Hamiltonicity in locally finite graphs: two extensions and a counterexample

Hamiltonicity in locally finite graphs: two extensions and a counterexample

Karl Heuer Karl Heuer, Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
Abstract.

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs.

We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing or as a minor is Hamiltonian if and only if it is -connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the -contractible edges.

The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist.

1. Introduction

Results about Hamilton cycles in finite graphs can be extended to locally finite graphs in the following way. For a locally finite graph we consider its Freudenthal compactification  [7, 8], which is a topological space obtained by taking , seen as a -complex, and adding the ends of , which are the equivalence classes of the rays of under the relation of being inseparable by finitely many vertices, as additional points. Extending the notion of cycles, we define circles [9, 10] in as homeomorphic images of the unit circle in , and we call them Hamilton circles, if they additionally contain all vertices of . As a consequence of being a closed subspace of , Hamilton circles also contain all ends of . Following this notion we call Hamiltonian (or say that has a Hamilton circle) if there is a Hamilton circle in .

One of the first and probably one of the deepest results about Hamilton circles was Georgakopoulos’s extension of Fleischner’s theorem to locally finite graphs.

Theorem 1.1.

[13] The square of any finite -connected graph is Hamiltonian.

Theorem 1.2.

[14, Thm. 3] The square of any locally finite -connected graph is Hamiltonian.

Following this breakthrough, more Hamiltonicity theorems have been extended to locally finite graphs in this way [1, 4, 14, 15, 18, 19, 21].

The purpose of this paper is to prove more extensions of Hamiltonicity results about finite graphs to locally finite ones and to construct a graph which shows that another result does not extend.

The first result we consider is a corollary of the following theorem of Harary and Schwenk. A caterpillar is a tree such that after deleting its leaves only a path is left. Let denote the graph obtained by taking the star with three leaves, , and subdividing each edge once.

Theorem 1.3.

[16, Thm. 1] Let be a finite tree with at least three vertices. Then the following statements are equivalent:

  1. is Hamiltonian.

  2. does not contain as a subgraph.

  3. is a caterpillar.

Theorem 1.3 has the following obvious corollary.

Corollary 1.4.

[16] The square of every finite graph on at least three vertices that contains a spanning caterpillar is Hamiltonian.

While the proof of Corollary 1.4 is immediate, the proof of the following extension of it, which is the first result of this paper, needs more work. We call the closure in of a subgraph of a standard subspace of . Extending the notion of trees, we define topological trees as topologically connected standard subspaces not containing any circles. As an analogue of a path, we define an arc as a homeomorphic image of the unit interval in . For our extension we adapt the notion of a caterpillar to the space and work with topological caterpillars, which are topological trees such that , where denotes the set of vertices of degree in , is an arc.

Theorem 1.5.

The square of every locally finite connected graph on at least three vertices that contains a spanning topological caterpillar is Hamiltonian.

The other two results of this paper concern the uniqueness of Hamilton circles. The first is about finite outerplanar graphs. These are finite graphs that can be embedded in the plane such that all vertices lie on the boundary of a common face. Clearly, finite outerplanar graphs have a Hamilton cycle if and only if they are -connected. It is also easy to see that finite -connected outerplanar graphs have a unique Hamilton cycle consisting of precisely its -contractible edges, i.e., those edges each of whose contraction leaves the graph -connected (except for the case where the graph is a ), as pointed out by Sysło. We summarise this with the following proposition.

Proposition 1.6.
  1. A finite outerplanar graph is Hamiltonian if and only if it is -connected.

  2. [26, Thm. 6] Finite -connected outerplanar graphs have a unique Hamilton cycle, which consists precisely of the -contractible edges if the graph is not isomorphic to a .

Finite outerplanar graphs can also be characterised by forbidden minors, which was done by Chartrand and Harary.

Theorem 1.7.

[6, Thm. 1] A finite graph is outerplanar if and only if it contains neither a nor a as a minor.111Actually these statements can be strengthened a little bit by replacing the part about not containing a as a minor by not containing it as a subgraph. This follows from Lemma 4.1.

In the light of Theorem 1.7 we first prove the following extension of statement (i) of Proposition 1.6 to locally finite graphs.

Theorem 1.8.

Let be a locally finite connected graph. Then the following statements are equivalent:

  1. is -connected and contains neither nor as a minor.

  2. has a Hamilton circle and there exists an embedding of into a closed disk such that corresponds to the boundary of the disk.

Furthermore, if the statements (i) and (ii) hold, then has a unique Hamilton circle.

From this we then obtain the following corollary, which extends statement (ii) of Proposition 1.6. Given a circle in , we call the edge set whose closure in equals the circuit of .

Corollary 1.9.

The circuit of the Hamilton circle of a locally finite -connected graph not containing or as a minor consists precisely of the -contractible edges if the graph is not isomorphic to a ..

We should note here that parts of Theorem 1.8 and Corollary 1.9 are already known. Chan [5, Thm. 20 with Thm. 27] proved that the -contractible edges of a locally finite -connected graph not containing or as a minor form the circuit of a Hamilton circle. He deduces this from other general results about -contractible edges in locally finite -connected graphs. In our proof, however, we directly construct the Hamilton circle and show its uniqueness without working with -contractible edges. Afterwards, we deduce Corollary 1.9.

Our third result is related to the following conjecture Sheehan made for finite graphs.

Conjecture 1.10.

[25] For every there is no finite -regular graph with a unique Hamilton cycle.

This conjecture is still open, but some partial results have been proved [17, 28, 29]. For the statement of the conjecture was verified by Smith first. This was noted in an article of Tutte [30] where the statement for was published for the first time.

For infinite graphs Conjecture 1.10 is not true in this formulation. It fails already with . To see this consider the graph depicted in Figure 1, called the double ladder.

Figure 1. The double ladder

It is easy to check that the double ladder has a unique Hamilton circle, but all vertices have degree . Mohar has modified the statement of the conjecture and raised the following question. To state them we need to define two terms. For a graph we call the equivalence classes of rays under the relation of being inseparable by finitely many vertices the ends of . We define the vertex- or edge-degree of an end to be the supremum of the number of vertex- or edge-disjoint rays in , respectively. In particular, ends of a graph can have infinite degree, even if is locally finite.

Question 1.

[22] Does an infinite graph exist that has a unique Hamilton circle and degree at every vertex as well as vertex-degree at every end?

Our result shows in contrast to Conjecture 1.10 and its known cases that there are infinite graphs having the same degree at every vertex and end while being Hamiltonian in a unique way.

Theorem 1.11.

There exists an infinite connected graph with a unique Hamilton circle that has degree at every vertex and vertex- as well as edge-degree at every end.

So with Theorem 1.11 we answer Question 1 positively and therefore disprove the modified version of Conjecture 1.10 for infinite graphs in the way Mohar suggested by considering degrees of both vertices and ends.

The rest of this paper is structured as follows. In Section 2 we establish the notation and terminology we need for the rest of the paper. We also list some lemmas that will serve as auxiliary tools for the proofs of the main theorems. Section 3 is dedicated to Theorem 1.5 where at the beginning of that section we discuss how one can sensibly extend Corollary 1.4 and which problems arise when we try to extend Theorem 1.3 in a similar way. In Section 4 we present a proof of Theorem 1.8 and describe afterwards how a different proof of this theorem works that is copying the ideas of a proof of statement (i) of Proposition 1.6. The last section, Section 5, contains the construction of a graph witnessing Theorem 1.11.

2. Preliminaries

When we mention a graph in this paper we always mean an undirected and simple graph. For basic facts and notation about finite as well as infinite graphs we refer the reader to [7]. For a broader survey about locally finite graphs and a topological approach to them see [8].

Now we list important notions and concepts that we shall need in this paper followed by useful statements about them. In a graph with a vertex we denote by the set of edges incident with in . Similar for a subgraph of or just its vertex set we denote by the set of edges that have only one endvertex in . Although formally different, we will not always distinguish between a cut and the partition it is induced by. For two vertices let denote the distance between and in .

We call a finite graph outerplanar if it can be embedded in the plane such that all vertices lie on the boundary of a common face.

For a graph and an integer we define the -th power of as the graph obtained by taking and adding additional edges for any two vertices such that .

A tree is called a caterpillar if after the deletion of its leaves only a path is left.

We denote by the graph obtained by taking the star with three leaves and subdividing each edge once.

We call a graph locally finite if each vertex has finite degree.

A one-way infinite path in a graph is called a ray of , while we call a two-way infinite path in a double ray of . An equivalence relation can be defined on the set of rays of a graph by saying that two rays are equivalent if and only if they cannot be separated by finitely many vertices in . The equivalence classes of this relation are called the ends of .

For a locally finite and connected graph we can endow together with its ends with a topology that yields the space . A precise definition of can be found in [7, Ch. 8.5]. Let us point out here that a ray of converges in to the end of it is contained in. Another way of describing is to endow with the topology of a -complex and then forming the Freudenthal compactification [11].

For a point set in , we denote its closure in by .

We call a subspace of standard if for a subgraph of .

A circle in is the image of a homeomorphism having the unit circle in as domain and mapping into . Note that all finite cycles of a locally finite graph correspond to circles in , but there might also be infinite subgraphs of such that is a circle in . Similar to finite graphs we call a locally finite graph Hamiltonian if there exists a circle in which contains all vertices of . Such circles are called Hamilton circles of . Given a circle in , we call the edge set with the circuit of .

We call the image of a homeomorphism with the closed real unit interval as domain and mapping into an arc in . Given an arc in , we call a point of an endpoint of if or is mapped to by the homeomorphism defining . Similar as for paths, we call an arc an arc if and are the endpoints of the arc. The possibly simplest example of a nontrivial arc is a ray together with the end it converges to. However, the structure of arcs is more complicated in general and they might contain up to many ends. We call a subspace of arc-connected if for any two points and of there is an arc in .

Using the notions of circles and arc-connectedness we now extend trees in a similar topological way. We call an arc-connected standard subspace of a topological tree if it does not contain any circle. Generalizing the definition of caterpillars, we call a topological tree a topological caterpillar if is an arc, where denotes the set of all leaves of , i.e., vertices of degree in .

Now let be an end of a locally finite graph . We define the vertex- or edge-degree of in as the supremum of the number of vertex- or edge-disjoint rays in , respectively, which are contained in . By this definition ends may have infinite vertex- or edge-degree. Similar we define the vertex- or edge-degree of in a standard subspace of as the supremum of vertex- or edge-disjoint arcs in , respectively, that have as an endpoint. We should mention here that the supremum is actually an attained maximum in both definitions. Furthermore, these definitions coincide when we take . The proofs of these statements are nontrivial and since it is enough for us to work with the supremum, we will not go into detail here.

We make one last definition with respect to end degrees which allows us to distinguish the parity of degrees of ends when they are infinite. The idea of this definition is due to Bruhn and Stein [3]. We call the vertex- or edge-degree of an end  of in a standard subspace of even if there is a finite set such that for every finite set with the maximum number of vertex- or edge-disjoint arcs in , respectively, whose endpoints are and some is even. Otherwise, we call the vertex- or edge-degree of in , respectively, odd.

Next we collect some useful statements about the space for a locally finite graph .

Proposition 2.1.

[7, Prop. 8.5.1] If is a locally finite connected graph, then is a compact Hausdorff space.

Having Proposition 2.1 in mind the following basic lemma helps us to work with continuous maps and verify homeomorphisms, for example when considering circles or arcs.

Lemma 2.2.

Let be a compact space, be a Hausdorff space and be a continuous injection. Then is continuous too.

The following lemma tells us an important combinatorial property of arcs. To state the lemma more easily, let denote the set of inner points of edges in for an edge set .

Lemma 2.3.

[7, Lemma 8.5.3] Let be a locally finite connected graph and be a cut with sides and .

  1. If is finite, then , and there is no arc in with one endpoint in and the other in .

  2. If is infinite, then , and there may be such an arc.

The next lemma ensures that connectedness and arc-connectedness are equivalent for the spaces we are mostly interested in, namely standard subspaces, which are closed by definition.

Lemma 2.4.

[12, Thm. 2.6] If is a locally finite connected graph, then every closed topologically connected subset of is arc-connected.

Continuing with the idea of Lemma 2.3 of characterising important topological properties of the space in terms of combinatorial ones, the following lemma about arc-connected subspaces was obtained, which will be convenient for us to use in a proof later on.

Lemma 2.5.

[7, Lemma 8.5.5] If is a locally finite connected graph, then a standard subspace of is topologically connected (equivalently: arc-connected) if and only if it contains an edge from every finite cut of of which it meets both sides.

The next theorem is actually part of a bigger one containing more equivalent statements. Since we shall need only one equivalence, we reduced it to the following formulation. For us it will be helpful to check or at least bound the degree of an end in a standard subspace just by looking at finite cuts instead of dealing with the homeomorphisms that actually define the relevant arcs.

Theorem 2.6.

[8, Thm. 2.5] Let be a locally finite connected graph. Then the following are equivalent for :

  1. meets every finite cut in an even number of edges.

  2. Every vertex and every end of has even degree or edge-degree in , respectively.

The following lemma gives us a nice combinatorial description of circles and will be useful especially in combination with Theorem2.6 and Lemma 2.5.

Lemma 2.7.

[3, Prop. 3] Let be a subgraph of a locally finite connected graph . Then is a circle if and only if is topologically connected and every vertex or end of with has degree or edge-degree in , respectively.

We obtain the following corollary, which is a basic fact for finite graphs.

Corollary 2.8.

Every locally finite connected Hamiltonian graph is -connected.

Proof.

Let be a locally finite connected Hamiltonian graph and suppose for a contradiction that it is not -connected. Fix a subgraph of whose closure is a Hamilton circle of and a cut vertex of . Let and be two different components of . By Theorem 2.6 the circle uses evenly many edges of each of the finite cuts and . Since is a Hamilton circle and therefore topologically connected, we get also that it uses at least two edges of each of these cuts by Lemma 2.5. This implies that has degree at least in which contradicts Lemma 2.7. ∎

3. Topological caterpillars

In this section we close a gap with respect to the general question when the -th power of a graph has a Hamilton circle. Let us begin by summarizing the results in this field. We start with finite graphs. The first result to mention is the famous theorem of Fleischner, Theorem 1.1, which deals with -connected graphs.

For higher powers of graphs the following theorem captures the whole situation.

Theorem 3.1.

[20, 24] The cube of any finite connected graph on at least three vertices is Hamiltonian.

These theorems leave the question whether and when one can weaken the assumption of being -connected and still maintain the property of being Hamiltonian. Theorem 1.3 gives an answer to this question.

Now let us turn our attention towards locally finite infinite graphs. As mentioned in the introduction, Georgakopoulos has completely generalized Theorem 1.1 to locally finite graphs by proving Theorem 1.2. Furthermore, he also gave a complete generalization of Theorem 3.1 to locally finite graphs with the following theorem.

Theorem 3.2.

[14, Thm. 5] The cube of any locally finite connected graph on at least three vertices is Hamiltonian.

What is left and what we do in the rest of this section is to prove lemmas about locally finite graphs covering implications similar to those in Theorem 1.3, and mainly Theorem 1.5, which extends Corollary 1.4 to locally finite graphs. Note first that Theorem 1.3 remains true if we consider locally finite infinite trees and Hamilton circles in where the definition of a caterpillar should now include rays and double rays. Actually the same proof can be used to show this.

Corollary 1.4 is also true for locally finite graphs, but its proof is not trivial anymore. The problem is that for a spanning tree of a locally finite connected graph the topological spaces and might differ not only in inner points of edges but also in ends. More precisely, there might be two equivalent rays in that belong to different ends of . So the Hamiltonicity of does not directly imply the one of . However, for being a spanning caterpillar of , this problem can only occur when contains a double ray such that all subrays belong to the same end of . Then the same construction as in the proof for the implication from (iii) to (i) of Theorem 1.3 can be used to build a spanning double ray in which ends up being a Hamilton circle in . The idea for the construction which is used for this implication is covered in Lemma 3.4.

For an infinite graph the assumption of having a spanning caterpillar is quite restrictive. Such graphs can especially have at most two ends since having three ends would imply that the spanning caterpillar must contain three disjoint rays, which is impossible because it would force the caterpillar to contain a . For this reason we have defined a topological version of a caterpillar, which allows graphs with arbitrary many ends to have a spanning one and yields with Theorem 1.5 an extension of Corollary 1.4 for locally finite graphs. We recall the definition of a topological caterpillar being a topological tree such that is an arc, where denotes the set of all leaves of , i.e., vertices of degree in .

The following basic lemma about topological caterpillars is easy to show and so we omit its proof. It is an analogue of the equivalence of the statements (ii) and (iii) of Theorem 1.3 for topological caterpillars.

Lemma 3.3.

A topological tree is a caterpillar if and only if does neither contain as a subgraph nor an end of vertex-degree at least in it.

Note that we do not get a full extension of Theorem 1.3 to locally finite graphs because has a Hamilton circle if and only if is a topological caterpillar with at most two ends, as noted above.

We continue with another basic lemma, which covers the idea of the proof that statement (iii) of Theorem 1.3 implies statement (i) of Theorem 1.3. We shall also need this in the proof of Theorem 1.5.

Lemma 3.4.

Let be a topological caterpillar in for a locally finite connected graph . Then there exists a partition of the vertices of and a linear order of the partition classes such that:

  1. Any two different vertices belonging to the same partition class have distance  from each other in .

  2. For consecutive partition classes , and with there is a unique vertex in that is not a leaf of and has distance to every vertex of .

Proof.

If has only two vertices, the statement is obvious. So we may assume that  has at least three vertices. Let be the set of leaves of . We know by definition that is an arc . This arc induces a linear order of the vertices of . Using this linear order we define the desired partition of . For consecutive vertices with we define the set (cf. Figure 2). If has a maximal element with respect to , we define an additional set . Should have a minimal element with respect to , we define another additional set . By definition of topological caterpillars, the sets together with and form a partition of where all vertices in a partition class have distance in . This proves part (i).

Note for statement (ii) that the linear order induces a linear order on the partition classes in the following way. For vertices with set . If (resp. ) exists, set (resp. ) for every . Now the definition of the partition classes ensures that for consecutive partition classes , and with the vertex has distance in to every vertex of . For the same is true with the unique vertex by definition. ∎

Figure 2. Partition classes as in Lemma 3.4.

Referring to statement (ii) of Lemma 3.4 let us call the vertex in a partition class  that is not a leaf of the jumping vertex of .

We still need a bit of notation and preparation work before we can prove the main theorem of this section.

Let be a topological spanning caterpillar of a locally finite graph . Next take a partition and a linear order on its classes as in Lemma 3.4. For a vertex  let be the partition class containing . For two vertices  with let .

Now let denote a topological caterpillar with only one graph-theoretical component. Let be a bipartition of the partition classes such that consecutive classes with respect to lie not both in or . Furthermore, let be two vertices, say with , whose distance is even in . We define a square string in as a path in which uses only vertices of partitions that lie in the bipartition class in which and lie and which contains all vertices of partition classes for , but only and from and , respectively. Similarly, we define , and square strings in , but with the difference that they should also contain all vertices of , and , respectively. We call the first two types of square strings that were defined left open and the latter ones left closed. The notion of being right open and right closed is analogously defined. Lemma 3.4 contains the idea of how to construct square strings.

The next lemma gives us two possibilities to decompose a graph-theoretical component of a topological caterpillar that contains a double ray into two, possibly infinite, paths of . Later on we will use these decompositions to connect the parts of all graph-theoretical components of in a certain way such that a Hamilton circle of is formed in the end.

Lemma 3.5.

For a locally finite connected graph , let be a topological caterpillar in with only one graph-theoretical component and which contains a double ray. Furthermore, let and be vertices of with .

  1. If is even, then can disjointly be decomposed into a path and a double ray of as well as into two rays and of with endvertices and , respectively, such that for every and for every .

  2. If is odd, then can disjointly be decomposed into two rays and of with endvertices and , respectively, such that for every and for every as well as into two rays and of with endvertices and , respectively, such that for every and for every .

Proof.

We sketch the proof of statement (i). As path for the first decomposition, we take a square string in with and as endvertices. Depending whether is a jumping vertex or not we take a left open or closed square string, respectively. Depending on we take a right closed or open square string if is a jumping vertex or not, respectively. Since is even, we can find such square strings. To construct a corresponding double ray start with a square string in where and denote the jumping vertices in the partition classes proceeding and , respectively. Using Lemma 3.4 the square string can be extend to a desired double ray containing all vertices of that do not lie in .

For the second decomposition, we start for the definition of with a square string having as one endvertex. For the definition of we distinguish four cases. If and are jumping vertices, we set as a path obtained by taking a square string and deleting from it. If is not a jumping vertex, but  is one, take a square string, delete from it and set the remaining path as . In the case that is a jumping vertex, but is none, is defined as a path obtained from a deleting from a square string. In the case that neither nor is a jumping vertex, we take a square string, delete from it and set the remaining path as . Next we extend using a square string to a path with as one endvertex containing all vertices in partition classes with . We extend the remaining path to a ray that contains also all vertices in partition classes with , but none from partition classes for . The desired second ray can now easily be build in .

The decompositions for statement (ii) are defined in a very similar way (cf. Figure 3). Therefore, we omit their definitions here. ∎

Figure 3. Examples for decompositions as in Lemma 3.5.

The following lemma is essential for connecting parts of decomposed graph-theoretical components of . Especially, here we make use of the structure of instead of arguing only inside of . This allows us basically to build a Hamilton circle using square strings and to “jump over” an end to avoid producing an edge-degree bigger than at that end.

Lemma 3.6.

Let be a topological spanning caterpillar of a locally finite connected graph and where . Then for any two vertices with and there exists an path in .

Proof.

Let the vertices and be as in the statement of the lemma. Now suppose for a contradiction that there is no path in . Then we can find an empty cut of with sides and such that and lie on different sides of it. Since contains an arc, there must exist an end . By definition,  contains an arc with all vertices of that are no leaves in , and every vertex of is only adjacent to finitely many leaves, because is locally finite. Therefore, we can find an open set in containing , but no vertex of . Inside we can find a basic open set around , which contains a graph-theoretical connected subgraph with all vertices of . Now contains vertices of and as well as a path between them, which must then also exist in . Such a path would have to cross contradicting the assumption that is an empty cut in . ∎

To figure out which parts of which decomposed graph-theoretical components of  we can connect such that afterwards we are still able to extend this construction to a Hamilton circle of , we shall use the next lemma. For the formulation of the lemma, we use the notion of splits.

Let be a multigraph and . Furthermore, let such that where for . Now we call a multigraph a -split of if

with and

We call the vertices and replacement vertices of .

Lemma 3.7.

Let be a finite Eulerian multigraph and be a vertex of degree in . Then there exist two -splits and of which are Eulerian too.

Proof.

There are possible non-isomorphic -splits of such that and have degree in the -split. Assume that one of them, call it , is not Eulerian. This can only be the case if is not connected. Let be an empty cut of . Note that has precisely two components and since is Eulerian and has degree in . So and must lie in different sides of , say . Since was connected, we get that and lie in different sides of the cut , say . Therefore, and . If and , set and as -splits of such that the inclusions and hold. Now and are Eulerian, because every vertex has even degree in each of those multigraphs and both multigraphs are connected. To see the latter statement, note that any empty cut of for would need to have and on different sides. If also and are on different sides, we would have , which does not define an empty cut of by definition of . But having and on the same side of the cut , this would induce an empty cut in after identifying and in and yield a contradiction to the assumption that is Eulerian and therefore especially connected. ∎

Now we have all tools together to proof Theorem 1.5.

Proof of Theorem 1.5.

Let be a graph as in the statement of the theorem and let be a topological spanning caterpillar of . We fix a partition of and an order on it as in Lemma 3.4 with respect to where shall denote the partition class containing a vertex . We may assume by Corollary 1.4 that has infinitely many vertices. Now let us fix an enumeration of the vertices, which is possible since every locally finite connected graph is countable. We build a Hamilton circle of inductively in at most many steps where we have two disjoint arcs and in in each step whose endpoints are vertices of subgraphs and of , respectively. Let and (resp.  and ) denote the endvertices of (resp. ) such that (resp. ). For the construction we ensure the following properties in each step :

  1. The vertices and are the jumping vertices of and , respectively.

  2. The partition sets and as well as and are consecutive with respect to .

  3. If holds for any vertex , then .

  4. If for any vertex there are vertices such that and , then is true.

  5. and , but contains the least vertex with respect to the fixed vertex enumeration that was not already contained in .

We start the construction by picking two adjacent vertices and in that are no leaves in . Then and are consecutive with respect to . Since and are cliques by statement (i) of Lemma 3.4, we set to be a Hamilton path of with endvertex and to be one of with endvertex . This completes the first step of the construction.

Suppose we have already constructed and . Let be the least vertex with respect to the fixed vertex enumeration that is not already contained in . We know by our construction that either or for every vertex . Consider the second case, since the argument for the first works analogously. Let be a vertex such that is the predecessor of with respect to and be a vertex such that and is the successor of either or , say . By Lemma 3.6 there exists a path in . We may assume that does not contain an edge whose endvertices lie in the same graph-theoretical component of and that every graph-theoretical component of is incident with at most two edges of . Otherwise we could use square strings to reduce the situation to the assumptions we made.

Next we inductively define a finite sequence of finite Eulerian auxiliary multigraphs for some where every vertex has either degree or in each of these multigraphs and we obtain from as a -split for some vertex of degree until we end up with a multigraph that is a cycle.

As take the set of all graph-theoretical components of that are incident with an edge of . Two vertices and are adjacent if either there is an edge in whose endpoints lie in and or there is a arc in for a subgraph of and vertices and such that no endvertex of any edge of lies in . Since is a topological spanning caterpillar, the multigraph is connected and by definition of it is also Eulerian where all vertices have either degree or .

Now suppose we have already constructed and there exists a vertex with degree in . Since is obtained from via repeated splitting operations, we know that is incident with two edges in that correspond to edges  of and with two edges that correspond to arcs and , respectively, of for subgraphs and of such that neither nor contain an endvertex of an edge of . Let be the graph-theoretical component of in which each of and has an endvertex, say and , respectively. Here we consider two cases:

Case 1.

The distance in between and is even.

In this case we define as a Eulerian -split of such that the edge in corresponding to is either adjacent to the one corresponding to or to the one corresponding to either or with the property that the path in connecting  and (resp.  ) does not contain . This is possible since two of the three possible non-isomorphic -splits of are Eulerian by Lemma 3.7.

Case 2.

The distance in between and is odd.

Here we set as a Eulerian -split of such that the edge in corresponding to is not adjacent to the one corresponding to . As in the first case, this is possible because two of the three possible non-isomorphic -splits of are Eulerian by Lemma 3.7. This completes the definition of the sequence of auxiliary multigraphs.

Now we use the last auxiliary multigraph of the sequence to define the arcs and . Note that is a path in where and lie in the same graph-theoretical components and of as and , respectively. Since we may assume that holds, let denote the edge which contains one endvertex in . Then either the distance between and or between and is even, say the latter one holds. Now we first extend via a square string in and by a square string in  where is the successor of with respect to and is the jumping vertex of . Then we further extend using a ray to contain all vertices of partition classes with for . This is possible by Lemma 3.4.

Next let and be the two edge-disjoint paths in . Since every edge of corresponds to an edge of , we get that corresponds either to or , say to the former one. Therefore, we will use to obtain arcs to extend  and for arcs extending . The way we have defined via splittings ensures that for any vertex of of degree we have performed a -split such that the partition of the edges incident with into pairs of edges incident with a replacement vertex of corresponds to a decomposition of as in Lemma 3.5. So for every vertex of of degree we take such a decomposition. For every graph-theoretical component of such that there exist two consecutive edges and of or that do not correspond to edges of and or holds for every choice of , , and , we take a spanning double ray of . We can find such spanning double rays using Lemma 3.4. Since is a cycle, we can use these decompositions and double rays to extend and to be disjoint arcs  and with endvertices on . With the same construction that we have used for extending and on , we can extend and to have endvertices  and which are the jumping vertices of and , respectively, and containing all vertices of partition classes for and . Then we take these arcs as and where and are the corresponding subgraphs of whose closures give the arcs. By setting and to be and , depending on which of the two arcs or ends in these vertices, we have guaranteed all properties from to for the construction.

Now the properties yield not only that and are disjoint arcs for and , but also that . If there exists neither a maximal nor minimal partition class with respect to , the union  forms a Hamilton circle of by Lemma 2.7. Should there exist a maximal partition class, say for some with jumping vertex , the vertex will also be an endvertex of . In this case we connect the endvertices and of and  via an edge. Such an edge exists since and are consecutive with respect to by property and as well as are jumping vertices by property . Analogously, we add an edge if there exists a minimal partition class. Therefore, we can always obtain the desired Hamilton circle of . ∎

4. Graphs without or as minor

We begin this section with a small observation which allows to strengthen Theorem 1.8 a bit by forbidding subgraphs isomorphic to a instead of minors.

Lemma 4.1.

For graphs without as a minor it is equivalent to contain a as a minor or as a subgraph.

Proof.

One implication is clear. So suppose for a contradiction, we have a graph without a as a minor that does not contain as a subgraph but as a subdivision, which is equivalent to containing a as a minor since is cubic. Consider a subdivided where at least one edge of the corresponds to a path in the subdivision whose length is at least two. Let be an interior vertex of and be the endvertices of . Let the other two branch vertices of the subdivision of be called and . Now we take as branch vertex set of a subdivision of . The vertices and can be joined to and by internally disjoint paths using the ones of the subdivision of except the path . Furthermore, the vertex can be joined to and using the paths and . So we can find a subdivision of in the whole graph, which contradicts our assumption. ∎

Before we start with the proof of Theorem 1.8 we need to prepare two structural lemmas. The first one will be very convenient to control end degrees because it bounds the size of certain separators.

Lemma 4.2.

Let be a -connected graph without as a minor and be a connected subgraph of . Then holds for every component of .

Proof.

Let , and be defined as in the statement of the lemma. Since is -connected, we know that holds. Now suppose for a contradiction that contains three vertices, say and . Pick neighbours and of and , respectively, in for . Furthermore, take a finite tree in whose leaves are precisely , and for . This is possible because and are connected. Now we have a contradiction since the graph with and forms a subdivision of . ∎

For a connected graph with a subgraph let denote the graph which is formed by taking and contracting all components of where we delete multiple edges or loops. Obviously is connected if was connected. We can push this observation a bit further towards -connectedness with the following lemma.

Lemma 4.3.

Let be a connected subgraph with at least three vertices of a -connected graph . Then is -connected.

Proof.

Suppose for a contradiction that is not -connected for some and as in the statement of the lemma. Since has at least three vertices, we obtain that has at least three vertices too. So there exists a cut vertex in . If is also a vertex of and therefore does not correspond to a contracted component of , then would also be a cut vertex of , which contradicts the assumption that is -connected.

Otherwise corresponds to a contracted component of . Since vertices of that correspond to contracted components of are not adjacent by definition of and , as a cut vertex in , must have at least one neighbour in each component of , we get in particular that separates two vertices, say and , of that do not correspond to contracted components of . This yields a contradiction because is connected and therefore contains an path, which still exists in