′ + u′v′ = F ′ andG′ + u′v′ = G′. So when|V (F ′)|�3, we drawu′v′ in the infinite face ofG′ so thatF ′ +u′v′ is the outer cycle ofG′ +u′v′. Because the face width ofG is 2,F ′ ∩F ′′ = ∅ and(G′+u′v′)−F ′′ has a cycle containingF ′+u′v′. LetL denote the block of(G′+u′v′)−F ′′containing one such cycle, and letD′ denote the outer cycle ofL. ThenD′ = F ′ + u′v′if |V (F ′)|�3, and otherwise,F ′ ⊆ D′. Let D be the cycle bounding the face ofL whichcontainsF ′′ (as a subset of the plane). Letu∗, v∗ ∈ V (F ′′)with u∗F ′′v∗ maximal such thatu′′, u∗, v∗, v′′ occur onF ′′ in this order andu∗ andv∗ are attachments of some(L ∪ F ′′)-bridges ofG′ + u′v′ which also have attachments onL. Let x, z ∈ V (D) such that{x, u∗}is contained in a(L∪F ′′)-bridge ofG′ + u′v′, {z, v∗} is contained in a(L∪F ′′)-bridge ofG′ + u′v′, and subject to these conditions,zDx is minimal. ThenzDx − {z, x} contains noattachment of any(L ∪ F ′′)-bridge ofG′ + u′v′. See Fig.5. Possibly,z = x.

Next we define a graphM and a subgraphPM of M, according to whether or not wehavex = z. First, assumex �= z. We letu1 = u′′ andv1 = v′′, let V (M) = {x, u′′, v′′}


andE(M) = ∅, and letPM = ∅. Now assumex = z. Let u1, v1 ∈ V (F ′′) such thatu′′, u1, v1, v

′′ occur onF ′′ in order,{x, u1} is contained in a(L∪F ′′)-bridge ofG′ + u′v′,{x, v1} is contained in a(L ∪ F ′′)-bridge ofG′ + u′v′, and subject to these conditions,u1F

′′v1 is minimal. Whenu1 �= v1, let M denote the union ofu1F′′u′′ ∪ v′′F ′′v1 and

those(L∪ F ′′)-bridges ofG′ + u′v′ whose attachments are all contained inV (u1F′′u′′)∪

V (v′′F ′′v1) ∪ {x}. Whenu1 = v1, let M be obtained from the union ofF ′′ and those(L ∪ F ′′)-bridges ofG′ + u′v′ whose attachments are all contained inV (F ′′) ∪ {x} bysplitting the vertexu1 = v1 to two verticesu1 andv1 in a natural way. We apply Lemma2.3 toM+{u1x, v1x, u

′′v′′} to find a(u′′F ′′u1 ∪v′′F ′′v1)-Tutte pathP ′M fromu′′ to v′′ and

throughu1x andv1x. Let PM = P ′M − x, which consists of disjoint paths fromu′′, v′′ to

u1, v1, respectively.We divide the remainder of the proof into two cases.Case1: x /∈ {u′, v′}, and both edges ofD incident withx are incident with the edgeu′v′.In this case, we must havex = z, for otherwise, the edge ofzDx incident withx is not

incident withu′ or v′ (becausezDx ⊆ C′′ andu′, v′ /∈ V (C′′)). Also the two edges ofDincident withx must bexu′ andxv′.

Thereforeu1 �= u′′ andv1 �= v′′, for otherwise, there is a non-null homotopic simpleclosed curve in the torus intersectingG only at u or v, contradicting the assumption thatthe face width ofG is at least 2. Moreover, becauseG is (4, C)-connected,{u′, v′, x}induces a facial triangle inG′ + u′v′. Hence,x has exactly two neighbors inL, namelyu′ andv′.

LetL′ denote the plane graph obtained fromG′ + u′v′ by deletingM − {u1, v1}, addingedgesu1u

′ andv1v′, and deletingu′v′ when|V (F ′)| > 2, such thatQ := (u1F

′′v1 ∪F ′)+{u1u

′, v1v′} is the outer cycle ofL′ on whichv′, v′v1, u1u

′, u′, y occur in clockwise order.By applying the mirror image version of Lemma 2.2 (withL′,Q, v′, y, u′u1, v

′v1 asG,C, x, y, e, f , respectively), there exist aC-flap H in L′ with attachmentsa, b, c (a =b = c = y if H is null) and a(Q− I (H))-Tutte pathP ′ from b to v′ in L′ − I (H) such thatu′, b, y, a, v′ occur onQ (and hence, onD′) in clockwise order,y ∈ (V (H)− {a}) ∪ {b},u1u

′, v1v′ ∈ E(P ′), anda, c ∈ V (P ′). (Note thatH is null when|V (F ′)| = 2.) By planarity,

noP ′-bridge ofL′ contains vertices of bothF ′ andu1F′′v1.

Let P denote the path inG induced by(E(P ′)−{u′u1, v′v1})∪E(PM)∪{xv′}. ThenP is

a path fromy tox,u, v, x, u1, v1 ∈ V (P ), andb, y, a occur onC in clockwise order. Hence,|V (P )∩V (C)|�2 and|V (P )|�4 (becausex /∈ {u′, v′}, u′′ �= u1 andv′′ �= v1). Clearly,His aC-flap inG. Moreover, everyP-bridge ofG is either aP ′-bridge ofL′, or aP ′

M -bridgeof M, or a bridge induced by the edgeu′x. Hence,P is a(C − I (H))-Tutte path fromb inG − I (H) such thata, c ∈ V (P ). It is also clear that everyP-bridge ofG containing anedge ofC is either aP ′

M -bridge ofM containing an edge ofM ∩ F ′′, or aP ′-bridge ofL′containing an edge ofF ′ or u1F

′′v1 but not both. Hence, everyP-bridgeB of G containingan edge ofC has a plane representation withB ∩ (C ∪ P) on its outer walk. SoP gives thedesired path.

Case2: Eitherx ∈ {u′, v′} or there is an edgef ∈ E(D) incident withx such that{f, u′v′} is a matching.

Whenx ∈ {u′, v′}, we pick an arbitrary vertexx′ ∈ V (D)−{u′, v′}. Next, we show that wemay apply Lemma 3.2 toL,D′,D, y, x′, u′v′ (asG,C,D, y, x, e, respectively) and, in thecase when the edgef exists, apply Lemma 3.3 toL,D′,D, y, f, u′v′ (asG,C,D, y, f, e,


respectively). When|V (F ′)| = 2, we apply a mirror image version of Lemma 3.2 orLemma 3.3, and the hypothesis of Lemma 3.2 or Lemma 3.3 holds becausey = u′. When|V (F ′)|�3, we claim that there do not exist verticesp, q ∈ V (D′) and a 2-separation(G′

1,G′2) in L such thatV (G′

1)∩V (G′2) = {p, q},p, y, u′v′, q occur onD′ in this clockwise

order,pD′q ⊆ G′1, andqD′p∪D ⊆ G′

2. This is obvious ifu′ = y. So assume thatu′ �= y.Suppose the abovep, q, and(G′

1,G′2) do exist. Then we can choose a non-null homotopic

simple closed curve�1 in the torus such that�1 meetsGonly inpandqand passes throughR.Becausey ∈ pCq−q and becausepCyis properly contained inuCy, (�1, p, q) contradictsthe choice of(�, u, v). So we may apply Lemma 3.2 toL,D′,D, y, x′, u′v′ or apply Lemma3.3 toL,D′,D, y, f, u′v′.

By Lemma 3.2 (whenx ∈ {u′, v′}) and Lemma 3.3 (whenf exists), there are two possi-bilities, and we treat them in two separate cases.

Subcase2.1: There exist aC-flapH in L with attachmentsa, b, c (a = b = c = y if H isnull) and a((D′ − I (H))∪D)-Tutte pathP ′ from b in L− I (H) such thatD ⊆ L− I (H),u′v′, b, y, a occur onD′ in this clockwise order (or when|V (F ′)| = 2,u′v′, a, y = b occuronD′ in this clockwise order),y ∈ (V (H) − {a}) ∪ {b}, x′ ∈ V (P ′) whenx ∈ {u′, v′},f ∈ E(P ′) when the edgef exists,u′v′ ∈ E(P ′), a, c ∈ V (P ′), and noP ′-bridge ofLcontains vertices of bothD′ − P ′ andD − P ′.

Note that when|V (F ′)| = 2,H must be null. For otherwise,H −{a, b, c} is a componentof G − {a, b, c} containing no vertex ofC, contradicting the assumption thatG is (4,C)-connected. See Fig.5.

Next we apply Lemma 2.5 to find a pathP1 from u1 to v1. For this purpose, we view(G′ +u′v′)− (M−{u1, v1, x}), L, u1, v1, x, u1F

′′v1,D, P′ asK,L, p, q, u,Q,Q′, T in

Lemma 2.5, respectively. It is straightforward to verify that the conditions of Lemma 2.5are satisfied. (In particular,|V (P ′) ∩ V (D)|�2 becausef ∈ E(D) ∩ E(P ′) or becausex, x′ ∈ V (P ′) ∩ V (D).) By Lemma 2.5, there is a pathP1 (asS in Lemma 2.5) betweenu1 andv1 in ((G′ + u′v′)− (M − {u1, v1, x}))− P ′ such thatP ′ ∪ P1 is au1F

′′v1-Tuttesubgraph of(G′ +u′v′)− (M−{u1, v1, x}) and everyP ′-bridge ofL containing no edge ofD is also a(P ′ ∪P1)-bridge of(G′ +u′v′)− (M−{u1, v1, x}). By planarity, no(P ′ ∪P1)-bridge of(G′ + u′v′)− (M − {u1, v1, x}) contains edges of bothF ′ andu1F

′′v1. Hence,P ′ ∪ P1 ∪ PM is a((D′ − I (H)) ∪ F ′′)-Tutte subgraph ofG′ + u′v′.

Clearly,H is aC-flap inG, andE((P ′ − u′v′) ∪ P1 ∪ PM) induces a(C − I (H))-TuttepathP from b in G − I (H) such thata, c ∈ V (P ). It is also clear that everyP-bridge ofG containing an edge ofC is a (P ′ ∪ P1 ∪ PM)-bridge ofG′ + u′v′ containing an edgeof F ′ or F ′′ but not both. Hence, everyP-bridgeB of G containing an edge ofC has aplane representation withB ∩ (C ∪ P) on its outer walk. Becauseu, v ∈ V (P ) ∩ V (C),|V (P ) ∩ V (C)|�2.

We conclude this case by showing|V (P )|�4. This is obvious when the edgef exists,becausef ∈ E(P ∩ D) and {f, u′v′} is a matching. Now assumex ∈ {u′, v′}. Then|V (P )|�3 (becauseu, v, x′ ∈ V (P )). Suppose|V (P )| = 3. Then|V (PM ∪P1)| = 2. Thisimplies thatu′′ = u1, v′′ = v1, andu′′v′′ ∈ E(G′). Therefore, sinceG is (4,C)-connected,{x, u′′, v′′} is not a cut inG. HenceV (M) = {x, u′′, v′′} andE(M) = {xu′′, xv′′}. Thisshowsx ∈ C′′, contradicting the fact thatu′, v′ /∈ C′′. So we have|V (P )|�4.

Subcase2.2: There exista′, b′ ∈ V (D′)∩V (D), a separation(H ′, H ∗) in L withV (H ′)∩V (H ∗) = {a′, b′}, and a(D′ ∪ D)-Tutte pathP ′ from b′ in L such thatu′v′, b′, y, a′


s

a ′

b ′

c

H

y

a

b

v ′u ′

H ′

u ′′

f

v ′′F ′′

b ′ = y = u ′

tv ′

a ′

s

P ′

H*

H ′

u ′′

H*

f

F ′′v ′′

P ′

t

(B)(A)

Fig. 6. Subcase 2.2.

(u′v′, a′, y, b′ when|V (F ′)| = 2) occur onD′ in this clockwise order,b′D′a′∪b′Da′ ⊆ H ′(or a′D′b′ ∪ a′Db′ ⊆ H ′ when|V (F ′)| = 2),a′D′b′ ∪ a′Db′ ⊆ H ∗ (or b′D′a′ ∪ b′Da′ ⊆H ∗ when|V (F ′)| = 2),P ′ ⊆ H ∗, y ∈ (V (H ′)− {a′}) ∪ {b′} (y = b′ when|V (F ′)| = 2),x′ ∈ V (P ′)whenx ∈ {u′, v′},f ∈ E(P ′)when the edgef exists,u′v′ ∈ E(P ′),a′ ∈ V (P ′),and noP ′-bridge ofL other thanH ′ contains vertices of bothD′ −P ′ andD−P ′ (Fig.6).

Note that there are vertices ofu1F′′v1 which are co-facial witha′ or b′. Let s, t ∈

V (u1F′′v1) such that (a)u′′, u1, s, t, v1, v

′′ occur onF ′′ in this order, (b)s is co-facial withb′ (or a′ when|V (F ′)| = 2) andt is co-facial witha′ (or b′ when|V (F ′)| = 2), and (c)subject to (a) and (b),sF ′′t is maximal. LetJ denote the union ofH ′, sF ′′t , and those(L ∪ F ′′)-bridges ofG′ + u′v′ whose attachments are all contained inV (sF ′′t) ∪ V (H ′).

SinceG is 2-connected,J ∗ := J +{a′t, sb′} (orJ ∗ := J +{a′s, b′t} when|V (F ′)| = 2)is 2-connected.Without loss of generality, assume thatC∗ := (b′D′a′∪sF ′′t)+{a′t, sb′} (orC∗ := (a′D′b′ ∪ sF ′′t)+{a′s, tb′} when|V (F ′)| = 2) is the outer cycle ofJ ∗. By Lemma2.2, with J ∗, C∗, a′, y, sb′, a′t (or a′, y, b′t, a′s when |V (F ′)| = 2) asG,C, x, y, e, f ,respectively, there exist aC∗-flap H with attachmentsa, b, c (a = b = c = y if His null) and a(C∗ − I (H))-Tutte pathP ∗ betweenb and a′ in J ∗ − I (H) such thatsb′, b, y, a, a′ (or a, y = b, b′t, sa′ when |V (F ′)| = 2) occur onC∗ in this clockwiseorder,y ∈ (V (H) − {a}) ∪ {b}, a, c ∈ V (P ∗), and{a′t, sb′} ⊆ E(P ∗) (or {a′s, b′t} ⊆E(P ∗) when |V (F ′)| = 2). Note when|V (F ′)| = 2, y = b andH is null becauseG is(4,C)-connected. SoH is also aC-flap inG. By planarity, noP ∗-bridge ofJ ∗ contains edgesof bothD′ andF ′′.

Next we apply Lemma 2.5 to find a pathP1 from u1 to s and a pathP2 from t to v1.Note that, sincex ∈ V (D) ∩ V (P ′), x divides the attachments onD ∩ H ∗ of (L ∪ F ′′)-bridges so that theP ′-bridges ofH ∗ used in constructingP1 are different from those used inconstructingP2.

LetL1 denote the union ofL,u1F′′s, and those(L∪F ′′)-bridges ofGwhose attachments

are all contained inV (H ∗) ∪ V (u1F′′s). ThenL1 is connected becausex ∈ V (H ∗). We

viewL1, L, u1, s, x, u1F′′s,D, P ′ asK,L, p, q, u,Q,Q′, T in Lemma 2.5, respectively.

It is straightforward to verify that the conditions of Lemma 2.5 hold. (Recall that|V (P ′)∩V (D)|�2 becausef ∈ E(D ∩ P) or x, x′ ∈ V (D ∩ P ′). By Lemma 2.5,L1 − P ′ has a


z

a c

bs

t

a ′

b ′y

c ′

S

T

H ′

Fig. 7. An illustration of Lemma 4.2.

pathP1 (asSin Lemma 2.5) betweenu1 andssuch thatP1 ∪P ′ is au1F′′s-Tutte subgraph

ofL1 and everyP ′-bridge ofL containing no edge ofD is also a(P1 ∪P ′)-bridge ofL1. Byplanarity, no(P1 ∪P ′)-bridge ofL1 contains edges of bothu1F

′′s andD′. Hence,P1 ∪P ′is a(D′ ∪ u1F

′′s)-Tutte subgraph ofL1.LetL2 denote the union ofL, edgeta′ (or tb′ when|V (F ′)| = 2), tF ′′v1, and those(L∪

F ′′)-bridges ofGwhose attachments are all contained inV (H ∗)∪V (tF ′′v1). Note thatL2 isconnected because of the edgeta′ (or tb′ when|V (F ′)| = 2).We viewL2, L, t, v1, x, tF

′′v1,

D, P ′ asK,L, p, q, u,Q,Q′, T in Lemma 2.5, respectively. By Lemma 2.5,L2 − P ′ hasa pathP2 betweent andv1 such thatP2 ∪ P ′ is a tF ′′v1-Tutte subgraph ofL2 and everyP ′-bridge ofL containing no edge ofD is also a(P2 ∪ P ′)-bridge ofL2. By planarity,no (P2 ∪ P ′)-bridge ofL2 contains edges of bothtF ′′v1 andD′. Hence,P2 ∪ P ′ is a(D′ ∪ tF ′′v1)-Tutte subgraph ofL2.

NowE(P ′ ∪ P ∗ ∪ P1 ∪ P2 ∪ PM)− {u′v′, a′t, sb′} (orE(P ′ ∪ P ∗ ∪ P1 ∪ P2 ∪ PM)−{u′v′, a′s, b′t} when|V (F ′)| = 2) induces a(C− I (H))-Tutte pathP from b inG− I (H)such thatb, y, a occur onC in this clockwise order,y ∈ (V (H) − {a}) ∪ {b}, anda, c ∈V (P ). It is clear that everyP-bridge ofG is one of the following: a(P ′ ∪ P1)-bridge ofL1, or (P ′ ∪ P2)-bridge ofL2, or aP ∗-bridge ofJ ∗, or aP ′

M -bridge ofM. Hence, everyP-bridgeB of G containing an edge ofC has a plane representation withB ∩ (C ∪ P) onits outer walk. Becauseu, v ∈ V (P ) ∩ V (C), |V (P ) ∩ V (C)|�2.

By exactly the same argument as in the end of Subcase 2.1, we can show|V (P )|�4. �

Before we proceed to the general case, let us prove the following lemma which will beused to extend Tutte paths throughC-flaps. See Fig.7 for an illustration.

Lemma 4.2. Let G be a2-connected plane graph with outer cycle C, lets, t, b, c, a ∈ V (C)be distinct such thattCs = tbcas, and lety ∈ V (sCt − s). Suppose thatG− ((sCt − s)∪{a, c}) contains a path from b to s. Then there exist an sCt-flapH ′ in G with attachmentsa′, b′, c′ (a′ = b′ = c′ = y if H ′ is null) and disjoint paths S and T in(G−{a, c})− I (H ′)such thats, a′, y, b′, t occur on C in this clockwise order, y ∈ (V (H ′)− {a′}) ∪ {b′}, S isfrom s to b, T is from t tob′, a′, c′ ∈ V (S ∪ T ), andS ∪ T ∪ {a, c} is a (C − I (H ′))-Tuttesubgraph of G.


Proof. BecauseG− ((sCt − s)∪{a, c}) contains a path betweenb ands, sCbis containedin a cycle inG− {a, c}. LetG′ denote the block ofG− {a, c} containingsCb, and letC′denote the outer cycle ofG′. ThussC′b = sCb. Note that every(G′ ∪ {a, c})-bridge ofGhas at most one attachment onG′ (becauseG′ is a block ofG) and at least one attachmentin {a, c} (becauseG is 2-connected).

By planarity ofG, there existsz ∈ V (bC′s) such that, for each(G′ ∪ {a, c})-bridgeB ofG, B ∩ C′ ⊆ zC′s if a ∈ V (B), andB ∩ C′ ⊆ bC′z if c ∈ V (B). See Fig.7.

By Lemma 2.2 (withG′, C′, s, y, tb, z asG,C, x, y, e, f , respectively), there exist aC′-flapH ′ with attachmentsa′, b′, c′ (a′ = b′ = c′ = y if H ′ is null) and a(C′ − I (H ′))-Tutte pathP betweenb′ ands in G′ − I (H ′) such thats, a′, y, b′, tb, z occur onC′ in thisclockwise order,y ∈ (V (H ′)− {a′}) ∪ {b′}, tb ∈ E(P ), anda′, c′, z ∈ V (P ). Let SandTdenote the disjoint paths inP − tb such thatSis betweensandb andT is betweent andb′.

Note thata′, c′ ∈ V (S∪T )andH ′ is ansCt-flap inG.Also note that every(S∪T ∪{a, c})-bridge of G other than those contained inH ′ is either aP-bridge ofG′ − I (H ′), or a(G′ ∪{a, c})-bridge ofG, or induced by the edgetb. BecausetCs = tbcas, no(G′ ∪{a, c})-bridge of G with three attachments contains an edge ofC. Hence,S ∪ T ∪ {a, c} is a(C − I (H ′))-Tutte subgraph ofG. �

We now prove Theorem 1.2 when theR-width is even for some faceR.

Lemma 4.3. Let G be a2-connected graph embedded in the torus with face width at least2. Let R be a face of G, let C be the subgraph of G consisting of vertices and edges of Gincident with R, and lety ∈ V (C). Assume that the R-width of G is a positive even integer,and G is(4,C)-connected. Then there exist a C-flapH ′ in G with attachmentsa′, b′, c′ (a′ =b′ = c′ = y if H ′ is null) and a(C − I (H ′))-Tutte path P fromb′ in G− I (H ′) such thatb′, y, a′ occur on C in this clockwise order, y ∈ (V (H ′)− {a′})∪ {b′}, a′, c′ ∈ V (P ), andevery P-bridge B of G containing an edge of C has a plane representation withB ∩ (C∪P)on its outer walk. Moreover, |V (P ) ∩ V (C)|�2 and|V (P )|�4.

Proof. Because the face width ofG is at least 2,C is a cycle inG. By Lemma 4.1, we mayassume that theR-width of G is at least 4. Hence,G−C contains a cycle which bounds anopen disc in the plane containingR. LetL be the block ofG−C containing one such cycle.ThenL has a faceR′ which containsR (as subsets of the torus). SinceL is 2-connected andtheR′-width of L is at least 2,R′ is bounded by a cycle, sayD. SinceG has face width atleast 2 and becauseD bounds a disc in the torus,L has face width at least 2. SinceG is(4,C)-connected,L is (4,D)-connected.

Clearly, every(L∪C)-bridge ofG has at most one attachment onL and all these attach-ments are contained inV (D). LetL∗ be the union ofC,D, and all(L ∪ C)-bridges ofG.ThenL∗ is contained in the closed disc in the torus bounded byD. Hence, we viewL∗ asa plane graph such thatD is its outer cycle andC is a facial cycle. See Fig.8.

Let v1, . . . , vn be the attachments onD of (L ∪ C)-bridges ofG, and occur onD in thisclockwise order. Letpi, qi ∈ V (C) such thatqiCpi is maximal subject to the followingconditions: (a){pi, vi} is contained in a(L ∪ C)-bridge ofG, (b) {qi, vi} is contained in a(L∪C)-bridge ofG, and (c) no(L∪C)-bridge ofG with an attachment inV (D)−{vi} hasan attachment inV (qiCpi)−{pi, qi}. SinceG is (4,C)-connected,pi andqi are well defined


L

a b

qmy

DC t = pm-1

s = pl

H

vk+1

c

Fig. 8. The graphG.

and there is somepj �= pi (because otherwiseG − {vi, pi} has a component containingno vertex ofC, contradicting(4, C)-connectivity ofG). For i = 1, . . . , n, let Ji denotethe union ofqiCpi and those(L ∪ C)-bridges ofG whose attachments are all containedin V (qiCpi) ∪ {vi}. Note thaty ∈ V (pkCpk+1)− {pk+1} for somek ∈ {1, . . . , n}, wherepn+1 = p1, qn+1 = q1 andvn+1 = v1. In particular,pk �= pk+1.

Because theR′-width of L is both even and less than theR-width of G, by inductionhypothesis (withL,D,R′, vk+1 asG,C,R, y, respectively), there exist aD-flap H withattachmentsa, b, c (a = b = c = vk+1 if H is null) and a(D − I (H))-Tutte pathTin L − I (H) from b such thatb, vk+1, a occur onD in this clockwise order,vk+1 ∈(V (H) − {a}) ∪ {b}, a, c ∈ V (T ), and everyT-bridgeB of L containing an edge ofDhas a plane representation withB ∩ (D ∪ P) on its outer walk. Note that|V (T )|�4 and|V (T )∩ V (D)|�2 by Lemma 4.1 or by induction hypothesis when theR′-width of L is atleast 4.

We distinguish two cases.Case1: H is null.In this case,T is aD-Tutte path fromvk+1 in L.Let J denote the union ofpkCpk+1 and those(L ∪ C)-bridges ofG whose attach-

ments are all contained inV (pkCpk+1) ∪ {vk+1}. SinceG is 2-connected,J + pkvk+1is also 2-connected. We can viewJ + pkvk+1 as a 2-connected plane graph such thatpkCpk+1 and pkvk+1 are contained in its outer cycleCk. By Lemma 2.1 (withJ +pkvk+1, Ck, pk+1, y, pkvk+1 asG,C, x, y, e, respectively),J + pkvk+1 has aCk-TuttepathRbetweenpk+1 andy such thatpkvk+1 ∈ E(R).

To find a pathS from pk+1 to pk, we letK = G − (V (J ) − {pk, pk+1, vk+1}). Weview pk+1Cpk,D, vk+1 asQ,Q′, u in Lemma 2.5, respectively. It is straightforward tocheck that the conditions of Lemma 2.5 are satisfied (using the plane representation ofL∗). By Lemma 2.5, there is a pathS in K − T betweenpk andpk+1 such thatS ∪ Tis a pk+1Cpk-Tutte subgraph ofK and everyT-bridge ofL containing no edge ofD isalso an(S ∪ T )-bridge of K. By the plane representation ofL∗, every (S ∪ T )-bridgeB of K containing an edge ofC has a plane representation withB ∩ (S ∪ T ∪ C) onits outer walk.


J J ′

a b

c

H

a b

c

H

C

s t

Cy y

pl = pm-1

vk+1 vk+1

Fig. 9. The graphJ ′ whenpm−1 = pl .

Let P := S ∪ T ∪ (R − pkvk+1). Then everyP-bridge ofG is either an(S ∪ T )-bridgeof K or aR-bridge ofJ + pkvk+1. HenceP is aC-Tutte path fromy in G such that everyP-bridgeB of G containing an edge ofC has a plane representation withB ∩ (C ∪P) on itsouter walk. Clearly,|V (P )∩V (C)|�2 (becausepk, pk+1 ∈ V (P )∩V (C)) and|V (P )|�4(because|V (T )|�4).

Case2: H is non-null.Let vl, vm ∈ I (H) ∩ V (D) such thatb, vm, vk+1, vl, a occur onD in this clockwise

order and such thatvmDvl is maximal. See Fig.8. Note that ifpm−1 = pl then we do nothavepm−1 = pm = · · · = pk+1 = · · · = pl becausepk+1 �= pk. Let J denote the unionof pm−1Cpl (whenpm−1 �= pl) or C (whenpm−1 = pl), H, and those(L ∪ C)-bridgesof G whose attachments are all contained inV (pm−1Cpl) ∪ I (H) (whenpm−1 �= pl) orV (C) ∪ I (H) (whenpm−1 = pl), wherep0 = pn.

Next we defineJ ′. If pm−1 �= pl , then letJ ′ = J (and so,J ′ has a plane representationwith pm−1Cpl and{a, b, c} on its outer walk), and lett := pm−1 ands := pl . See Fig.8. Now assume thatpm−1 = pl . Thenpk �= pl or pk+1 �= pl (becausepk �= pk+1).SinceG is (4, C)-connected, every(L∪C)-bridge ofG with an attachment inD− I (H) isinduced by a single edge and haspm−1 = pl as an attachment. Alsovm �= vl ; for otherwise,G− {vl, pl} has a component containing no vertex ofC, contradicting(4, C)-connectivityof G. LetJ ′ be the plane graph obtained fromJ by splitting the vertexpm−1 = pl to sandtin a natural way such thatC becomes a path and the neighbors ofpm−1 = pl in J containedin Jl (respectively, not contained inJl) become the neighbors ofs (respectively,t) and suchthatJ ′ has a plane representation withE(C) and{a, b, c} on its outer walk. See Fig. 9.

Let J ′′ := J ′ + {tb, bc, ca, as}, and letC′′ be the cycle ofJ ′′ induced byE(C) ∪{tb, bc, ca, as}. We can viewJ ′′ as a 2-connected plane graph with outer cycleC′′ suchthat tC′′s = tbcas. It is easy to see thaty ∈ V (sC′′t − s) (sincey ∈ pkCpk+1 − pk+1)andJ ′′ − ((sC′′t − s) ∪ {a, c}) has a path fromb to s.

By Lemma 4.2 (withJ ′′, C′′ asG,C, respectively), there exist ansC′′t-flapH ′ in J ′′with attachmentsa′, b′, c′ (a′ = b′ = c′ = y if H ′ is null) and disjoint pathsS′, T ′ in(J ′′ − {a, c}) − I (H ′) such thats, a′, y, b′, t occur onC′′ in this clockwise order,y ∈


(V (H ′) − {a′}) ∪ {b′}, S′ is betweens andb, T ′ is betweenb′ andt, a′, c′ ∈ V (S′ ∪ T ′),andS′ ∪ T ′ ∪ {a, c} is an(C′′ − I (H ′))-Tutte subgraph ofJ ′′. Note thatt, b′, y, a′, s occuronC in this clockwise order,

If pm−1 = pl , then identifyingt and s to pm−1 = pl in T ∪ S′ ∪ T ′ gives the de-sired pathP in G. Note that|V (P )|�4 because|V (T )|�4. Moreover,|V (P )∩V (C)|�2,for otherwise, C is contained in aP-bridge of G whose attachments onP are acut of size at most 3 inG (because|V (T )|�4) showing thatG is not (4, C)-connected,a contradiction.

So assume thatpm−1 �= pl . LetK = G− (V (J )− (V (H)∪ {s, t})). We views, t, sCt ,D, a asp, q,Q,Q′, u in Lemma 2.5, respectively. It is easy to verify that the conditionsof Lemma 2.5 are satisfied (using the plane representation ofL∗). By Lemma 2.5, thereis a pathS in K − T betweens and t such thatS ∪ T is an sCt-Tutte subgraph ofKand everyT-bridge ofL containing no edge ofD is also an(S ∪ T )-bridge ofK. Notethat each(S ∪ T )-bridgeB of K containing an edge ofC has a plane representation withB∩(S∪T ∪C) on its outer walk. Because|V (T )|�4, |V (P )|�4. Sincepm−1, pl ∈ V (P ),we have|V (P )∩V (C)|�2. Hence,P := S ∪ T ∪ S′ ∪ T ′ gives the desired path inG. �

Proof of Theorem 1.2. Let G be a 4-connected graph embedded in the torus, and let� bethe face width ofG. If � = 0, thenG is a 4-connected planar graph, and so, has a Hamiltoncycle by a theorem of Tutte[8]. So assume that��1.

Suppose� = 1. Then there is a non-null homotopic simple closed curve� meetingGonly in one vertex, sayu. Cutting the torus open along�, we obtain a plane graphG′with two verticesu′, u′′ such thatG can be obtained fromG′ by identifying u′ andu′′as u. SinceG is 4-connected, the blocks ofG′ can be labeled asB1, . . . , Bn such thatBi ∩ Bi+1 consists of a single vertexvi , Bi ∩ Bj = ∅ if j > i + 1, andv0 := u′ ∈V (B1) − {v1} andvn := u′′ ∈ V (Bn) − {vn−1}. BecauseG is 4-connected,n = 1 orn = 2, and ifn = 2 then exactly one ofB1 or B2 is induced by an edge. Without lossof generality, assume thatB1 is 2-connected andv0 is contained in the outer cycleC1 ofB1. We may assumev1 /∈ V (C1); as otherwise,G is planar andG has a Hamilton cycleby a theorem of Tutte. By Lemma 2.1 (withB1, C1, v0, v1 asG,C, x, y, respectively),B1has aC1-Tutte pathP1 betweenv0 andv1. SinceG is 4-connected andP1 is aC1-Tuttepath inB1, V (C1) ⊆ V (P1). Hence|V (P1)|�4 (becausev1 /∈ V (C1)). Let P ′ := P1if n = 1, and otherwise letP ′ := P1 ∪ B2. ThenP ′ is a Tutte path inG′ betweenu′andu′′. BecauseG is 4-connected and|V (P ′)|� |V (P1)|�4, P ′ is a Hamilton path inG′ betweenu′ andu′′. Clearly,E(P ′) induces a Hamilton cycleT in G. Hence,G has aHamilton path.

Therefore we may assume that��2. Let� be a non-null homotopic simple closed curvein the torus that meetsG exactly� times (only at vertices).

Case1: � is even.Let R be a face ofG that � passes through, and letC be the cycle ofG consisting of

vertices and edges ofG incident withR.Assume� = 2. Then theR-width of G is 2. By Lemma 4.1,G has a Tutte pathP from

someb ∈ V (C) such that|V (P )|�4. SinceG is 4-connected,P is a Hamilton path inG.Now assume��4. By Lemma 4.3,G has a Tutte pathP from someb′ ∈ V (C) such that

|V (P )|�4. SinceG is 4-connected and|V (P )|�4, P is a Hamilton path inG.


Case2: � is odd.Let u ∈ � ∩ V (G). ThenG − u has face width� − 1�2. Let R be the face ofG − u

which containsu (as a subset of the torus). LetC denote the cycle inG − u consisting ofvertices and edges incident withR, and choose a vertexy ∈ V (C) such thatyu ∈ E(G).ThenG−u is (4,C)-connected. Applying Lemma 4.1 or Lemma 4.3 toG−u,R, y, we canshow that there is aC-flapH ′ inG−uwith attachmentsa′, b′, c′ (a′ = b′ = c′ = y if H ′ isnull) and a(C − I (H ′))-Tutte pathP ′ from b′ in (G− u)− I (H ′) such thatb′, y, a′ occuronC in this clockwise order,y ∈ (V (H ′)− {a′}) ∪ {b′}, a′, c′ ∈ V (P ′), and|V (P ′)|�4.

If H ′ is null, thenP ′ is aC-Tutte path inG−u from y. In this case,P := (P ′ ∪ {u})+yuis a Tutte path inG. BecauseG is 4-connected and|V (P )|�4, P is a Hamilton path inG.

Therefore we may assume thatH ′ is non-null. LetH ∗ denote the union ofH ′, u, andall edges ofG with both ends inI (H ′) ∪ {u}. BecauseG is 4-connected,H ∗ is connected.In fact,H ∗ + {a′u, b′c′} is 2-connected. Assume without loss of generality thata′u andb′c′ are contained in the outer cycleC∗ of H ∗ + {a′u, b′c′}. By applying Lemma 2.3 (withH ∗ + {a′u, b′c′}, C∗, c′, a′, a′u, b′c′ asG,C, x, y, e, f , respectively),H ∗ + {a′u, b′c′}contains a Tutte pathP ∗ betweena′ and c′ such that{a′u, b′c′} ⊆ E(P ∗). Let P :=P ′ ∪ (P ∗ − {a′, c′}). Then everyP-bridge ofG is either aP ′-bridge ofG − u, or aP ∗-bridge ofH ∗ + {a′u, b′c′}. Hence,P is a Tutte path ofG. BecauseG is 4-connected and|V (P )|�4, P is a Hamilton path inG. �

Acknowledgments

Part of the work was done while the second author was visiting the Department ofMathematics at the University of Hong Kong. The authors thank both the referees for manyhelpful suggestions that led to the elimination of several mistakes in the previous manuscript.In particular, we thank the referee who pointed out a missing case (Case 1) in the proof ofLemma 4.1.

References

[1] A. Altshuler, Hamiltonian circuits in some maps on the torus, Discrete Math. 1 (1972) 299–314.[2] R. Brunet, R.B. Richter, Hamiltonicity of 5-connected toroidal triangulations, J. Graph Theory 20 (1995)

267–286.[3] B. Grünbaum, Polytopes, graphs, and complexes, Bull. Amer. Math. Soc. 76 (1970) 1131–1201.[4] C.St.J.A. Nash-Williams, Unexplored and semi-explored territories in graph theory, in: F. Harary (Ed.), New

Directions in Graph Theory, Academic Press, New York, 1973, pp. 169–176.[5] R. Thomas, X.Yu, 4-connected projective-planar graphs are hamiltonian, J. Combin. Theory, Ser. B 62 (1994)

114–132.[6] R. Thomas, X. Yu, 5-connected toroidal graphs are hamiltonian, J. Combin. Theory, Ser. B 69 (1997) 79–96.[7] C. Thomassen, A theorem on paths in planar graphs, J. Graph Theory 7 (1983) 169–176.[8] W.T. Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956) 99–116.[9] H. Whitney, A theorem on graphs, Ann. of Math. 32 (1931) 378–390.

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Hamilton paths in toroidal graphs