long cycles in 3-connected graphspeople.math.gatech.edu/~yu/papers/longcycle.pdft x x x w t 1 0 2 1...

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Long Cycles in 3-Connected Graphs * Guantao Chen Department of Mathematics & Statistics Georgia State University Atlanta, GA 30303 Xingxing Yu School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332 Abstract Moon and Moser in 1963 conjectured that if G is a 3-connected planar graph on n vertices, then G contains a cycle of length at least Ω(n log 3 2 ). In this paper, this conjecture is proved. In addition, the same result is proved for 3-connected graphs embeddable in the pro- jective plane, or the torus, or the Klein bottle. * MSC Primary 05C38 and 05C50 Secondary 57M15 Partially supported by NSA grant MDA904-97-1-0101 Partially supported by NSF grant DMS-9970527 1

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Page 1: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

Long Cycles in 3-Connected Graphs∗

Guantao Chen †

Department of Mathematics & StatisticsGeorgia State University

Atlanta, GA 30303

Xingxing Yu ‡

School of Mathematics

Georgia Institute of TechnologyAtlanta, Georgia 30332

Abstract

Moon and Moser in 1963 conjectured that if G is a 3-connectedplanar graph on n vertices, then G contains a cycle of length at leastΩ(nlog3 2). In this paper, this conjecture is proved. In addition, thesame result is proved for 3-connected graphs embeddable in the pro-jective plane, or the torus, or the Klein bottle.

∗MSC Primary 05C38 and 05C50 Secondary 57M15†Partially supported by NSA grant MDA904-97-1-0101‡Partially supported by NSF grant DMS-9970527

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Page 2: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

1 Introduction and notation

A graph is Hamiltonian if it contains a cycle using all vertices, and such acycle is called a Hamilton cycle. A planar graph is a graph which can beembedded in the plane without crossing edges, and such an embedding iscalled a plane graph.

In 1931, Whitney [11] proved that every 4-connected triangulation ofthe plane contains a Hamilton cycle. In 1956, Tutte [10] proved a moregeneral result: every 4-connected planar graph contains a Hamilton cycle.However, 3-connected planar graphs need not contain Hamilton cycles. Forsuch examples, see Holton and McKay [7].

The circumference of a graph G, denoted by circ(G), is the length ofa longest cycle in G. In 1963, Moon and Moser [9] implicitely made thefollowing conjecture by giving 3-connected planar graphs G with circ(G) ≤9|V (G)|log3

2.

Conjecture 1.1 If G is a 3-connected planar graph on n vertices, thencirc(G) ≥ Ω(nlog3 2).

We mention here that Grunbaum and Walther [6] made the same conjec-ture for a family of 3-connected cubic planar graphs.

Barnette [1] showed that every 3-connected planar graph with n verticescontains a cycle of length at least

√lg n and Clark [3] later improved this

lower bound to e√

lg n. In [8], Jackson and Wormald obtained a polynomiallower bound βnα, where β is some constant and α ≈ 0.207. Recently, Gaoand Yu [5] improved α to 0.4 and extended the result to 3-connected graphsembeddable in the projective plane, or the torus, or the Klein bottle.

The main result of this paper is the following.

Theorem 1.2 Let G be a 3-connected graph with n vertices, and supposethat G is embeddable in the sphere, or the projective plane, or the torus, orthe Klein bottle. Then circ(G) ≥ Ω(nlog3 2).

Throughout this paper, we consider finite graphs with no loops or multipleedges. For a graph G, we use V (G) and E(G) to denote its vertex set andedge set, respectively. A graph H is a subgraph of G, denoted by H ⊂ G,if V (H) ⊂ V (G) and E(H) ⊂ E(G). We shall use ∅ to denote the emptygraph (as well as the empty set).

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Page 3: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

For S ⊂ V (G), the subgraph of G induced by S, denoted by G[S], is thegraph whose vertex set is S and whose edge set consists of the edges in Gwith both incident vertices in S. For S ⊂ E(G), the subgraph of G inducedby S, denoted by G[S], is the graph whose edge set is S and whose vertexset consists of the vertices in G incident with edges in S. Let e be an edgeof G with incident vertices x and y; then we write e = xy, and write G[xy]or G[e] instead of G[xy] or G[e].

Let X ⊂ V (G), or X ⊂ E(G), or X ⊂ G; then G −X denotes the graphobtained from G by deleting X and the edges of G incident with a vertex inX. If X = x, then we write G − x instead of G − x.

Let H ⊂ G; then G/H denotes the graph with V (G/H) = V (G−H)∪h(where h /∈ V (G)) and E(G/H) = E(G − H) ∪ hy : y ∈ V (G − H) andyy′ ∈ E(G) for some y′ ∈ V (H). We say that G/H is obtained from Gby contracting H to the vertex h. If G is a graph, and x, y ∈ V (G), thenG + xy = G if xy ∈ E(G); otherwise, G + xy denotes the graph obtainedfrom G by adding the edge xy.

Let G and H be subgraphs of a graph. Then G∩H (respectively, G∪H)is the graph with vertex set V (G) ∩ V (H) (respectively, V (G) ∪ V (H)) andedge set E(G) ∩ E(H) (respectively, E(G) ∪ E(H)). We shall use G − Hinstead of G − (H ∩ G).

A block of a graph G is a maximal 2-connected subgraph of G. (Thecomplete graph on two vertices is 2-connected.) Let G be a connected graphand X ⊂ V (G), where |X| = k and k is a positive integer; then X is calleda k-cut of G if G− X is not connected. If X = x is a cut set of G, then xis a cut vertex of G.

Let G be a plane graph, a plane subgraph of G is a subgraph of G inheritingthe embedding of G. The faces of G are the connected components (intopological sense) of the complement of G in the plane. The outer face ofa plane graph G is the unbounded face; the bounded faces are inner faces.The boundary of the outer face is called the outer walk of the graph, orthe outer cycle if it is a cycle. A cycle is a facial cycle in a plane graph ifit bounds a face of the graph. An open disc (respectively, closed disc) inthe plane is a homeomorphic image of (x, y) : x2 + y2 < 1 (respectively,(x, y) : x2 + y2 ≤ 1).

For u, v ∈ V (G), a u − v path in a graph G is a path with end verticesu and v. For any path P and x, y ∈ V (P ), xPy denotes the subpath of Pbetween x and y. Given two vertices x and y on a cycle C in a plane graph,

3

Page 4: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

we use xCy to denote the path in C from x to y in clockwise order.Let H be a subgraph of a graph G. An H-bridge of G is a subgraph of G

which either (1) is induced by an edge of E(G) − E(H) with both incidentvertices in H or (2) is induced by the edges in a component of G − H andthe edges of G from H to that component. For any H-bridge B of G, theattachments of B (on H) are the vertices in V (B) ∩ V (H).

Although Theorem 1.2 is stated for 3-connected graphs, we need to workwith certain 2-connected graphs. The following concepts will serve this pur-pose.

A circuit graph is a pair (G, C), where G is a 2-connected plane graphand C is a facial cycle of G, such that, for any 2-cut S of G, every componentof G − S contains a vertex of C.

An annulus graph is a triple (G, C1, C2), where G is a 2-connected planegraph and C1 and C2 are facial cycles of G, such that, for any 2-cut S of G,every component of G − S contains a vertex of C1 ∪ C2.

The rest of the paper is organized as follows. In Section 2, we givethe Moon-Moser example which shows that the exponent log3 2 in Theo-rem 1.2 cannot be improved. (In fact, we slightly improve their bound of9|V (G)|log3 2.) In Section 3, we prove a result for weighted graphs, fromwhich Conjecture 1.1 follows as a consequence. In Section 4, we prove The-orem 1.2 for graphs embeddable in the projective plane, or the torus, or theKlein bottle.

2 An example

In this section, we use the Moon-Moser example to illustrate that the boundin Conjecture 1.1 (and hence, Theorem 1.2) is in a sense best possible.

First, we define a sequence of 3-connected plane graphs Tk as fol-lows. Let T1 be a plane graph isomorphic to K4. Further, let V (T1) =w, x0, x1, x2 and let x0x1x2x0 be the outer cycle of T1. Suppose that Tk isdefined for some k ≥ 1. Let Tk+1 be the graph obtained from Tk as follows:in each inner face of Tk, add a new vertex and join the new vertex to thevertices of Tk incident with that face. The graphs T1 and T2 are shown inFigure 1.

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Page 5: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

T

x

x x

w

T 1

0

1 2

x 0

x 1x 2

2

w

Figure 1:

By the above construction, for any k ≥ 1, Tk is a 3-connected plane graphwith outer cycle x0x1x2x0. We shall show that circ(Tk) < 7

2nlog3 2).

Let αk be the length of a longest x1−x2 path in Tk and βk be the length ofa longest x1−x2 path in Tk−x0. By the construction of Tk, for i, j ∈ 0, 1, 2and i 6= j, the length of a longest xi − xj path in Tk is αk and the length ofa longest xi − xj path in Tk − (x0, x1, x2 − xi, xj) is βk.

Proposition 2.1 For k ≥ 1, αk = 3 · 2k−1 and βk = 2k.

Proof: It is easy to check that α1 = 3 and β1 = 2. Assume that αk = 3 ·2k−1

and βk = 2k. We need to show that αk+1 = 3 · 2k and βk+1 = 2k+1.For i ∈ 0, 1, 2, let Di denote the open disc in the plane bounded by the

triangle in Tk+1 induced by w, x0, x1, x2 − xi. Let V i denote the set ofvertices in Tk+1 contained in Di, and let T i be the plane subgraph of Tk+1

induced by V i ∪ (w, x0, x1, x2 − xi).Clearly, T i is a 3-connected plane graph isomorphic to Tk. We shall

proceed with claims (a) - (d).

(a) αk+1 ≤ 3 · 2k.Let P be an x1 − x2 path in Tk+1. Since the outer cycle of each T i

is a triangle, P ∩ T i consists of a path, or a path and an isolated ver-tex in w, x0, x1, x2 − xi, or isolated vertices which are contained inw, x0, x1, x2 − xi.

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Page 6: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

If P ∩ T 0 is a path, then P ⊂ T 0. In this case, |E(P )| ≤ αk = 3 · 2k−1 ≤3 · 2k.

If P ∩T 1 consists of a path and w, x0, x2 ⊂ P ∩T 1, then either V (P )∩V 2 = ∅ or V (P )∩V 0 = ∅. In this case, P ⊂ T 1 ∪T j for some j ∈ 0, 2, andP ∩T j consists of a path and an isolated vertex. By induction, |E(P ∩T 1)| ≤αk and E(P ∩ T j)| ≤ βk. Hence, |E(P )| ≤ αk + βk = 3 · 2k−1 + 2k ≤ 3 · 2k.

Similarly, if P ∩ T 2 consists of a path and w, x0, x1 ⊂ P ∩ T 2, then|E(P )| ≤ 3 · 2k.

Thus, we may assume that, for i ∈ 0, 1, 2, P∩T i is not a path containingw, x0, x1, x2−xi. Hence, for i ∈ 0, 1, 2, P∩T i consists of a single vertex,or a vertex and a path, or two isolated vertices. Therefore, by induction,|E(P ∩ T i)| ≤ βk, and so, |E(P )| ≤ 3βk = 3 · 2k. Hence, αk+1 ≤ 3 · 2k.

(b) αk+1 ≥ 3 · 2k.By induction, T 0 − x2 contains an x1 − w path P0 with |E(P0)| = βk,

T 2 − x1 contains a w − x0 path P1 with |E(P1)| = βk, and T 1 − w containsan x0 − x2 path with |E(P2)| = βk. Hence, P = P0 ∪ P1 ∪ P2 is an x1 − x2

path in Tk+1 with length 3βk = 3 · 2k. Hence, αk+1 ≥ 3 · 2k

By (a) and (b), αk+1 = 3 · 2k.

(c) βk+1 ≤ 2k+1.Let Q be an x1 − x2 path in Tk+1 − x0. If V (Q) ∩ V 0 = ∅, then V (Q) ⊂

(T 1 ∪ T 2) − x0. In this case, by induction, |E(Q ∩ T i)| ≤ 2k, and so,|E(Q)| ≤ 2 · 2k = 2k+1.

So assume that Q∩V 0 6= ∅. If Q ⊂ T 0, then |E(Q)| ≤ αk = 3·2k−1 ≤ 2k+1.So assume that Q 6⊂ T 0. Since x0 /∈ Q, either V (Q)∩V 1 = ∅ or V (Q)∩V 2 =∅. By symmetry assume that V (Q) ∩ V 1 = ∅. Then Q ∩ T 2 6= ∅ is a w − x1

path in T 2 − x0, and Q ∩ T 0 consists of a w − x2 path in T 0 − x1 and theisolated vertex x1. Hence, |E(Q)| = |E(Q∩T 0)|+ |E(Q∩T 2)| ≤ 2βk = 2k+1.Thus, βk+1 ≤ 2k+1.

(d) βk+1 ≥ 2k+1.For i ∈ 1, 2, T i − x0 has an xi − w path Pi of length βk = 2k. Hence,

P1 ∪ P2 is an x1 − x2 path in Tk+1 − x0 such that |E(P1 ∪ P2)| = 2k+1. Thus,βk+1 ≥ 2k+1.

By (c) and (d) βk+1 = 2k+1. 2

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Page 7: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

By the definition of Tk+1, we have n = |V (Tk+1)| = 4 + 3 + 32 + · · · 3k =3 + (3k+1 − 1)/2.

Next, we show that circ(Tk+1) ≤ αk + 2βk. Let C be a longest cycle inTk+1, and let T i be defined as in the proof of Proposition 2.1. If C uses anedge of the outer cycle of Tk+1, then |E(C)| ≤ αk+1 +1 ≤ αk +2βk. If C ⊂ T i

for some i, then by induction, |E(C)| ≤ αk−1 +2βk−1 ≤ αk +2βk. So assumethat C 6⊂ T i for any i ∈ 0, 1, 2. Note that E(C ∩ T i) induces a path in T i

between two vertices on the outer cycle of T i, and at most one of these C∩T i

contains all vertices of the outer cycle of T i. Hence, |E(C)| ≤ αk +2βk. Sincen = 3 + (3k+1 − 1)/2 and by a simple calculation, |E(C)| < 7

2nlog3 2).

Therefore, the bound in Conjecture 1.1 is best possible.

3 Planar graphs

In this section, we prove Conjecture 1.1. The following elementary result isneeded.

Lemma 3.1 Let m, n, k be non-negative real numbers. Then

(1) mr + nr ≥ (m + n)r for 0 < r < 1.

(2) mr + nr ≥ (m + n + k)r if 0 ≤ r ≤ log3 2 and k = minm, n, k.

Proof: To prove (1), it is sufficient to show that f(x) = xr + (1 − x)r ≥ 1,where 0 < r < 1 and 0 < x < 1. This can be done by showing that f(x) hasa unique critical point x = 1/2 in [0, 1] and f(1/2) > 1 (and so, the absoluteminimum of f(x) on [0,1] is 1).

Next, we prove (2). Without loss of generality, assume that m ≥ n ≥ k.If m ≥ n ≥ (m+n+k)/3, then mr+nr ≥ (2/3r)(m+n+k)r ≥ (m+n+k)r

(since 2/3r ≥ 1 when 0 ≤ r ≤ log3 2).Hence, we may assume that n = ( 1

3− t)(m + n + k), where 0 < t < 1/3.

Since m ≥ n ≥ k, m ≥ (m + n + k) − 2n = ( 13

+ 2t)(m + n + k). Thus,mr + nr ≥ [(1

3+ 2t)r + (1

3− t)r](m + n + k)r. Hence, it is sufficient to show

that f(t) = ( 13

+ 2t)r + (13− t)r ≥ 1 for 0 ≤ t ≤ 1

3.

Note that f(1/3) = 1. Since 0 ≤ r ≤ log3 2, f(0) ≥ 1. Also, a simplecalculation shows that f(1/6) > 1. Since f ′(t) = 2r(1

3+2t)r−1−r(1

3−t)r−1 =

0 has a unique solution, f(t) ≥ 1 when 0 ≤ t ≤ 1/3. 2

7

Page 8: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

In order to prove Conjecture 1.1, we need to work with a class of 2-connected planar graphs which includes all 3-connected planar graphs.

Definition 3.2 Let (G, C) be a circuit graph, and let x, y ∈ V (C). We saythat (G, xCy) is a strong circuit graph if, for any 2-cut S of G, S ∩V (yCx−x, y) 6= ∅.

We also need to work with graphs in which vertices are assigned non-negative weights. Let R

+ denote the set of non-negative real numbers. Let Gbe a graph, and w : V (G) → R

+. For H ⊂ G, we write w(H) = Σv∈V (H)w(v).Define w(∅) = 0.

Theorem 3.3 Let (G, xCy) be a strong circuit graph, and let w : V (G) →R

+. Then G contains an x − y path P such that

Σv∈V (P−y)[w(v)]log3 2 ≥ [w(G − y)]log3 2.

Remark: In the inequality of Theorem 3.3, y is not included. This is for atechnical reason which facilitates counting. Also note the symmetry betweenx and y (we can always re-embed G so that the clockwise direction of Cbecomes the counter clockwise direction).

Proof: We use induction on |V (G)| + |E(G)|. Since G is 2-connected andsince G contains the cycle C, |V (G)| + |E(G)| ≥ 6. If |V (G)| + |E(G)| = 6,then G is the complete graph on three vertices, and the inequality in Theorem3.3 follows from (1) of Lemma 3.1. So assume that |V (G)|+ |E(G)| > 6. Forconvenience, let r = log3 2, and assume that C is the outer cycle of G. Weshall proceed with claims (a) - (g).

(a) We may assume that xy /∈ E(G).Suppose that xy ∈ E(G). Since (G, xCy) is a strong circuit graph, x, y

is not a cut set of G, and so, xy ∈ E(C). Since G is 2-connected, we can labelthe cut vertices of G−xy as v1, · · · , vm−1 and blocks of G−xy as B1, · · · , Bm

such that Bi∩Bi+1 = vi for i ∈ 1, 2, · · · , m−1, Bi∩Bj = ∅ for |i−j| ≥ 2and i, j ∈ 1, · · · , m, and v0 = x ∈ B1 − v1 and vm = y ∈ Bm − vm−1.See Figure 2. In the left figure xCy = G[xy] and v0, v1, · · · , vm occur onC in counter clockwise order, and in the right figure yCx = G[xy] and

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Page 9: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

m-1

B

y=v

v v

x=v

i

1

0 m

m-1

vv i-1 i

Bv

x=v y=v

v v

i

m 0

1

v i-1

iBB 1 m

BB 1 m

Figure 2:

v0, v1, · · · , vm occur on C in clockwise order. We view each Bi as a planesubgraph of G.

Next, we shall find a vi−1 − vi path Pi in Bi such that⋃m

i=1 Pi gives thedesired path P .

If |V (Bi)| = 2, then let Pi = Bi. It is easy to see that

Σv∈V (Pi−vi)[w(v)]r ≥ [w(Bi − vi)]r.

So assume that |V (Bi)| ≥ 3. Then |V (Bi)| + |E(Bi)| ≥ 6. Let Ci denotethe outer cycle of Bi.

We claim that (Bi, vi−1Civi) is a strong circuit graph. Let S be an ar-bitrary 2-cut of Bi. If Bi − S contains a component T with T ∩ Ci = ∅,then T is also a component of G − S with T ∩ C = ∅, and so, (G, C) isnot a circuit graph, a contradiction. Hence, any component of Bi − S con-tains a vertex of Ci, and so, (Bi, Ci) is a circuit graph. Now assume thatS∩(viCivi−1−vi, vi−1) = ∅. Then S ⊂ vi−1Civi. If xCy = G[xy], then G−Scontains component T with T ∩ C = ∅, a contradiction. So yCx = G[xy].Then S is a 2-cut of G such that S ∩ (yCx−x, y) = ∅, and so, (G, xCy) isnot a strong circuit graph, a contradiction. Hence (Bi, vi−1Civi) is a strongcircuit graph.

By induction, Bi contains a vi−1 − vi path Pi such that

Σv∈V (Pi−vi)[w(v)]r ≥ [w(Bi − vi)]r.

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Page 10: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

Now let P =⋃m

i=1 Pi. Then P is an x − y path in G. Moreover,

Σv∈V (P−y)[w(v)]r = Σmi=1(Σv∈V (Pi−vi)[w(v)]r)

≥ Σmi=1[w(Bi − vi)]

r

≥ [w(G − y)]r.

The first inequality follows from the previous inequalities about Pi. SinceV (G − y) is the disjoint union of V (Bi − vi) for i ∈ 1, · · · , m, the secondinequality follows from (1) of Lemma 3.1. This completes the proof of (a).

Since (G, xCy) is a strong circuit graph, x is not a cut vertex of G − y.Hence, G−y contains a unique block, say H, containing x. Since xy /∈ E(G)and since H is a block of G − y, |V (H)| + |E(H)| ≥ 6. Let D denote theouter cycle of H.

Let y1, y2, · · · , ym ∈ V (D) denote the attachments of (H ∪ y)-bridgesof G, and assume that x, y1, · · · , ym occur on D in this clockwise order. SeeFigure 3.

H

y

yy

y

x

1

2

m

Figure 3:

(b) For i ∈ 1, · · · , m, there are exactly two y, yi-bridges of G; and fori ∈ 1, · · · , m− 1, yyi ∈ E(G) and the y, yi-bridge of G not containing xis G[yyi].

Since (G, xCy) is a strong circuit graph, for any 2-cut S of G, S∩ (yCx−x, y) 6= ∅. Hence, for i ∈ 1, · · · , m − 1, y, yi is not a cut set of G, and

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Page 11: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

so, G has exactly two y, yi-bridges, yyi ∈ E(G), and the y, yi-bridge ofG not containing x is G[yyi]. If G has at least three y, ym-bridges, thenG−y, ym contains a component T with T ∩C = ∅, a contradiction. Thus,(b) follows.

Let X = yCx−yCym. Let x′ = x if X = x; otherwise, let x′ be the endvertex of X other than x. Let B denote the minimal subgraph of H−y1Dym

such that X ⊂ B and B is a union of blocks of H − y1Dym.Let w1, · · · , wn ∈ V (B) be the attachments of (B ∪ y1Dym)-bridges of

H. For k ∈ 1, · · · , n, define sk, tk ∈ V (y1Dym) as follows: y1, sk, tk, ym

occur on D in this clockwise order, sk, wk and tk, wk are contained in(B∪y1Dym)-bridges of H, and subject to these conditions, skDtk is maximal.

We can choose the notation of w1, · · · , wn so that y1, s1, t1, s2, t2, · · · , sk, tk, sk+1, tk+1,· · · , sn, tn, ym occur on D in this clockwise order. Then y1 = s1, ym = tn,and wn = x′. See Figure 4.

y

yy

x’

x

w =w

= ts =

ts

w

B

1 1 m n

n1

M

k k

k

Figure 4:

(c) There is some k ∈ 1, · · · , n such that B contains an x − wk pathPB and

Σv∈V (PB)[w(v)]r ≥ [w(B)]r.

Let Y = G − B and B∗ = G/Y , and let y∗ denote the unique vertex inB∗ − B.

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Page 12: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

We may assume that B 6= x. Otherwise, V (B∗) = x, y∗, E(B∗) =xy∗, n = 1 and w1 = x = x′. Let PB = x. Clearly, PB is an x−w1 pathin B such that

Σv∈V (PB)[w(v)]r ≥ [w(B)]r.

Hence, |V (B∗)|+|E(B∗)| ≥ 6. Let C∗ denote the outer cycle of B∗, whereE(C∗) = E(wnDw1) ∪ w1y

∗, y∗wn.We claim that (B∗, xC∗y∗) is a strong circuit graph. Suppose that S

is an arbitrary 2-cut of B∗. By the construction of B∗, if y∗ ∈ S, thenS − y∗ ⊂ X − x ⊂ y∗C∗x − y∗, x and the vertex in S − y∗ is a cut vertexof B, and so, each component of B∗ − S contains a vertex of C∗. Hence, wemay assume that y∗ /∈ S. If T is a component of B∗ − S, then T ∩ C∗ 6= ∅;otherwise, T is a component of G−S with T ∩C = ∅, a contradiction. Thus,since y∗ /∈ S, S ∩ V (y∗C∗x − y∗, x) 6= ∅. Hence (B∗, xC∗y∗) is a strongcircuit graph.

Let w∗ : V (B∗) → R+ be defined as follows: w∗(v) = w(v) if v ∈ V (B),

and w∗(y∗) is an arbitrary non-negative integer. By induction, B∗ containsan x − y∗ path P ∗ such that

Σv∈V (P ∗−y∗)[w∗(v)]r ≥ [w∗(B∗ − y∗)]r.

Let PB = P ∗ − y∗. Then PB is a path in B from x to wk for some k ∈1, · · · , n. Since B∗ − y∗ = B, and by the definition of w∗, we have

Σv∈V (PB)[w(v)]r ≥ [w(B)]r.

This completes (c).

Let M be the union of skDtk and those (B ∪ y1Dym)-bridges of H whoseattachments are contained in skDtk ∪ wk, where k is given in (c). SeeFigure 4. Let L = ∅ if sk = y1; otherwise, let L denote the component ofH − (B ∪ M) containing y1. Let R = G − (B ∪ L ∪ M ∪ y) (possiblyR = ∅). Note that V (G− y) is the disjoint union of V (B), V (L), V (R), andV (M − wk).

(d) We may assume that w(B) > minw(L), w(R).Suppose that w(B) ≤ minw(L), w(R). Let L∗ = G[V (L ∪ M ∪ B)]/B,

and let x∗ denote the unique vertex of L∗− (L∪M). Thus x∗ is the result ofthe contraction of B. Let R∗ = G[V (R) ∪ y, tk]. We shall find an x∗ − tk

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path PL in L∗ and a tk − y path PR in R∗, and use PL and PR to constructthe desired path P .

(d1) L∗ contains an x∗ − tk path PL such that

Σv∈V (PL−x∗,tk)[w(v)]r ≥ [w((L ∪ M) − wk, tk)]r.

First, assume that tk = y1. Since (G, xCy) is a strong circuit graph,L = ∅, tkwk ∈ E(G), M = G[tkwk], V (L∗) = x∗, tk, and E(L∗) = x∗tk.Let PL = L∗. Clearly,

Σv∈V (PL−x∗,tk)[w(v)]r = [w(∅)]r ≥ [w((L ∪ M) − wk, tk)]r.

So assume that tk 6= y1. Then L∗ is a 2-connected plane graph with atleast three vertices. Hence, |V (L∗)| + |E(L∗)| ≥ 6. Let CL denote the outercycle of L∗ such that w1Dtk is the clockwise segment of CL from x∗ = w1 totk.

Next, we show that (L∗, tkCLx∗) is a strong circuit graph. Let S be anarbitrary 2-cut of L∗. First, assume that L∗ − S has a component T withT ∩ CL = ∅. Then x∗ ∈ S; otherwise, T is also a component of G − S withT ∩ C = ∅, a contradiction. Thus, by the construction of L∗, there is somewi, where 1 ≤ i ≤ k, such that S ′ = (S − x∗) ∪ wi is a 2-cut of G,and T is a component of G − S ′ with T ∩ C = ∅, a contradiction. Hence,(L∗, CL) is a circuit graph. Now, assume that S ∩ (x∗CLtk − x∗, tk) = ∅.Then S ⊂ tkCLx∗. Thus, G−S contains a component U with U ∩ (tkCLx∗−tk, x∗) 6= ∅. Hence, wk 6= x′; otherwise, tk = ym and tkCLx∗ − tk, x∗ = ∅,a contradiction. Let S ′ = S if x∗ /∈ S; otherwise, let S ′ = (S −x∗)∪ wk.Thus, U is a component of G − S ′ with U ∩ C = ∅, a contradiction. Hence,(L∗, tkCLx∗) is a strong circuit graph.

Let w∗ : V (L∗) → R+ be defined as follows: w∗(v) = w(v) for v ∈ L∗−x∗,

and w∗(x∗) = 0. By induction, L∗ contains an x∗ − tk path PL such that

Σv∈V (PL−tk)[w∗(v)]r ≥ [w∗(L∗ − tk)]

r.

By the definition of w∗, we have

Σv∈V (PL−x∗,tk)[w(v)]r ≥ [w((L ∪ M) − wk, tk)]r.

This proves (d1).

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(d2) R∗ contains a tk − y path PR such that

Σv∈V (PR−y)[w(v)]r ≥ [w(R ∪ tk)]r.

If |V (R∗)| = 2, then tk = ym, tky ∈ E(G), and the y, ym-bridge of Gnot containing x is G[tky]. In this case, R = ∅. Let PR = R∗. Clearly,

Σv∈V (PR−y)[w(v)]r ≥ [w(R∗ − y)]r = [w(R ∪ tk)]r.

So assume that |V (R∗)| ≥ 3. Hence, |V (R∗ + tky)| + |E(R∗ + tky)| ≥ 3.Without loss of generality, assume that R∗ + tky is a plane graph with outercycle CR such that tky ∪ E(yCym) ⊂ E(CR), and E(tkCRy) = tky.

Next we show that (R∗ + tky, tkCky) is a strong circuit graph. Supposethat S is an arbitrary 2-cut of R∗ + tky. If (R∗ + tky) − S has a componentT with T ∩ CR = ∅, then T is a component of G − S with T ∩ C = ∅, acontradiction. Hence, (R∗ + tky, CR) is a circuit graph, and so, S ⊂ CR.Since E(tkCRy) = tky, S ∩ (yCRtk − y, tk) 6= ∅. Thus, (R∗ + tky, tkCky)is a strong circuit graph.

By induction, R∗ contains atk − y path PR such that

Σv∈V (PR−y)[w(v)]r ≥ [w(R∗ − y)]r = [w(R ∪ tk)]r.

Note that we can always select PR so that E(PR) 6= tky. Hence, PR is apath in R∗. This proves (d2).

Finally, we find the desired path P as follows. Without loss of generality,assume that the edge of PL incident with x∗ is x∗v, and v is incident withwl, where 1 ≤ l ≤ k. Let Q be a path in B from x to wl, and let P =((PL − x∗) ∪ Q ∪ PR) + vwl. Then P is an x − y path in G such that

Σv∈V (P−y)[w(v)]r ≥ Σv∈V (PL−x∗,tk)[w(v)]r + Σv∈V (PR−y)[w(v)]r

≥ [w((L ∪ M) − wk, tk)]r + [w(R ∪ tk)]r≥ [w(G − y)]r.

Here, the first inequality is obvious, and the second inequality follows from(d1) and (d2). By the assumption that w(B) ≤ minw(L), w(R) and sinceV (G−y) is the disjoint union of V ((L∪M)−wk, tk), V (B), and V (R∪tk),the third inequality follows from (2) of Lemma 3.1. This completes the proofof (d).

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(e) We may assume that w(L) < w(R).Otherwise, assume that w(L) ≥ w(R). Let L∗ = G[V (L) ∪ y, sk]. We

shall extend PB in (c) to the desired path P by finding an sk − y path PL inL∗ and a wk − sk path PM in M .

(e1) L∗ contains an sk − y path PL such that

Σv∈V (PL−sk,y)[w(v)]r ≥ [w(L)]r.

If y1 = sk, then L = ∅ and L∗ = G[yy1]. In this case, let PL = L∗.Clearly,

Σv∈V (PL−sk,y)[w(v)]r = [w(L∗ − sk, y)]r = [w(∅)]r = [w(L)]r.

So assume that y1 6= sk. Then L∗ + ysk is a 2-connected graph with atleast three vertices. Hence, |V (L∗ + ysk)|+ |E(L∗ + ysk)| ≥ 6. Without lossof generality, we may assume that L∗ + ysk is embedded in the plane withouter cycle CL such that y1CLsk = y1ysk (and hence, E(yCLsk) = ysk).

Next we show that (L∗ + ysk, yCLsk) is a strong circuit graph. Let Sbe an arbitrary 2-cut of L∗ + ysk. If (L∗ + ysk) − S has a component Twith T ∩ CL = ∅, then T is a component of G − S with T ∩ C = ∅, acontradiction. Hence, (L∗ + ysk, CL) is a circuit graph, and so, S ⊂ CL.Since E(yCLsk) = ysk, (L∗ + ysk, yCLsk) is a strong circuit graph.

Let w∗ : V (L∗ + ysk) → R+ be defined as follows: w∗(v) = w(v) for

v ∈ L∗ − sk, and w∗(sk) = 0. By induction and by the definition of w∗,L∗ + ysk contains an sk − y path PL such that

Σv∈V (PL−sk,y)[w(v)]r = Σv∈V (PL−y)[w∗(v)]r

≥ [w∗(L∗ − y)]r

= [w(L)]r.

Note that we can always select PL so that E(PL) 6= ysk. Hence, PL is apath in L∗. This proves (e1).

(e2) M contains a path PM from wk to sk such that

Σv∈V (PM−wk)[w(v)]r ≥ [w(M − wk)]r.

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If sk = tk, then wksk ∈ E(G) and M = G[wksk]. In this case, let PM = M .Clearly,

Σv∈V (PM−wk)[w(v)]r ≥ [w(M − wk)]r.

So assume that sk 6= tk. Then M + wksk is a 2-connected graph with atleast three vertices. Hence, |V (M + wksk)| + |E(M + wksk)| ≥ 6. Withoutloss of generality, assume that M +wksk is embedded in the plane with outercycle CM such that E(skDtk) ∪ wksk ⊂ CM , and E(wkCMsk) = wksk.

Next we show that (M + wksk, wkCMsk) is a strong circuit graph. LetS be an arbitrary 2-cut of M + wksk. If (M + tksk) − S has a componentT with T ∩ CM = ∅, then T is a component of G − S with T ∩ C = ∅, acontradiction. Thus, (M + wksk, CM) is a circuit graph, and so, S ⊂ CM .Since E(wkCMsk) = wksk, (M + wksk, wkCMsk) is a strong circuit graph.

By induction, M + wksk contains a wk − sk path PM such that

Σv∈V (PM−wk)[w(v)]r ≥ [w(M − wk)]r.

Note that we can always select PM so that E(PM) 6= wksk. Hence, PM isa path in M . This proves (e2).

Now let P = PB ∪ PM ∪ PL. Then P is an x − y path in G. Moreover,

Σv∈V (P−y)[w(v)]r = Σv∈V (PB)[w(v)]r + Σv∈V (PM−wk)[w(v)]r + Σv∈V (PL−sk,y)[w(v)]r

≥ [w(B)]r + [w(M − wk)]r + [w(L)]r

≥ [w(B ∪ L ∪ R)]r + [w(M − wk)]r

≥ [w(G − y)]r.

Here, the first inequality follows from (c), (e1) and (e2). By (d) and theassumption that w(L) ≥ w(R), the second inequality follows from (2) ofLemma 3.1. The third inequality follows from (1) of Lemma 3.1. This proves(e).

By (e), R 6= ∅, and so tk 6= ym. We shall extend PB in (c) to the desiredpath P by finding a tk − y path PR and a wk − tk path PM . The argumentis similar to that for (e). Since it is not too long and is not completelysymmetric to (e), we provide the details. Let R∗ = G[V (R) ∪ y, tk].

(f) R∗ contains a tk − y path PR such that

Σv∈V (PR−tk,y)[w(v)]r ≥ [w(R)]r.

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If |V (R∗)| = 2, then R = ∅, tk = ym, yym ∈ E(G), and the y, ym-bridgeof G not containing x is G[yym]. In this case, let PR = R∗. Clearly,

Σv∈V (PR−tk,y)[w(v)]r = [w(∅)]r = [w(R)]r.

So assume that |V (R∗)| ≥ 3. Then, |V (R∗ + tky)| + |E(R∗ + tky)| ≥ 6.Without loss of generality, assume that R∗+tky is embedded in the plane withouter cycle CR such that tky ∪E(yCym) ⊂ E(CR) and E(tkCRy) = tky.

Now we show that (R∗ + tky, tkCRy) is a strong circuit graph. Let Sbe an arbitrary 2-cut of R∗ + tky. If (R∗ + tky) − S contains a componentT with T ∩ CR = ∅, then T is a component of G − S with T ∩ C = ∅, acontradiction. Hence, (R∗ + tky, CR) is a circuit graph, and so, S ⊂ CR.Since E(tkCRy) = tky, (R∗ + tky, tkCRy) is a strong circuit graph.

Let w∗ : V (R∗) → R+ be defined as follows: w∗(v) = w(v) for v ∈

V (R∗ − tk), and w∗(tk) = 0. By induction and by the definition of w∗,R∗ + tky contains a tk − y path PR such that

Σv∈V (PR−tk,y)[w(v)]r = Σv∈V (PR−y)[w∗(v)]r

≥ [w∗(R∗ − y)]r

= [w(R)]r.

Note that we can always select PR so that E(PR) 6= tky. Hence, PR ⊂ R∗.This completes (f).

(g) M contains a wk − tk path PM such that

Σv∈V (PM−wk)[w(v)]r ≥ [w(M − wk)]r.

If sk = tk, then tkwk ∈ E(G) and M = G[tkwk]. In this case, let PM = M .Clearly,

Σv∈V (PM−wk)[w(v)]r ≥ [w(M − wk)]r.

So assume that sk 6= tk. Then M + tkwk is a 2-connected graph with atleast three vertices. Hence, |V (M + tkwk)| + |E(M + tkwk)| ≥ 6. Withoutloss of generality, assume that M + tkwk is embedded in the plane with outercycle CM such that E(skDtk)∪ tkwk ⊂ E(CM) and E(tkCMwk) = tkwk.

Next, we show that (M + tkwk, tkCMwk) is a strong circuit graph. LetS be an arbitrary 2-cut of M + tkwk. If (M + tkwk) − S has a componentT with T ∩ CM = ∅, then T is a component of G − S with T ∩ C = ∅,

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a contradiction. Recall that, by (e), tk 6= ym, and so, wk 6= x′. Hence,(M + tkwk, CM) is a circuit graph, and so, S ⊂ CM . Since E(tkCMwk) =tkwk, (M + tkwk, tkCMwk) is a strong circuit graph.

By induction, M + tkwk contains a tk − wk path PM such that

Σv∈V (PM−wk)[w(v)]r ≥ [w(M − wk)]r.

Note that we can always select PM so that E(PM) 6= tkwk. Hence, PM ⊂ M .This proves (g).

Now let P = PB∪PM ∪PR. Then P is a path in G from x to y. Moreover,

Σv∈V (P−y)[w(v)]r = Σv∈V (PB)[w(v)]r + Σv∈V (PM−wk)[w(v)]r + Σv∈V (PR−tk ,y)[w(v)]r

≥ [w(B)]r + [w(R)]r + [w(M − wk)]r

≥ [w(B ∪ L ∪ R)]r + [w(M − wk)]r

≥ [w(G − y)]r.

Here, the first inequality follows from (c), (f) and (g). By (d) and by (e),the second inequality follows from (2) of Lemma 3.1. The third inequalityfollows from (1) of Lemma 3.1. 2

Corollary 3.4 If (G, C) is a circuit graph and e ∈ E(C). Then G containsa cycle T through e such that |E(T ) ≥ |V (G)|log3 2.

Proof: Let e = xy such that xCy = G[xy]. Then (G, xCy) is a strong circuitgraph. Let w : V (G) → R

+ such that w(v) = 1 for all v ∈ V (G).By Theorem 3.3, G contains an x − y path P such that

|V (P )| − 1 = Σv∈V (P−y)[w(v)]log3 2

≥ [w(G − y)]log3 2

= (|V (G)| − 1)log3 2.

Hence, by (1) of Lemma 3.1,

|V (P )| ≥ 1 + (|V (G)| − 1)log3 2 ≥ |V (G)|log3 2.

Thus P + xy gives the desired cycle. 2

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Corollary 3.5 Let G be a 3-connected planar graph, and let e ∈ E(G). ThenG contains a cycle C through e such that |E(C)| ≥ |V (G)|log3 2.

Proof: Without loss of generality, we assume that G is embedded in theplane such that e ∈ E(C), where C is a facial cycle of G. Then (G, C) is acircuit graph. Hence, Corollary 3.5 follows from Corollary 3.4. 2

4 Graphs on other surfaces

In this section, we prove that Conjecture 1.1 also holds for 3-connected graphsembedded in the projective plane, or the torus, or the Klein bottle.

Definition 4.1 Given an embedding σ : G → Π of a graph G into a surfaceΠ, the representativity of σ is defined to be the number min|σ(G)∩Γ| : Γ isa non-null homotopic simple closed curve in Π.

For graphs embedded in the projective plane, Fiedler et al proved thefollowing result ([4], Proposition 1). Note that in [4], a cycle in G is called apolygon in G.

Lemma 4.2 Let σ : G → Π be an embedding of a 3-connected graph Ginto the projective plane Π, let P1 be a cycle in G such that σ(P1) is nullhomotopic, and let D1 be the open disc in Π bounded by σ(P1). Then thereis a cycle P in G such that σ(P ) is null homotopic and bounds a disc Dcontaining D1, and the closure of D contains σ(V (G)).

Note that the condition about P1 and D1 in Lemma 4.2 holds if therepresentativity of σ is at least 2. Also note that the subgraph H of Gcontained in the closure of D is a 2-connected spanning subgraph of G, andσ(H) can be viewed as a plane graph embedded in the closed disc boundedby D.

Lemma 4.3 Let G be a 3-connected graph embeddable in the projective plane.Then G contains a 2-connected spanning planar subgraph H. Moreover, Hcan be embedded in the plane with a facial cycle C such that (H, C) is acircuit graph.

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Proof: Let σ : G → Π be an embedding of G in the projective plane Π. If therepresentativity of σ is at least 2, then by the remarks following Lemma 4.2,G has a cycle C, such that the subgraph H of G contained in the closed discin Π bounded by σ(C) is a spanning subgraph of G. Since G is 3-connected,it is easy to see that (H, C) is a circuit graph.

So assume that the representativity of σ is at most 1. Then G is a 3-connected planar graph. Let H be a plane embedding of G and let C befacial cycle of H. Then (H, C) is a circuit graph. 2

Now we are ready to prove Conjecture 1.1 for 3-connected graphs embed-dable in the projective plane.

Theorem 4.4 Let G be a 3-connected graph embeddable in the projectiveplane. Then circ(G) ≥ |V (G)|log3 2.

Proof: By Lemma 4.3, let H be a 2-connected plane graph with a facialcycle C such that H is a spanning subgraph of G and (H, C) is a circuitgraph. By Corollary 3.4, H, and hence G, has a cycle of length at least|V (H)|log3 2 = |V (G)log3 2. 2

For graphs embeddable in the torus or the Klein bottle, we need thefollowing result ([2], Theorems 2 and 3).

Lemma 4.5 Let σ : G → Π be an embedding of a 3-connected graph Gin a surface Π, where Π is the torus or the Klein bottle. Suppose σ hasrepresentativity at least 1. Then there is a spanning subgraph H of G andeither (1) there is a cycle C of H such that (H, C) is a circuit graph or (2)there are two cycles C1 and C2 of H such that (H, C1, C2) is an annulusgraph.

Theorem 4.6 Let G be a 3-connected graph embeddable in the torus or theKlein bottle. Then circ(G) ≥ (|V (G)|/2)log3 2.

Proof: Let σ : G → Π be an embedding of G into a surface Π, where Πis the torus or the Klein bottle. If the representativity of σ is 0, then G isembeddable in the projective plane, and Theorem 4.6 follows from Theorem4.4. So we may assume that the representativity of σ is at least 1. Then byLemma 4.5, there is a spanning subgraph H of G and either (1) there is a

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cycle C of H such that (H, C) is a circuit graph or (2) there are cycles Cand D of H such that (H, C, D) is an annulus graph.

If there is a cycle C of H such that (H, C) is a circuit graph, then by Corol-lary 3.4, H, and hence, G, contains a cycle of length at least (|V (G)|/2)log3 2.

So assume that there are cycles C and D of H such that (H, C, D) isan annulus graph. Without loss of generality, we may assume that C is theouter cycle of H. We consider two cases: C ∩ D = ∅ and C ∩ D 6= ∅.

Case 1. C ∩ D = ∅.Since H is 2-connected, H contains vertex disjoint paths P1, P2 from

v1, v2 ∈ V (C) to u1, u2 ∈ V (D), respectively, such that, for i ∈ 1, 2,(Pi − ui, vi) ∩ (C ∪ D) = ∅. Let C1 = v1Cv2 ∪ P2 ∪ u1Du2 ∪ P1, and letC2 = v2Cv1 ∪ P1 ∪ u2Du1 ∪ P2. For i ∈ 1, 2, let Hi be the subgraph ofH contained in the closed disc in the plane bounded by Ci. Then (Hi, Ci),i ∈ 1, 2, are circuit graphs. See Figure 5.

H

H

v vu u

D

P P

2

1

1 1 1 2 2 2

C

Figure 5:

Since |V (H1)| + |V (H2)| > |V (H)|, we may assume, without loss of gen-erality, that |V (H1)| > |V (H)|/2. Since (H1, C1) is a circuit graph, by Corol-lary 3.4, circ(H1) ≥ |V (H1)|log3 2. Hence, circ(G) ≥ (|V (G)|/2)log3 2.

Case 2. C ∩ D 6= ∅.

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Assume that u ∈ V (C ∩ D). Let u1, · · · , un be the neighbors of u inclockwise order around u such that u1, un ∈ V (C) and uk, uk+1 ∈ V (D),where 1 ≤ k < n. Let H ′ be the plane graph obtained from H −u by addingtwo vertices u′ and u′′, and adding edges u′ui for i ∈ 1, · · · , k and edgesu′′ui for i ∈ k + 1, · · · , n. See Figure 6.

Since H is 2-connected, we can label the cut vertices of H ′ as v1, · · · , vm−1

and blocks of H ′ as B1, · · · , Bm such that u, v1, · · · , vm−1 occur on C in thisclockwise order, Bi ∩ Bi+1 = vi for i ∈ 1, · · · , m − 1, Bi ∩ Bj = ∅ fori, j ∈ 1, · · · , m with |i − j| ≥ 2, and v0 = u′′ ∈ B1 − v1 and vm = u′ ∈Bm − vm−1.

m 0

u’=vu’’=v

uu u

u u

1

k k+1

n

B

v v

u

u

u

u

k+1

n

i

i-1 i

k

1

Figure 6:

Let WC ⊂ V (H) (respectively, WD ⊂ V (H)) be defined as follows: Forv ∈ V (H), v ∈ WC (respectively, v ∈ WD) if, and only if, H has a 2-cut Sv ⊂V (C) (respectively, Sv ⊂ V (D)) such that v and D−Sv (respectively, C−Sv)are contained in different components of H − Sv. Note that WC ∩ WD = ∅and u /∈ WC ∪ WD.

We may assume that WC 6= ∅ and WD 6= ∅. Otherwise, by symmetryassume that WD = ∅. Then (H, C) is a circuit graph. Hence, by Corollary3.4, circ(H) ≥ |V (H)|log3 2, and so, circ(G) ≥ (|V (G)|/2)log3 2.

Without loss of generality, assume that |WD| ≤ |WC |. Then, |WD| <|V (H)|/2 (since u /∈ WC ∪ WD).

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Next, for i ∈ 1, · · · , m, we find a vi−1 − vi path Pi in Bi.If |V (Bi)| = 2, then let Pi = Bi.So assume that |V (Bi)| ≥ 3. Then let Ci denote the outer cycle of Bi.

Note that vi−1Civi = viDvi−1 and viCivi−1 = viCvi−1 (where we view v0 andvm as u). We construct a new graph B∗

i from Bi as follows.If WD ∩ Bi = ∅, then let B∗

i = Bi and let C∗i = Ci.

If WD∩Bi 6= ∅, then WD∩V (Bi) can be partitioned into sets W i1, · · · , W i

mi

with the following properties: (1) for j ∈ 1, · · · , mi, Bi has a 2-cut Sij =

sij, t

ij ⊂ V (D ∩ Bi) and G[W i

j ] is the component of Bi − Sij not containing

C ∩ Bi, and (2) Bi has no 2-cut S ⊂ V (D ∩ Bi) such that the componentof Bi − S not containing C ∩ Bi properly contains G[W i

j ]. Let B∗i denote

the graph obtained from Bi by deleting G[W ij ] and adding the edges si

jtij for

j ∈ 1, · · · , mi. Let C∗i be obtained from Ci by deleting W i

j ∩D and addingedges si

jtij. Let the edges si

jtij be added so that C∗

i is the outer cycle of B∗i

and viC∗i vi−1 = viCivi−1.

By the above construction, (B∗i , viC

∗i vi−1) is a strong circuit graph. Let

w : V (B∗i ) → R

+ with w(v) = 1 for v ∈ V (B∗i ). By Theorem 3.3, B∗

i containsa vi−1 − vi path P ∗

i such that

|V (P ∗i )| − 1 = Σv∈V (P ∗

i−vi)[w(v)]log3 2

≥ [w(B∗i − vi)]

log3 2

= (|V (B∗i )| − 1)log3 2.

If sijt

ij ∈ E(P ∗

i ), then we replace sijt

ij in P ∗

i by a path in G[W ij ∪ si

j, tij]

from sij to tij, and let Pi denote the resulting path. Then |V (Pi)| − 1 ≥

|V (P ∗i )| − 1 ≥ (|V (B∗

i )| − 1)log3 2.Now let P =

⋃mi=1 Pi. Then

|V (P )| − 1 = Σmi=1(|V (Pi)| − 1)

≥ Σmi=1(|V (B∗

i )| − 1)log3 2

≥ [|V (H) − WD|]log3 2

≥ (|V (G)|/2)log3 2.

Here, the first inequality follows from previous inequalities. Since V (H)−WD

is the disjoint union of V (B∗i − vi) for i = 1, · · · , m, the second inequality

follows from (1) of Lemma 3.1. The third inequality follows from the factthat |WD| < |V (G)|/2.

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Page 24: Long Cycles in 3-Connected Graphspeople.math.gatech.edu/~yu/Papers/longcycle.pdfT x x x w T 1 0 2 1 x 0 x 1 x 2 2 w Figure 1: By the above construction, for any k 1, Tk is a 3-connected

Hence, |V (P )| ≥ (|V (G)|/2)log3 2 + 1. Therefore, identifying v′ and v′′ inP gives the desired cycle. 2

It is easy to see that our main result, Theorem 1.2, follows from Corollary3.5, Theorem 4.4 and Theorem 4.6.

Acknowledgment

The authors would like to thank Ron Gould and Robin Thomas for helpfuldiscussions.

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References

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