K a,k minors in graphs of bounded tree-width *

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1 K a,k minors in graphs of bounded tree-width * Thomas Böhme Institut für Mathematik Technische Universität Imenau Imenau, Germany E-mai: tboehme@theoinf.tu-imenau.de and John Maharry Department of Mathematics The Ohio State University Coumbus, Ohio E-mai: maharry@math.ohio-state.edu and Bojan Mohar Department of Mathematics University of Ljubjana Ljubjana, Sovenia E-mai: bojan.mohar@uni-j.si It is shown that for any positive integers k and w there exists a constant N = N(k, w) such that every 7-connected graph of tree-width ess than w and of order at east N contains K 3,k as a minor. Simiar resut is proved for K a,k minors where a is an arbitrary fixed integer and the required connectivity depends ony on a. These are the first resuts of this type where fixed connectivity forces arbitrariy arge (nontrivia) minors. Key Words: Graph theory, graph minor, connectivity. * This paper appeared as T. Böhme, J. Maharry, B. Mohar, K a,k minors in graphs of bounded tree-width, J. Combin. Theory, Ser. B 86 (2002) Supported in part part by the Internationa Research Project SLO-US-007 on Graph Minors. Supported in part by the Ministry of Science and Technoogy of Sovenia, Research Project J , and the Internationa Research Project SLO-US-007 on Graph Minors. 1

2 2 T. BÖHME, J. MAHARRY, AND B. MOHAR 1. INTRODUCTION In this paper, a graphs are finite and may have oops and mutipe edges. A graph H is a minor of a graph G, H m G, if H can be obtained from a subgraph of G by contracting connected subgraphs. There are many resuts concerning the structure of graphs that do not contain a certain graph as a minor. These excuded graphs incude K 5 and K 3,3 [13], V 8 [8], the 3-cube [6] and the octahedron [7]. See aso [2] and [12]. There are we-known structures which guarantee a certain minor exists for arge graphs. For instance, any 5-connected graph on at east 11 vertices contains the 3-cube as a minor [6]. Any 5-connected non-panar graph on at east 8 vertices contains a V 8 minor [8]. In addition, there are Ramsey-type resuts simiar to the fact that any sufficienty arge connected graph contains either a k-path or a k-star. Oporowski, Oxey and Thomas [11] proved that any arge 4-connected graph must have a arge minor from a set of four famiies of graphs. Ding [3] has characterized arge graphs that do not contain a K 2,k minor. A coroary of his resut is that any arge 5-connected graph contains a K 2,k minor. Our resuts are a cross section of a of these types of resuts: Theorem 1.1. For any positive integers k and w there exists a constant N = N(k, w) such that every 7-connected graph of tree-width at most w and of order at east N contains K 3,k as a minor. Theorem 1.2. There is a function c : N N such that for any a 3 the foowing hods. For any positive integers k and w there exists a constant N = N(k, w) such that every c(a)-connected graph of tree-width at most w and of order at east N contains K a,k as a minor. Theorem 1.1 is sharp in the sense that the 7-connectivity condition cannot be reaxed. Moreover, the function c(a) in Theorem 1.2 must be at east 2a + 1. These facts foow from the foowing construction of a famiy of arbitrariy arge 2a-connected graphs (of tree-width 3a 1) none of which contain a K a,2a+1 -minor. Let m and a be integers greater than 3. Define the graph N m,a as foows. Let the vertices be indexed v x,y where 1 x m and 1 y a. The vertex v x,y is adjacent to another vertex v w,z if and ony if w {x 1, x, x + 1} where x ± 1 is considered moduo m. Proposition 1.1. For any integers a 3 and m 3, K a,2a+1 m N m,a.

3 MINORS IN GRAPHS OF BOUNDED TREE-WIDTH 3 Proof. Suppose the theorem is fase for some a 3. Let m be the east integer such that N m,a m K a,2a+1. Let the casps of N m,a be defined as CL i = {v i,y y = 1, 2,..., a} for i = 1, 2,..., m. As N m,a m K a,2a+1, there is a set of 2a + 1 connected subgraphs, S = {S 1, S 2,..., S 2a+1 }, and a set of a connected subgraphs of N m,a, T = {T 1, T 2,..., T a }, such that for every i, j there is an edge from some vertex in T i to some vertex in S j and such that a these subgraphs are pairwise disjoint. Assume that the S i and T i are chosen with := 2a+1 i=1 V (S i) + a i=1 V (T i) minimum. Then it is easy to see that each of the subgraphs in S T is a path meeting each casp in at most one vertex. Let S 1 be the set of singe vertex subgraphs contained in S. It is easy to see that T cannot contain any singe vertex subgraphs. Caim 1: For every 1 i m, there is a subgraph S j S 1 such that S j CL i. Suppose CL i does not contain any of the subgraphs in S 1. Then contracting a matching of size a between CL i and CL i 1 CL i+1 (indices taken moduo m) using as many edges of S T as possibe gives a subgraph of N m 1,a that sti contains K a,2a+1 as a minor. This contradiction to the minimaity of m proves the caim. Caim 2: If there is a subgraph in S that contains at east two vertices, then there is a casp that contains no member of S 1. Suppose S 1 (say) intersects CL 1 and CL 2. By the minimaity of, we may assume that S 1 CL m =. Moreover, there is a subgraph T j that does not intersect CL 1 CL 2 CL 3. Otherwise, the intersection of S 1 with CL 1 coud be removed from S 1. Therefore, a singe vertex subgraph S i S 1 contained in CL 2 woud not be adjacent to T j. Hence, the casp CL 2 is as stated in the caim. Caims 1 and 2 impy that a subgraphs in S are singe vertices. To compete the proof, notice that if every casp of N m,a contains one of the singe vertex subgraphs of S 1, then each T j must must contain at east m 2 vertices in order to be adjacent to a of the subgraphs in S. Hence V (S) + V (T ) S +(m 2) T 2a+1+(m 2)a > ma = V (N m,a ). This contradiction competes the proof. In our proof of Theorem 1.2, c(3) = 7 and c(a) = 264a + 1 for a 4, and we have no intention to find the best possibe vaue for c(a). However, the previous exampe shows that c(a) must be at east 2a + 1 for a 3. It is worth remarking that our proof of Theorem 1.2 works aso for c(a) = 3a 1 if we assume that the minimum degree is at east 264a + 1.

4 4 T. BÖHME, J. MAHARRY, AND B. MOHAR 2. BOUNDED TREE-WIDTH STRUCTURE A tree decomposition of a graph G is a pair (T, Y ), where T is a tree and Y is a famiy {Y t t V (T )} of vertex sets Y t V (G), such that the foowing two properties hod: (W1) t V (T ) Y t = V (G), and every edge of G has both ends in some Y t. (W2) If t, t, t V (T ) and t ies on the path in T between t and t, then Y t Y t Y t. The width of a tree decomposition (T, Y ) is max t V (T ) ( Y t 1). It was shown in [11] that if a graph G has a tree decomposition of width at most w then G has a tree decomposition of width at most w that further satisfies: (W3) For every two vertices t, t of T and every positive integer k, either there are k disjoint paths in G between Y t and Y t, or there is a vertex t of T on the path between t and t such that Y t < k. (W4) If t, t are distinct vertices of T, then Y t Y t. (W5) If t 0 V (T ) and B is a component of T t 0, then t V (B) Y t\y t0. In the rest of the paper we give the proof of Theorems 1.1 and 1.2. We et a 3, k, and w be given positive integers. Let G be an c(a)-connected graph with a tree decomposition (T, Y ) of width at most w that satisfies (W1) (W5). We wi deveop a structure that is simiar to that used in [11]. First, we define the constants that wi be used in the proofs. ( ) w + 1 n 5 = r n4, where r = (k 1) a n 4 = n w+1 3 n 3 = (2n 2 ) p, where p = 2 w+1 n 2 = n q 1, where q = 2(w+1 { 2k(2w + 3) 2 if a = 3 n 1 = 2k(c(a) + 2a + 2) 4a 2 if a 4 2 ) We assume that V (G) = N (w +1)n 5 and that G has no K a,k -minor. By (W1) we have Caim 2.1. V (T ) n 5.

5 MINORS IN GRAPHS OF BOUNDED TREE-WIDTH 5 Caim 2.2. Every vertex of T has degree at most r = (k 1) ( ) w+1 a. Proof. Suppose t 0 V (T ) has degree at east r + 1. Let C be the set of components of G Y t0. By (W2) and (W5), it is cear that C r + 1. For C C, et X(C) be the set of vertices of Y t0 adjacent to some vertex of C. Ceary, X(C) a for every C C since G is c(a)-connected and c(a) a. By the Pigeonhoe Principe, there is a set C C of k components for which C C X(C) contains a (or more) vertices of Y t 0. By contracting B to a vertex for each B C, we see that G contains a K a,k minor, a contradiction. From this it foows that Caim 2.3. T contains a path R of ength E(R) n 4. The proof of the foowing caim can be found in [11]. Caim 2.4. There is a subsequence of ength n 3 of the vertices of V (R), r 1, r 2,..., r n3, such that for some s 1, Y ri = s for i = 1, 2,..., n 3 and for every vertex of R between r 1 and r n3, Y ri s. From now on we repace R by the subpath from r 1 to r n3. Note that because of the c(a)-connectivity and (W5), c(a) s w + 1. By (W3) and Caim 2.4, there are s disjoint paths in G from Y r1 to Y rn3. Fix these paths, denote them by P 1, P 2,..., P s, and put Z = P 1 P s. Since G is 3-connected, these paths can be chosen such that every Z-bridge in G is attached to at east two of the paths (cf., e.g., [4]), which we assume henceforth. Notice that for any t, t {r 1,..., r n3 } and for every j {1,..., s} there is a unique subpath of P j with one end in Y t and the other end in Y t. Denote this subpath by P j (t, t ). The path P j is said to be trivia if it consists of a singe vertex, and it is said to be everywhere nontrivia (amost nontrivia) w.r.t. the sequence r 1,..., r n3 if P j (r i, r i+1 ) contains at east three (respectivey, at east two) vertices for each i = 1,..., n 3 1. Caim 2.5. There is a subsequence q 1, q 2,..., q n2 of r 1,..., r n3 of ength n 2 such that for each j = 1,..., s, P j (q 1, q n2 ) is either trivia or everywhere nontrivia (w.r.t. the subsequence). Proof. Ceary, there is a subsequence of r 1,..., r n3 of ength n 3 such that the corresponding segment of P 1 is either trivia or everywhere amost nontrivia with respect to the subsequence. By repeating this argument

6 6 T. BÖHME, J. MAHARRY, AND B. MOHAR on the subsequence for P 2,..., P s, respectivey, we end up with a sequence of ength at east 2n 2 such that every path is either trivia or everywhere amost nontrivia. By taking every second eement of this sequence, the required subsequence q 1, q 2,..., q n2 is obtained. The paths P j and P are said to be everywhere bridge connected (resp. everywhere bridge disconnected) with respect to a sequence p 1,..., p n of vertices of R if for every i = 1,..., n 1, there exists (resp. does not exist) a Z-bridge which has a vertex of attachment in P j (p i, p i+1 ) and a vertex of attachment in P (p i, p i+1 ). Caim 2.6. There is a subsequence p 1, p 2,..., p n1 of q 1,..., q n2 of ength n 1 such that for every distinct pair of indices j, {1,..., s}, P j (p 1, p n1 ) and P (p 1, p n1 ) are either everywhere bridge connected or everywhere bridge disconnected (w.r.t. the new subsequence). Proof. The proof is simiar to the proof of Caim 2.5 except that we have to repeat the subsequence argument ( ) ( s 2 w+1 ) 2 times. 3. THE AUXILIARY GRAPH A Our next goa is to examine the structure of the auxiiary graph A which contains information about which pairs of the paths are everywhere bridge connected. The graph A has vertex set V (A) = {P 1,..., P s }, and the paths P j and P are adjacent vertices in A if they are everywhere bridge connected w.r.t. p 1,..., p n1 (cf. Caim 2.6). Caim 3.1. Suppose that U V (A) contains ony everywhere nontrivia paths. If the subgraph of A induced by U is connected, then V (A) \ U contains at most a 1 vertices that are adjacent to U in A. Proof. Suppose that P 1,..., P a are vertices in V (A) \ U adjacent to U in A. Contract each path P j (j = 1,..., a) in G to a singe vertex w j. Next, for i = 1, 3, 5,..., 2k 1, contract a segments P j (p i, p i+1 ), where P j U, and aso contract a edges in bridges connecting these segments in G, to get k vertices z 1, z 3,..., z 2k 1 in a minor of G. Ceary, n 1 2k, so z 1, z 3,..., z 2k 1 exist. Since U is adjacent to P 1,..., P a in A, it is easy to see that vertices w 1,..., w a and z 1, z 3,..., z 2k 1 give rise to a K a,k minor of G. We sha appy Caim 3.1 together with the hep of the foowing emma.

7 MINORS IN GRAPHS OF BOUNDED TREE-WIDTH 7 Lemma 3.1. Let H be a connected graph. If H has at east 2a 2 vertices of degree 3, then H contains a tree T with a vertices of degree 1. Proof. Let d be the maximum vertex degree in H, and et v 0 be a vertex of degree d. If d a, then T is the star centered at v 0. So, suppose that d < a. Then it is sufficient to prove the foowing. Assuming that H has at east 2a 2 (d 1) 2 vertices of degree 3, we sha prove by induction on a d that the tree T exists. Let N 1 be the set of a vertices of degree 3 which can be reached from v 0 on paths whose interna vertices a have degree 2. Then 1 N 1 d. Let N 2 be the second neighborhood of v 0, consisting of vertices of degree 3 which are not in N 1 {v 0 } and which can be reached from v 0 on paths for which exacty one interna vertex has degree 3. Simiary, et N 3 be the third neighborhood of v 0. Then 1 N 2 d(d 1) and N 3 1 since H is connected and 2a 2 (d 1) 2 > 1+d+d(d 1) 1+ N 1 + N 2. Let v 3 N 3, and et W be a path from v 0 to v 3 which contains precisey two other vertices of degree 3. Now, contract W to a vertex ṽ 0 and remove possibe parae edges. Denote the resuting graph by H. If a vertex of H has degree smaer than in H, then it was adjacent to two (or three) vertices of W. This impies that H has at east 2a 2 (d 1) 2 (2d 1) = 2a 2 ((d+1) 1) 2 vertices of degree 3. Since v 0 and v 3 have no common neighbors, ṽ 0 is its vertex of maximum degree d + 1. By the induction hypothesis, H contains a tree T with at east a vertices of degree 1. Ceary, T gives rise to the required tree T in H. At east one of the paths is everywhere nontrivia, say P 1. Let A 1 be the induced subgraph of A on the everywhere nontrivia paths. Let A 0 be the induced subgraph of A consisting of the connected component of A 1 containing P 1 together with (at most a 1) trivia paths adjacent to that component. From now on we sha assume that G is c(a)-connected, where c(3) = 7 and c(a) = 264a + 1 for a 4. Caim 3.2. A 0 A 1 has at east c(a) a+1 2 vertices. If a = 3, A 0 is isomorphic to a path or a cyce on at east four vertices. If a 4, then every vertex of A 0 A 1 has degree at most a 1 and at most 2a 2 of these vertices have degree more than 2 in A 0 A 1. Proof. Let U = V (A 0 A 1 ), x = U, and y = V (A 0 ) x. By Caim 3.1 we see that y a 1. Since the 2x + y endvertices of the paths in A 0 in Y p1 and Y p3 separate the graph G, we have 2x + y c(a). This impies that x (c(a) a + 1)/2, and proves the first part of the caim.

8 8 T. BÖHME, J. MAHARRY, AND B. MOHAR By Caim 3.1, every vertex in A 0 A 1 has degree at most a 1 in A. If a = 3, this impies that A 0 A 1 is a path or a cyce, and the trivia paths in V (A 0 ) can be adjacent ony to vertices of degree 1 in A 0 A 1. This and Caim 3.1 impy that A 0 is a path or a cyce. If V (A 0 ) 3, then the endpoints of the paths in V (A 0 ) woud give a 6-separator in G. Suppose now that a 4. By Caim 3.1 every vertex of A 0 A 1 has degree at most a 1. Suppose that there are more than 2a 2 vertices of degree 3. By Lemma 3.1, A 0 A 1 contains a tree T with a vertices of degree 1. Let U be the set of vertices of degree 2 in T. The subgraph of A induced by U is connected, and Caim 3.1 yieds a contradiction. This competes the proof. Denote by Z (i) the union of P j (p i, p i+1 ) where P j V (A 0 ), i = 1, 2,..., n 1 1. Let Z i be the subgraph of G obtained by taking the union of Z (i) and a those Z-bridges B that have a vertices of attachment in Z (i) such that there is no i < i for which B woud have a its vertices of attachment in Z (i ). 4. FINDING K 3,K MINORS In this section we consider the case when a = 3 since the best possibe connectivity 7 requires more eaborate techniques than the genera case treated in the next section. For i = 1, 2,..., n 1 2w 2, et H i = 2w k=0 Z i+k. Let R, R V (A 0 ) be paths which are adjacent in A 0. For i = 1, 2,..., n 1 2w 2 define the graph D i = D i (R, R ) as foows. First, take S = (R R ) H i together with a Z-bridges in H i that have vertices of attachment on R and on R. Finay, add two edges e 1, e 2, where e 1 joins the eft endvertices, λ in R H i and λ in R H i, and e 2 joins the right endvertices, ρ and ρ, of these two paths. Then S + e 1 + e 2 =: C is a cyce in D i. If R (R ) is everywhere trivia, then λ = ρ (λ = ρ ). Caim 4.1. Suppose that a = 3. Then for every i, there are adjacent vertices R, R of A 0 such that D i (R, R ) has no embedding in the pane where the vertices λ, λ, ρ, ρ woud ie on the outer face in the prescribed order. Proof. Suppose that H i is a panar graph. Let v j be the number of vertices of degree j in H i. By Euer s formua and standard counting arguments it foows that L := j 0(6 j)v j 12. (1)

9 MINORS IN GRAPHS OF BOUNDED TREE-WIDTH 9 Observe that H i has at most 2s vertices of degree 6 since the minimum degree in G is at east 7 (by the 7-connectivity of G). On the other hand, since at east three of the paths in H i are nontrivia, these paths contain at east 3(2(2w + 1) 1) = 12w + 3 vertices of degree 7 in H i. Therefore, L 6 2s (12w + 3) 12(w + 1) 12w 3 = 9. This contradiction to (1) shows that H i is not panar. Reca that A 0 is a path or a cyce on at east 4 vertices, R 1,..., R d, d 4. This impies, in particuar, that no Z-bridge in H i is attached to more than two of the paths (otherwise, there woud be a 3-cyce in A 0, and so A 0 woud be equa to the 3-cyce). Moreover, if every D i (R j, R j+1 ) (j = 1,..., d, indices taken moduo d) has an embedding in the pane with the corresponding cyce C j being the outer cyce, then d j=1 D i(r j, R j+1 ) H i woud be panar as we, contrary to the above. Hence, there is an index j such that D i (R j, R j+1 ) has no such embedding. Since there are no oca Z-bridges, D i (R j, R j+1 ) neither has an embedding in the pane where the vertices λ, λ, ρ, ρ are on the outer face in the prescribed order. We sha need a resut about crossing paths from from [9]. A separation of a graph G is a pair (A, B) of subraphs with A B = G and E(A B) =, and its order is V (A B). By a society we mean a pair (G, Ω), where G is a graph and Ω a cycic permutation of a subset Ω of V (G). A cross in (G, Ω) is a pair of disjoint paths in G with ends s 1, t 1 and s 2, t 2, respectivey, a in Ω, such that s 1, s 2, t 1, t 2 occur in Ω in that order (but not necessariy consecutive). The foowing formuation of a theorem of Robertson and Seymour [9] appears in [10]. Theorem 4.1 (Robertson and Seymour). Let (G, Ω) be a society such that there is no separation (A, B) of G of order 3 with Ω V (A) V (G). Then the foowing are equivaent: (a)there is no cross in (G, Ω). (b)g can be drawn in a disc with the vertices in Ω drawn on the boundary of the disc in order given by Ω. Caim 4.2. If D i (R, R ) is nonpanar, then one of the foowing hods: (a) D i (R, R ) contains disjoint paths Q 1, Q 2 connecting λ with ρ and λ with ρ, respectivey. (b) D i (R, R ) contains a path Q (resp., Q ) disjoint from R (resp., R) which connects λ and ρ (resp., λ and ρ ) such that after repacing R (resp., R ) by Q (resp., Q ), there is a Z-bridge in H i which is attached to more than two of the paths P 1,..., P s.

10 10 T. BÖHME, J. MAHARRY, AND B. MOHAR Proof. Let H = D i (R, R ). Let C be the cyce of H defined before Caim 4.1. Let Ω be the set of vertices of C which are incident with an edge in E(G)\E(H). The cycic order of Ω on C defines the society (H, Ω). Since G is 4-connected and no vertex in V (H) \ Ω is incident with an edge in E(G) \ E(H), there is no separation (A, B) of H of order 3 with Ω V (A) V (G). Since H is nonpanar, Theorem 4.1 impies that there is a cross R 1, R 2 in (H, Ω). Let α i, β i be the endvertices of R i (i = 1, 2). We may assume that: (i) None of the vertices λ, λ, ρ, ρ is an interna vertex of R 1 or R 2. Subject to (i) choose the cross R 1, R 2 such that (ii) {α 1, α 2, β 1, β 2 } contains as many vertices in {λ, λ, ρ, ρ } as possibe and, subject to (i) and (ii) (iii) as few edges in E(H) \ E(R R ) as possibe. If λ, λ, ρ, ρ are a endvertices of R 1, R 2, then we have (a). Hence we may assume that λ is not an endvertex of R 1, R 2. If R (R 1 R 2 ), et v be the first vertex of R 1 R 2 on R (starting at λ towards ρ). We may assume that v V (R 1 ). Let R 1 = R 1 R 1 where V (R 1) V (R 1) = {v}. By repacing one of the segments R 1 or R 1 in R 1 by a segment from v to λ on R, a new cross is obtained which contradicts (ii) or (iii), except when R 1 or R 1 is the segment of R from v to ρ. In particuar, three of the endvertices of R 1, R 2 are on R. The above proof impies that λ and ρ are the endvertices of the paths. Since R 1, R 2 cross, R 1 joins a vertex x V (R ) \ {λ, ρ } with ρ, and R 2 joins λ and ρ, where R 2 is disjoint from R. It is easy to see, that this gives (b). Suppose now that R (R 1 R 2 ) =. Condition (ii) impies that λ and ρ are the endvertices of R 1 and R 2, respectivey. There is a C-bridge B in H such that E(R 1 R 2 ) E(B). Since B is not a oca bridge, it is attached to R as we. Therefore, there is a path L in B from R to R 1 R 2 (say to R 2 ) which is internay disjoint from C R 1 R 2. Let y be the vertex of R 1 which is as cose as possibe to ρ on R. Let R 2 be the segment of R 2 from R 2 L to the end of R 2 distinct from ρ. By (iii), R 2 is disjoint from the segment Q of R from y to ρ. Therefore, the path Q composed of the segment of R 1 from λ to y and Q can be taken as the path Q in (b). Note that, after repacing R by Q, the Z-bridge containing L R 2 wi be attached to at east three paths in {P 1,..., P s }. We are ready to compete the proof of Theorem 1.1. Suppose that a = 3 and that A 0 is a path or a cyce on consecutive vertices R 1,..., R d, where 4 d w + 1. Let D j i = D i (R j, R j+1 ), j = 1,..., d. We sha ony

11 MINORS IN GRAPHS OF BOUNDED TREE-WIDTH 11 consider the indices i of the form i = 1 + t(2w + 2), t = 0, 1,..., and we ca them admissibe indices. Let us first assume that the case (b) of Caim 4.2 occurs ess than 2kd times at admissibe indices i. Since there are at east 4kd admissibe indices, Caim 4.2(a) impies that there is an index j {1,..., d}, and there are admissibe indices 1 i 1 < i 2 < < i k n 1 2w 2 such that (i) each of D j i 1, D j i 2,..., D j i k contains paths as stated in Caim 4.2(a), and (ii) for = 1,..., k 1, i +1 i 2w + 2. We can exchange the segments of the paths R j and R j+1 in H i by the two paths Q 1, Q 2 of Caim 4.2(a). In this way the new paths in H i Z i +2w+2 woud no onger satisfy the condition of Caim 3.1. Namey, if R j and R j+1 have degrees d 1, d 2 in A 0, then they woud be everywhere bridge connected (w.r.t. the sequence p i1 1, p i2 1,..., p ik 1) with d 1 + d 2 1 other paths. If d 1 = d 2 = 2, this gives a K 3,k minor in the same way as in the proof of Caim 3.1 (since one of R j or R j+1 is everywhere nontrivia). If d 1 = 1 (say), then the path R j+2 has degree 2 in A 0 by Caim 3.2 and (in addition to R j+3 ) it becomes everywhere bridge connected to the two new paths (w.r.t. the sequence p i1 1, p i2 1,..., p ik 1). It is easy to see from the definition of A 0 that R j+2 cannot be trivia, so the proof of Caim 3.1 appies again. Let us now assume that the case (b) of Caim 4.2 occurs 2kd or more times (for admissibe indices i). Then there is an index j {1,..., d}, and there are admissibe indices 1 i 1 < i 2 < < i k n 1 2w 2 such that (i) each of D j i 1, D j i 2,..., D j i k contains a path Q (or each of D j i 1, D j i 2,..., D j i k contains a path Q ) as stated in Caim 4.2(b), and (ii) for = 1,..., k 1, i +1 i 2w + 2. For any D j i we repace the segment of R j (resp., R j+1 ) by the corresponding path Q (resp., Q ) such that there is a Z-bridge (where Z is defined as the union of the new paths) attached to R j, R j+1, and R j+2 (or R j 1 ). We may assume that k of these bridges, B 1,..., B k are attached to R j, R j+1, and R j+2. Now, there is a K 3,k -minor obtained by contracting R j, R j+1, R j+2 into singe vertices and adding paths in B 1,..., B k to these vertices. This competes the proof of Theorem FINDING K A,K MINORS FOR A 4 Suppose now that a 4 and c(a) = 264a + 1. Let r = 2c(a) + 2. For i = 1, 2,..., n 1 r, et H i = r 1 j=0 Z i+j. We aso write S i = Y pi. Caim 5.1. For every 1 i n 1 r, the average degree of vertices in H i is at east c(a) 1 2.

12 12 T. BÖHME, J. MAHARRY, AND B. MOHAR Proof. Every vertex of G has degree at east c(a). Let s 0 = V (A 0 A 1 ) be the number of everywhere nontrivia paths in V (A 0 ). Then V (H i ) s 0 (2r + 1) > 4s 0 c(a). (2) Each trivia path in V (A 0 ) is everywhere bridge connected to some nontrivia path. Hence, the degree of the corresponding vertex in H i is at east r/2 c(a). Ony the ends of nontrivia paths can have degree ess than c(a) in H i. This fact and inequaity (2) impy that 2 E(H i ) c(a)( V (H i ) 2s 0 ) (c(a) 1 2 ) V (H i). This competes the proof. A graph L is said to be q-inked if it has at east 2q vertices and for any ordered q-tupes (s 1,..., s q ) and (t 1,..., t q ) of 2q distinct vertices of L, there exist pairwise disjoint paths P 1,..., P q such that for i = 1,..., q, the path P i connects s i and t i. Such coection of paths is caed a inkage of (s 1,..., s q ) and (t 1,..., t q ). Caim 5.2. For every 1 i n 1 r, there exists a subgraph L i of H i which is 3a-inked. Proof. Mader [5] proved that every graph of average degree at east 4c contains a c-connected subgraph. Therefore, since H i has average degree at east c(a) 1 264a, H i contains a 66a-connected subgraph L i. Boobás and Thomason [1] have shown that every 22t-connected graph is t-inked. Hence, the graph L i is 3a-inked. We wi now construct a disjoint paths P 1,..., P a by routing the paths P 1,..., P s through L i in at east k pairwise disjoint subgraphs H i. In each graph L i, there wi aso be an extra vertex inked to each of the a paths. Contracting these paths wi then give a K a,k -minor in G. Caim 5.3. In H i, there exist 2a pairwise disjoint paths, Q (i) 1,..., Q(i) a and Q 1 (i),..., Q a (i) such that the foowing hod: (a) For = 1, 2,..., a, the path Q (i) starts in L i and ends in S i+r. (b) For = 1, 2,..., a, the path Q (i) starts in S i and ends in L i. (c) Every path Q (i) and Q (i) ( = 1, 2,..., a) has ony its endvertices in S i S i+r V (L i ).

13 MINORS IN GRAPHS OF BOUNDED TREE-WIDTH 13 Proof. Let Π 0 = V (A 0 )\V (A 1 ) be the set of vertices of H i corresponding to the trivia paths in A 0. Let W = {W 1,..., W 2a } be a set of 2a pairwise disjoint paths joining V (L i ) with S i S i+r such that: (1) W H i Π 0 for every = 1, 2,..., 2a. (2) The number of edges in 2a =1 E(W ) \ r 1 j=0 E(Z (i + j)) is minimum. (3) Subject to (2), if n L is the number of paths W ending in S i, and n R is the number of paths W ending in S i+r, n L n R is minimum. Disjoint paths satisfying (1) exist by arge connectivity: Since c(a) 3a 1, and V (L i ) > 3a, and S i S i+r 3a 1, there exist 3a 1 disjoint paths from V (L i ) to S i S i+r+1 by Menger s theorem. Since there are at most a 1 vertices in Π 0, the remova of those paths which intersect Π 0 eaves at east 2a paths satisfying condition (1). If at east two paths of W intersect a path P j, then et W and W be the paths that intersect P j as cose as possibe (on P j ) to S i and S i+r, respectivey. If W = W, suppose that the intersection u of W with P j nearest S i (say) comes before the intersection nearest S i+r. By (2), W ends at S i, i.e., its segment from u to its end coincides with the segment P j (u, S i ) of P j. This shows that W W. Then the path W (resp. W ) must end at S i (resp. S i+r ) by (2). Suppose that precisey one path, say W W, intersects a path P j. In this case we can eect to have W ending at P j S i or at P j S i+r by foowing the path P j. This impies that the vaue n L n R in (3) can be made to be zero. Then n L = n R = a. Now et the a paths in W that end in S i be caed Q 1 (i), Q 2 (i),..., Q a (i) and the a paths in W that end in S i+r be caed Q (i) 1, Q(i) 2,..., Q(i) a. It is easy to see that (c) may be requested. This competes the proof. Let T be a spanning tree of A 0 A 1. By Caim 3.2, V (T ) a. This impies the foowing caim. Caim 5.4. There are vertices t 1, t 2,..., t a of T such that for = 1, 2,..., a, the vertex t is a eaf of the subtree T \ {t 1,..., t 1 }. For each i = 1, 2,..., n 1 r and each = 1, 2,..., a, et J (i) {P 1,..., P s } be the vertex of T such that Q (i) ends up on the corresponding path in G. Choose an enumeration of Q (i) 1, Q(i) 2,..., Q(i) a such that, for = 1, 2,..., a, the distance from J (i) to t in T is minimum (where smaer vaues of have preference over the arger vaues). Choose a simiar enumeration of Q 1 (i),..., Q a (i). Define α = r + 4a + 2 and for t = 1,..., k set i t = 1 + (t 1)α. Observe that i k = n 1 r.

14 14 T. BÖHME, J. MAHARRY, AND B. MOHAR To construct the path P, we first ink Q(it) to Q (it+1) for every t = 1,..., k 1. Then each Q (it) is inked to Q (it) inside L it (t = 1,..., k). We do this as described beow. Let i = i + α. Link Q (i) with Q (i ) as foows: Foow the path J (i) S i+r through 2 segments to the separator S i+r+2. Continue from J (i) the path within Z i+r+2 to the path t. This can be done by foowing the bridges between paths corresponding to the path in the spanning tree T from J (i) to t. Construct a simiar path from Q (i ) to t using bridges in Z i 2. Then connect these paths aong t, and denote by P i Q (i) with Q (i ). Caim 5.5. the resuting path joining The constructed paths P i ( = 1,..., a) are pairwise disjoint. Proof. Consider two of the paths, say P i and Pm, i where < m. There are four possibiities where these two paths may intersect: (1) P i intersects J (i) m then be coser to t in T, and the path Q (i) m inside Z i+r+2 : This is not possibe since J (i) m woud woud be indexed before Q (i). (2) P i m intersects t inside Z i+r+2m : This is not possibe since t is a eaf in T \ {t 1,..., t 1 }. The remaining cases, when P i intersects Pm i inside Z i 2 or inside Z i 2m (respectivey) are handed simiary. This competes the proof. Let v be the vertex of Q (i) in L i, and et u be the vertex of Q (i) in L i. Choose u to be a neighbor of u in L i \ {v 1,..., v a, u 1,..., u a }. Since L i is 3a-inked, the minimum degree of L i is at east 3a, so such neighbors exist. The vertices u may even be chosen so that they are pairwise distinct. Let v 1 = u 1, and et v 2,..., v a be distinct neighbors of v 1 in L i. We may assume that if v α = u β, then α = β. Since L i is 2a-inked, there is a inkage from (v 1,..., v a, v 1,..., v a) to (u 1,..., u a, u 1,..., u a). The resuting paths joining v and u ( = 1,..., a) are used to ink Q (i) and Q (i) inside L i, for i {i 1,..., i k }. Together with the paths P i, i {i 1,..., i k 1 }, this determines the path P. On the other hand, the paths in the inkage from (v 1,..., v a) to (u 1,..., u a) are disjoint from P1,..., Pa and can be used to ink v 1 to each of these paths. Now, it can be shown that G contains a K a,k minor: For each = 1,..., a, contract the path P to a singe vertex. For i {i 1,..., i k }, the vertex v 1 V (L i ) is joined to u 1,..., u a and hence to each of the a paths P1,..., Pa. Since this is repeated k times, we get a K a,k minor in G. The proof of Theorem 1.2 is compete.

15 MINORS IN GRAPHS OF BOUNDED TREE-WIDTH CONCLUSION Our more recent resuts show that the condition on bounded tree-width in Theorem 1.1 can be removed. The authors pan a second paper in which the arge tree-width case is handed. This wi prove the foowing, which was conjectured independenty by Ding [3] and the authors: There is a function f : N N such that any 7-connected graph on at east f(k) vertices contains a K 3,k minor. It seems reasonabe to the authors that this resut can be extended to K 4,k -minors and possiby even to K a,k -minors. The ogica conjectures woud be the foowing: Conjecture 6.1. There is a function f : N N such that any 9- connected graph on at east f(k) vertices contains a K 4,k minor. Conjecture 6.2. There are functions f : N N and c : N N such that any c(a)-connected graph on at east f(k) vertices contains a K a,k minor. Our fina remark is that the sequence of graphs K a,k, where a is fixed and k tends to infinity, is essentiay the ony famiy of graphs for which a resut ike our Theorem 1.2 hods. More precisey: Theorem 6.1. Let c and w c be positive integers, and et H k (k 1) be a sequence of graphs such that im k V (H k ) =. Suppose that for any positive integer k there exists an integer N(k) such that every c- connected graph of tree-width w and of order at east N(k) contains H k as a minor. Then H k m K c,n(k) for k 1. Proof. Ceary, the graph K c,n(k) is c-connected and has tree-width c w. By the assumption on the famiy H k, K c,n(k) contains H k as a minor. REFERENCES 1. B. Boobás, A. Thomason, Highy inked graphs, Combinatorica 16 (1996) R. Dieste, Graph Decompositions A Study in Infinite Graph Theory, Oxford University Press, Oxford, Guoi Ding, private communication. 4. M. Juvan, J. Marinček, B. Mohar, Eimination of oca bridges, Math. Sovaca 47 (1997) W. Mader, Existenz n-fach zusammenhängender Teigraphen in Graphen genügend grosser Kantendichte, Abh. Math. Sem. Univ. Hamburg 37 (1972)

16 16 T. BÖHME, J. MAHARRY, AND B. MOHAR 6. J. Maharry, A characterization of graphs with no cube minor, J. Combin. Theory Ser. B 80 (2000) J. Maharry, An excuded minor theorem for the octahedron, J. Graph Theory 31 (1999) N. Robertson, private communication. 9. N. Robertson, P. D. Seymour, Graph minors. IX. Disjoint crossed paths, J. Combin. Theory Ser. B 49 (1990) N. Robertson, P. D. Seymour, An outine of a disjoint paths agorithm, in: Paths, Fows, and VLSI-Layout, B. Korte, L. Lovász, H. J. Pröme, and A. Schrijver Eds., Springer-Verag, Berin, 1990, pp B. Oporowski, J. Oxey, R. Thomas, Typica subgraphs of 3- and 4-connected graphs, J. Combin. Theory Ser. B 57 (1993) R. Thomas, Recent excuded minor theorems for graphs, in Surveys in Combinatorics, 1999 (Canterbury), Cambridge Univ. Press, Cambridge, 1999, pp K. Wagner, Über eine Eigenschaft der ebenen Kompexe, Math. Ann. 114 (1937)