MINORS IN LARGE 6-CONNECTED GRAPHS
National Institute of Informatics
2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
Serguei Norine111Partially supported by NSF under Grants No. DMS-0200595 and DMS-0701033.
Department of Mathematics
Princeton, NJ 08544, USA
Robin Thomas222Partially supported by NSF under Grants No. DMS-0200595, DMS-0354742, and DMS-0701077.
School of Mathematics
Georgia Institute of Technology
Atlanta, Georgia 30332-0160, USA
Mathematisches Seminar der Universität Hamburg
D-20146 Hamburg, Germany
Jørgensen conjectured that every 6-connected graph with no minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.
8 April 2005, revised 9 March 2012.
Graphs in this paper are allowed to have loops and multiple edges. A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An minor is a minor isomorphic to . A graph is apex if it has a vertex such that is planar. (We use for deletion.) Jørgensen [Jor] made the following beautiful conjecture.
Every -connected graph with no minor is apex.
This is related to Hadwiger’s conjecture [Had], the following.
For every integer , if a loopless graph has no minor, then it is -colorable.
Hadwiger’s conjecture is known for . For it has been proven in [RobSeyThoHad] by showing that a minimal counterexample to Hadwiger’s conjecture for is apex. The proof uses an earlier result of Mader [MadHomkrit] that every minimal counterexample to Conjecture 1.2 is -connected. Thus Conjecture 1.1, if true, would give more structural information. Furthermore, the structure of all graphs with no minor is not known, and appears complicated and difficult. On the other hand, Conjecture 1.1 provides a nice and clean statement for -connected graphs. Unfortunately, it, too, appears to be a difficult problem. In this paper we prove Conjecture 1.1 for all sufficiently large graphs, as follows.
There exists an absolute constant such that every -connected graph on at least vertices with no minor is apex.
The second and third author recently announced a generalization [NorTho] of Theorem 1.3, where is replaced by an arbitrary integer . The result states that for every integer there exists an integer such that every -connected graph on at least vertices with no minor has a set of at most vertices whose deletion makes the graph planar. The proof follows a different strategy, but makes use of several ideas developed in this paper and its companion [KawNorThoWolbdtw].
We use a number of results from the Graph Minor series of Robertson and Seymour, and also three results of our own that are proved in [KawNorThoWolbdtw]. The first of those is a version of Theorem 1.3 for graphs of bounded tree-width, the following. (We will not define tree-width here, because it is sufficiently well-known, and because we do not need the concept per se, only several theorems that use it.)
For every integer there exists an integer such that every -connected graph of tree-width at most on at least vertices and with no minor is apex.
Theorem 1.4 reduces the proof of Theorem 1.3 to graphs of large tree-width. By a result of Robertson and Seymour [RobSeyGM5] those graphs have a large grid minor. However, for our purposes it is more convenient to work with walls instead. Let be even. An elementary wall of height has vertex-set
and an edge between any vertices and if either
and , or
, and and have the same parity.
Figure 1 shows an elementary wall of height . A wall of height is a subdivision of an elementary wall of height . The result of [RobSeyGM5] (see also [DieGorJenTho, ReedBCC, RobSeyThoQuickPlanar]) can be restated as follows.
For every even integer there exists an integer such that every graph of tree-width at least has a subgraph isomorphic to a wall of height .
The perimeter of a wall is the cycle that bounds the infinite face when the wall is drawn as in Figure 1. Now let be the perimeter of a wall in a graph . The compass of in is the restriction of to , where is the union of and the vertex-set of the unique component of that contains a vertex of . Thus is a subgraph of its compass, and the compass is connected. A wall with perimeter in a graph is planar if its compass can be drawn in the plane with bounding the infinite face. In Section 2 we prove the following.
For every even integer there exists an even integer such that if a -connected graph with no minor has a wall of height at least , then either it is apex, or has a planar wall of height .
Actually, in the proof of Theorem 1.6 we need Lemma 2.4 that is proved in [KawNorThoWolbdtw]. The lemma says that if a -connected graph with no minor has a subgraph isomorphic to subdivision of a pinwheel with sufficiently many vanes (see Figure 3), then it is apex.
By Theorem 1.6 we may assume that our graph has an arbitrarily large planar wall . Let be the perimeter of , and let be the compass of . Then separates into and another graph, say , such that , and . Next we study the graph . Since the order of the vertices on is important, we are lead to the notion of a “society”, introduced by Robertson and Seymour in [RobSeyGM9].
Let be a cyclic permutation of the elements of some set; we denote this set by . A society is a pair , where is a graph, and is a cyclic permutation with . Now let be as above, and let be one of the cyclic permutations of determined by the order of vertices on . Then is a society that is of primary interest to us. We call it the anticompass society of in .
We say that is a neighborhood if is a graph and are cyclic permutations, where both and are subsets of . Let be a plane, with some orientation called “clockwise.” We say that a neighborhood is rural if has a drawing in without crossings (so is planar) and there are closed discs , such that
(i) the drawing uses no point of outside , and none in the interior of , and
(ii) for , the point of representing in the drawing lies in (respectively, ) if and only if (respectively, , and the cyclic permutation of (respectively, obtained from the clockwise orientation of (respectively, ) coincides (in the natural sense) with (respectively, ).
We call a presentation of .
Let be a neighborhood, let be a society with , and let . Then is a society, and we say that is the composition of the society with the neighborhood . If the neighborhood is rural, then we say that is a planar truncation of . We say that a society is -cosmopolitan, where is an integer, if for every planar truncation of at least vertices in have at least two neighbors in . At the end of Section 2 we deduce
For every integer there exists an even integer such that if is a simple graph of minimum degree at least six and is a planar wall of height in , then the anticompass society of in is -cosmopolitan.
For a fixed presentation of a neighborhood and an integer we define an -nest for to be a sequence of pairwise disjoint cycles of such that , where denotes the closed disk in bounded by the image under of . We say that a society is -nested if it is the composition of a society with a rural neighborhood that has an -nest for some presentation of .
Let be a cyclic permutation. For we denote the image of under by . If , then we denote by the restriction of to . That is, is the permutation defined by saying that and is the first term of the sequence which belongs to . Let be distinct. We say that is clockwise in (or simply clockwise when is understood from context) if for all , where means and . For we define as the set of all such that either or or is clockwise in .
A separation of a graph is a pair such that and there is no edge with one end in and the other end in . The order of is . We say that a society is -connected if there is no separation of of order at most with and . A bump in is a path in with at least one edge, both ends in and otherwise disjoint from .
Let be a society and let be clockwise in . For let be a bump in with ends and , and let be either a bump with ends and , or the union of two internally disjoint bumps, one with ends and and the other with ends and . In the former case let , and in the latter case let be the subinterval of with ends and , including its ends. Assume that are pairwise disjoint. Let be distinct such that neither of the sets , includes both and . Let and be two not necessarily disjoint paths with one end in and the other end and , respectively, both internally disjoint from . In those circumstances we say that is a turtle in . We say that are the legs, is the neck, and is the body of the turtle. (See Figure 2(a),(b).)
Let be a society, let be clockwise in , and let be disjoint bumps such that has ends and . In those circumstances we say that are three crossed paths in .
Let be a society, and let be such that either or or is clockwise. For let be a bump with ends and such that these bumps are pairwise disjoint, except possibly for . In those circumstances we say that is a gridlet. (See Figure 2(c),(d).)
Let be a society and let be clockwise or counter-clockwise in . For let be a bump with ends and such that these bumps are pairwise disjoint, and let be a path with one end in , the other end in , and otherwise disjoint from . In those circumstances we say that is a separated doublecross.(See Figure 2(e),(f).)
A society is rural if can be drawn in a disk with drawn on the boundary of the disk in the order given by . A society is nearly rural if there exists a vertex such that the society obtained from by deleting is rural.
In Sections 4–LABEL:sec:lack we prove the following. The proof strategy is explained in Section LABEL:sec:transactions. It uses a couple of theorems from [RobSeyGM9] and Theorem 4.1 that we prove in Section 4.
There exists an integer such that for every integer and every -connected -nested -cosmopolitan society either is nearly rural, or has a triangle such that is rural, or has an -nested planar truncation that has a turtle, three crossed paths, a gridlet, or a separated doublecross.
Finally, we need to convert a turtle, three crossed paths, gridlet and a separated double-cross into a minor. Let be a -connected graph, let be a sufficiently high planar wall in , and let be the anticompass society of in . We wish to apply to Theorem 1.8 to . We can, in fact, assume that is a subgraph of a larger planar wall that includes concentric cycles surrounding and disjoint from , for some suitable integer , and hence is -nested. Theorem 1.8 guarantees a turtle or paths in forming three crossed paths, a gridlet, or a separated double-cross, but it does not say how the turtle or paths might intersect the cycles . In Section LABEL:sec:nest we prove a theorem that says that the cycles and the turtle (or paths) can be changed such that after possibly sacrificing a lot of the cycles, the remaining cycles and the new turtle (or paths) intersect nicely. Using that information it is then easy to find a minor in . We complete the proof of Theorem 1.3 in Section LABEL:sec:turtle.
2 Finding a planar wall
Let a pinwheel with four vanes be the graph pictured in Figure 3. We define a pinwheel with vanes analogously. A graph is internally -connected if it is simple, -connected, has at least five vertices, and for every separation of of order three, one of induces a graph with at most three edges.
The objective of this section is to prove the following theorem.
For every even integer there exists an even integer such that if is a wall of height at least in an internally -connected graph , then either
(1) has a minor, or
(2) has a subgraph isomorphic to a subdivision of a pinwheel with vanes, or
(3) has a planar wall of height .
In the proof we will be using several results from [RobSeyGM13]. Their statements require the following terminology: distance function, -star over , external -star over , subwall, dividing subwall, flat subwall, cross over a wall. We refer to [RobSeyGM13] for precise definitions, but we offer the following informal descriptions. The distance of two distinct vertices of a wall is the minimum number of times a curve in the plane joining and intersects the drawing of the wall, when the wall is drawn as in Figure 1. An -star over a wall in is a subdivision of a star with leaves such that only the leaves and possibly the center belong to , and the leaves are pairwise at distance at least . The star is external if the center does not belong to . A subwall of a wall is dividing if its perimeter separates the subwall from the rest of the wall. A cross over a wall is a set of two disjoint paths joining the diagonally opposite pairs of “corners” of the wall, the vertices represented by solid circles in Figure 1. A subwall is flat in if there is no cross over such that is a subgraph of the compass of in .
We begin with the following easy lemma. We leave the proof to the reader.
For every integer there exist integers and such that if a graph has a wall with an external -star, then it has a subgraph isomorphic to a subdivision of a pinwheel with vanes.
We need one more lemma, which follows immediately from [RobSeyGM13, Theorem 8.6].
Every flat wall in an internally -connected graph is planar.
Proof of Theorem 2.1. Let be given, let be as in Lemma 2.2, let , and let be as in [RobSeyGM13, Theorem 9.2]. If is sufficiently large, then has subwalls of height at least , pairwise at distance at least . If at least of these subwalls are non-dividing, then by [RobSeyGM13, Theorem 9.2] either has a minor, or an -star over , in which case it has a subgraph isomorphic to a pinwheel with vanes by Lemma 2.2. In either case the theorem holds, and so we may assume that at least two of the subwalls, say and , are dividing. We may assume that and are not planar, for otherwise the theorem holds. Let . By Lemma 2.3 the wall is not flat, and hence its compass has a cross . Since the subwalls and are dividing, it follows that the paths are pairwise disjoint. Thus has a minor isomorphic to the graph shown in Figure 4, but that graph has a minor isomorphic to a minor of , as indicated by the numbers in the figure. Thus has a minor, and the theorem holds. ∎
To deduce Theorem 1.6 we need the following lemma, proved in [KawNorThoWolbdtw, Lemma 5.3].
If a -connected graph with no minor has a subdivision isomorphic to a pinwheel with vanes, then is apex.
Proof of Theorem 1.6. Let be an even integer. We may assume that . Let be as in Theorem 2.1, and let be a -connected graph with no minor. From Theorem 2.1 we deduce that either satisfies the conclusion of Theorem 1.6, or has a subdivision isomorphic to a pinwheel with vanes. In the latter case the theorem follows from Lemma 2.4. ∎
We need the following theorem of DeVos and Seymour [DevSeyExt3col].
Let be a rural society such that is a simple graph and every vertex of not in has degree at least six. Then .
Proof of Theorem 1.7. Let be an integer, and let be an even integer such that if is the elementary wall of height and , then . Let be the compass of in , let be the anticompass society of in , let be a planar truncation of , and let . Thus is the composition of with a rural neighborhood . Then by Theorem 2.5 applied to the society , and hence . Let be the graph obtained from by adding a new vertex and joining it to every vertex of and by adding an edge joining every pair of nonadjacent vertices of that are consecutive in . Then is planar. Let be the number of vertices of with at least two neighbors in . Then all but vertices of have degree in at least six. Thus the sum of the degrees of vertices of is at least . On the other hand, the sum of the degrees is at most , because is planar, and hence , as desired. ∎
3 Rural societies
If is a path and , we denote by the unique subpath of with ends and . Let be a society. An orderly transaction in is a sequence of pairwise disjoint bumps such that has ends and and is clockwise. Let be the graph obtained from by adding the vertices of as isolated vertices. We say that is the frame of . We say that a path in is -coterminal if has both ends in and is otherwise disjoint from it and for every the following holds: if intersects , then their intersection is a path whose one end is a common end of and .
Let be a society, and let and be as in the previous paragraph. Let and let be a -coterminal path in with one end in and the other end in . In those circumstances we say that is a -jump over , or simply a -jump.
Now let and let be two disjoint -coterminal paths such that has ends and is clockwise in , where possibly , , , or , and means , means , means , and means . In those circumstances we say that is a -cross in region , or simply a -cross.
Finally, let and let , , be three paths such that has ends and is otherwise disjoint from all members of , , the vertices are internal vertices of , , , , and the paths , , are pairwise disjoint, except possibly . In those circumstances we say that is a -tunnel under , or simply a -tunnel.
Intuitively, if we think of the paths in as dividing the society into “regions”, then a -jump arises from a -path whose ends do not belong to the same region. A -cross arises from two -paths with ends in the same region that cross inside that region, and furthermore, each path in includes at most two ends of those crossing paths. Finally, a -tunnel can be converted into a -jump by rerouting along . However, in some applications such rerouting will be undesirable, and therefore we need to list -tunnels as outcomes.
Let be a subgraph of a graph . An -bridge in is a connected subgraph of such that and either consists of a unique edge with both ends in , or for some component of the set consists of all edges of with at least one end in . The vertices in are called the attachments of . Now let be such that no block of is a cycle. By a segment of we mean a maximal subpath of such that every internal vertex of has degree two in . It follows that the segments of are uniquely determined. Now if is an -bridge of , then we say that is unstable if some segment of includes all the attachments of , and otherwise we say that is stable.
A society is rurally -connected if for every separation of order at most three with the graph can be drawn in a disk with the vertices of drawn on the boundary of the disk. A society is cross-free if it has no cross. The following, a close relative of Lemma 2.3, follows from [RobSeyGM9, Theorem 2.4].
Every cross-free rurally -connected society is rural.
Let be a rurally -connected society, let be an orderly transaction in , and let be the frame of . If every -bridge of is stable and is not rural, then has a -jump, a -cross, or a -tunnel.
For let and be the ends of numbered as in the definition of orderly transaction, and for convenience let and be null graphs. We define cyclic permutations as follows. For let with the cyclic order defined by saying that is followed by in order from to , followed by followed by in order from to . The cyclic permutation is defined by letting be followed by in order from to , and is defined by letting be followed by in order from to .
Now if for some -bridge of there is no index such that all attachments of belong to , then has a -jump. Thus we may assume that such index exists for every -bridge , and since is stable that index is unique. Let us denote it by . For let be the subgraph of consisting of , the vertex-set and all -bridges of with . The society is rurally -connected. If each is cross-free, then each of them is rural by Theorem 3.1 and it follows that is rural. Thus we may assume that for some the society has a cross . If neither nor includes three or four ends of the paths and , then has a -cross. Thus we may assume that includes both ends of and at least one end of . Let be the ends of . Since the -bridge of containing is stable, it has an attachment outside , and so if needed, we may replace by a path with an end outside (or conclude that has a -jump). Thus we may assume that occur on in the order listed, and .
The -bridge of containing has an attachment outside . If it does not include and has an attachment outside , then has a -jump or -cross, and so we may assume not. Thus there exists a path with one end in the interior of and the other end with no internal vertex in . We call the triple a tripod, and the path the leg of the tripod. If is an internal vertex of , then we say that is sheltered by the tripod . Let be a path that is the leg of some tripod, and subject to that is minimal. From now on we fix and will consider different tripods with leg ; thus the vertices may change, but and will remain fixed as the ends of .
Let be such that they are sheltered by no tripod with leg , but every internal vertex of is sheltered by some tripod with leg . Let be the union of and all tripods with leg that shelter some internal vertex of , let and let . Since is rurally -connected we deduce that the set does not separate from in . It follows that there exists a path in with ends and . We may assume that has no internal vertex in . Let be a tripod with leg L such that either is sheltered by it, or . If , then by considering the paths it follows that either has a -jump or -tunnel. If , then there is a tripod whose leg is a proper subpath of , contrary to the choice of . Thus we may assume that , and that for every choice of the path as above. If then there is a tripod with leg that shelters or , a contradiction. Thus . Let be the -bridge containing . Since for all choices of it follows that the attachments of are a subset of . But is stable, and hence is an attachment of . The minimality of implies that includes a path from to , internally disjoint from . Using that path and the paths it is now easy to construct a tripod that shelters either or , a contradiction. ∎
4 Leap of length five
A leap of length in a society is a sequence of pairwise disjoint bumps such that has ends and and , is clockwise. In this section we prove the following.
Let be a -connected society with a leap of length five. Then is nearly rural, or has a triangle such that is rural, or has three crossed paths, a gridlet, a separated doublecross, or a turtle.
The following is a hypothesis that will be common to several lemmas of this section, and so we state it separately to avoid repetition.
Let be a society with no three crossed paths, a gridlet, a separated doublecross, or a turtle, let be an integer, let
be clockwise, and let be pairwise disjoint bumps such that has ends and . Let be the orderly transaction , let be the frame of and let
Let and .
If is a subgraph of , then an -path is a (possibly trivial) path with both ends in and otherwise disjoint from . This is somewhat non-standard, typically an -path is required to have at least one edge, but we use our definition for convenience. We say that a vertex of is exposed if there exists an -path with one end and the other in .
Assume Hypothesis 4.2 and let . Let be two disjoint -paths in such that has ends and , and assume that occur on in the order listed, where possibly , or , or both. Then either , or , or both. In particular, there do not exist two disjoint -paths from to .