Graph-Like Compacta: Characterizations and Eulerian Loops
aa r X i v : . [ m a t h . C O ] S e p GRAPH-LIKE COMPACTA:CHARACTERIZATIONS AND EULERIAN LOOPS
BENJAMIN ESPINOZA, PAUL GARTSIDE, AND MAX PITZ
Abstract. A compact graph-like space is a triple ( X, V, E ) where X is acompact, metrizable space, V ⊆ X is a closed zero-dimensional subset, and E is an index set such that X \ V ∼ = E × (0 , Introduction
Locally finite graphs can be compactified, to form the Freudenthal compactifi-cation, by adding their ends. This topological setting provides what appears tobe the ‘right’ framework for studying locally finite graphs. Indeed, many classicaltheorems from finite graph theory that involve paths or cycles have been shown togeneralize to locally finite infinite graphs in this topological setting, while failingto extend in a purely graph theoretic setting. See the survey series [10]. Morerecently, compact graph-like spaces were introduced by Thomassen and Vella, [22],as a natural class encompassing graphs, and in particular containing the standardsubspaces of Freudenthal compactification of locally finite graphs.A compact graph-like space is a triple (
X, V, E ) where: X is a compact, metriz-able space, V ⊆ X is a closed zero-dimensional subset, and E is a discrete indexset such that X \ V ∼ = E × (0 , V and E are the vertices and edges of X respectively. More generally, a topological space X is compact graph-like, if thereexists V ⊆ X and a set E such that ( X, V, E ) is a compact graph-like space. Recallthat connected compact metrizable spaces are called continua , and so a graph-likecontinuum is a continuum which is graph-like.Papers in which graph-like spaces have played a key role include [22] where sev-eral Menger-like results are given, and [8] where algebraic criteria for the planarityof graph-like continua are presented. In [2], aspects of the matroid theory for graphshave been generalized to infinite matroids on graph-like spaces.In this paper we present two groups of new results. The first group consists ofcharacterizations of compact graph-like spaces and continua. These connect graph-like continua to certain classes of continua which have been intensively studiedby continua theorists. We also establish that compact graph-like spaces are notsimply ‘like’ the Freudenthal compactifications of locally finite graphs, but in fact are standard subspaces of the latter. Our second group of results consists of various
Mathematics Subject Classification.
Key words and phrases.
Infinite graph; locally finite graph; end; Freudenthal compactification;graph-like space; topology; Eulerian. characterizations of when a graph-like continuum is Eulerian. These naturallyextend classical results for graphs.1.1.
The Main Theorems.
In Section 2 we give various characterizations and rep-resentations of compact graph-like spaces, and graph-like continua, which demon-strate that graph-like continua form a class of continua which are also of consid-erable interest from the point of view of continua theory. These results can besummarized as follows.
Theorem (A) . The following are equivalent for a continuum X :(i) X is graph-like,(ii) X is regular and has a closed zero-dimensional subset V such that all pointsoutside of V have order ,(iii) X is completely regular,(iv) X is a countable inverse limit of finite connected multi-graphs with onto,monotone, simplicial bonding maps with non-trivial fibres at vertices only, ( iv ) ′ X is a countable inverse limit of finite connected multi-graphs with onto,monotone bonding maps that project vertices onto vertices, and(v) X is homeomorphic to a connected standard subspace of a Freudenthal com-pactification of a locally finite graph. Here a continuum is regular if it has a base all of whose members have finiteboundary, and completely regular if all non-trivial subcontinua have non-empty inte-rior. A map is monotone if all fibres are connected, while a map between graph-likespaces is simplicial if it maps vertices to vertices, and edges either homeomorphi-cally to another edge, or to a vertex. A standard subspace of a compact graph-likespace is a closed subspace that contains all edges it intersects. The equivalence of (i)and (ii) is analogous to a well-known topological characterization of finite graphs,namely a continuum is a graph if and only if every point has finite order, and all butfinitely many points have order 2, [19, Theorem 9.10 & 9.13]. The equivalence of (i)and (iv) provides a powerful tool to lift results in finite graph theory to graph-likecontinua. Indeed this is key to our results on Eulerian paths and loops below. Italso is key to the equivalence of (i) and (v). The equivalence of (i) and (iii) yieldsa purely internal topological characterization of graph-like continua, without anyreference to distinguished points, ‘vertices’, or subsets, ‘edges’.We prove all of Theorem (A) taking ‘compact graph-like space’ as the basicnotion. Because ‘compact graph-like’ takes a middle ground between topology andgraph theory, our proofs are clean and efficient. However it is important to notethat the equivalence of (i) and (iii) follows, modulo some basic lemmas, from a resultof Krasinkiewicz, [16], while the implication (iii) implies (iv) is essentially shownby Nikiel in [20]. Nikiel also claimed, without proof, the converse implication.Regarding (v), Bowler et al. have claimed, without proof, the weaker assertion thatevery compact graph-like space is a minor (essentially: a quotient) of a Freudenthalcompactification of some locally finite graph, [2, p. 6].In Sections 3 and 4 we extend some well-known characterizations of Euleriangraphs to graph-like continua. Let G be a (multi-)graph. A trail in G is an edgepath with no repeated edges. It is open if the start and end vertices are distinct,and closed if they coincide. We also call closed trails circuits . A segment is a trailwhich does not cross itself. A cycle is a circuit which never crosses itself. A trailis Eulerian if it contains all edges of the graph. (Note that an
Eulerian circuit is a
RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 3 closed Eulerian trail.) The graph G is Eulerian (respectively, closed Eulerian ) if ithas an open (respectively, closed) Eulerian trail; and
Eulerian if it either open orclosed Eulerian. Call a vertex v of a graph G even (respectively, odd if the degreeof v in G is even (respectively, odd).Classical results of Euler and Veblen characterize multi-graphs with closed, andrespectively, open, Eulerian trails as follows. Let G be a connected multi-graphwith vertex set V , then the following are equivalent: (i) G is closed [open] Eulerian,(ii) every vertex is even [apart from precisely two vertices which are odd], (iii) [thereare vertices x = y such that] for every bi-partition of V , the number of cross edgesis even [if and only if x and y lie in the same part], and (iv) the edges of G can bepartitioned into edge disjoint cycles [and a non-trivial segment]. We extend theseresults to compact graph-like spaces, and prove the following result. Theorem (B) . Let X be a graph-like continuum with vertices V . The followingare equivalent:(i) X is closed [open] Eulerian,(ii) every vertex is even [apart from precisely two vertices which are odd],(ii) ′ every vertex has strongly even degree [apart from precisely two vertices whichhave strongly odd degree],(iii) [there are vertices x = y such that] for every partition of V into two clopenpieces, the number of cross edges is even [if and only if x and y lie in thesame part], and(iv) the edges of X can be partitioned into edge-disjoint circles [and a non-trivialarc].Further, if X is closed [open] Eulerian then either X has continuum many dis-tinct Eulerian loops [Eulerian paths], or has a finite number of distinct Eulerianloops [Eulerian paths], which occurs if and only if X is homeomorphic to a finiteclosed [open] Eulerian graph. Let X be a compact graph-like space with set of vertices V . A subspace of X iscalled an arc if it is homeomorphic to I = [0 , circle if it is homeomorphicto the circle, S . A (standard) path is a continuous map f : I → X such that f (0) and f (1) are vertices, f is injective on the interior of every edge and f − ( V )has empty interior. Note that every continuous map f : I → X with f (0) , f (1) asvertices is homotopy equivalent (with fixed endpoints) to a path. Also note that if X is a graph (with usual topology), then every path yields a corresponding trail,and every trail corresponds to a path. A path, f , is open if f (0) = f (1), and closed if f (0) = f (1). Closed paths are called loops . A path (or loop) is Eulerian if itsimage contains every edge. Note that in a graph with the usual topology there is anatural correspondence between Eulerian paths and Eulerian trails, and Euleriancircuits and Eulerian loops. We abbreviate ‘closed and open’ to ‘clopen’. A vertex v is odd (resp. even ) if and only if there exists a clopen subset A of V containing v ,such that for every clopen subset C of the vertices V with v ∈ C ⊆ A the numberof edges between C and V \ C is odd (resp. even).The equivalence of (i), (ii), (iii) and (iv) in Theorem (B) is established in Sec-tion 3.1. At the heart of our proof is our representation of compact graph-likespaces as inverse limits, and an induced inverse limit representation of all Eulerianloops (possibly empty, of course). The ‘further’ part of Theorem (B) follows inSection 3.2 from topological considerations of the space of all Eulerian loops. BENJAMIN ESPINOZA, PAUL GARTSIDE, AND MAX PITZ
Our definition of ‘even’ and ‘odd’ vertices is natural within the context of The-orem (B). An alternative approach to degree, due to Bruhn and Stein [3], leads tothe notions of ‘strongly even degree’ and ‘strongly odd degree’ appearing in item(ii) ′ of Theorem (B). See Section 4 for details and the proof that ‘(ii) implies (ii) ′ ’and ‘(ii) ′ implies (i)’. Alternative Paths.
Theorem (A) shines an unexpected light on connections be-tween concepts from continua theory (completely regular continua and inverse lim-its of graphs), and concepts arising from infinite graph theory (graph-like continua,Freudenthal compactifications of graphs, and their standard subspaces). As a re-sult we discover that the various parts of Theorem (B) generalize numerous resultsin the literature, and–with the considerable assistance of the machinery developedhere–Theorem (B) can be derived from older work.For Freudenthal compactification of graphs, the equivalence of (i), (iii) and (iv)is due to Diestel & K¨uhn, [11, Theorem 7.2], while the equivalence with (ii) ′ isdue to Bruhn & Stein, [3, Theorem 4]. For standard subspaces of Freudenthalcompactification of graphs, the equivalence of (iii) and (iv) is due to Diestel &K¨uhn, [12, Theorem 5.2], the equivalence of (i) and (iv) is due to Georgakopoulos,[14, Theorem 1.3], and the equivalence (ii) ′ and (iii) is due to Berger & Bruhn,[1, Theorem 5]. It should also be noted that the method of lifting Eulerian pathsand loops via inverse limits, used by Georgakopolous, was previously introduced byBula et al. in [4, Theorem 5].Thus an alternative path to proving Theorem (B) is as follows. Let X be agraph-like continuum. According to the equivalence of (i) and (v) in Theorem (A),which depends on the equivalence of (i) and (iv), X is homeomorphic to a standardsubspace of a Freudenthal compactification of a graph. Now equivalence of (i), (ii) ′ ,(iii) and (iv) follows from the results cited immediately above. To add equivalenceof (ii) apply Theorem 22 (which uses Theorem (A) (i) ⇐⇒ (iv), and a non-trivialinverse limit argument) and Lemmas 19 and 23. Although this alternative pathexists, the direct proofs given here in Sections 3 and 4, using compact graph-likespaces as the basic notion, are–in the authors’ view–much shorter and more natural.1.2. Examples.
Before proving our results on graph-like continua, we now intro-duce some examples. With these examples we have three objectives. First show alittle of the variety of graph-like continua. Second elucidate some of the less familiarterms in Theorem (B), in particular ‘even’ and ‘odd’ vertices. Third demonstratethe remarkable complexity of Eulerian loops and paths in graph-like continua. Thiscomplexity highlights the hidden depths of Theorem (B).
Example 1.
The two-way infinite ladder with single diagonals, which is the infinitegraph G shown below. Notice that all its vertices are even, but it has no Eulerianloop.The Freudenthal compactification, γG , of G adds two ends. Then, as shown inthe diagram, γG has an open Eulerian path from one end to the other. RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 5
It follows from Theorem (B) that the two ends are odd. We now demonstratethat the left end is odd from the definition. For the clopen neighborhood A take theleft end along with all vertices to the left of some rung of the ladder. Now consideran arbitrary clopen C containing the left end and contained in A (depicted by thegreen vertices in the diagram below). Then C is the disjoint union of a C whichcontains the end and all vertices to the left of a rung, and a C which is a finitesubset of A \ C . In the diagram we see that the number of edges from C to V \ C is 9, which is odd. In general, if we identify C (and all edges between membersof C ) to a vertex v and identify V \ A to a vertex w , then we get a finite graphwith exactly two odd vertices (namely, v and w , both of degree 3). Hence from theequivalence of (ii) and (iii) of the graph version of Theorem (B), we see that thenumber of edges from C to V \ C is odd – as required for the left end to be odd. A V \ AC For the next two examples let C denote the standard ‘middle thirds’ Cantorsubset of I . Example 2.
The Cantor bouquet of semi-circles, CBS. The vertices are C × { } inthe plane, along with semi-circular edges centered at the midpoint of each removedopen interval. Note that CBS is not a Freudenthal compactification of graph.12The vertex = (0 ,
0) is neither odd nor even, and hence CBS is not Eulerian.Indeed, as indicated on the diagram, there is one (odd) edge connecting all thevertices in the ‘left half’ of the vertices to its complement (the ‘right half’), but two(even) edges connecting the ‘left quarter’ to its complement. Similarly we see that
BENJAMIN ESPINOZA, PAUL GARTSIDE, AND MAX PITZ every clopen neighborhood of contains two clopen neighborhoods of of whichone has an odd number of edges to its complement, and the other an even number. Example 3.
The Cantor bouquet of circles, CBC, can be obtained from the Cantorbouquet of semi-circles by reflecting it in the real axis. One can check that allvertices are even. The diagram illustrates an Eulerian loop in CBC. I CBCSuppose f : I → X is a standard path in a graph-like continuum X with vertices V and (open) edges ( e n ) n . Then f − ( V ) is a closed nowhere dense subset of I , andits complement, f − ( S n e n ) is dense and a disjoint union of open intervals. Thiscountable family, { f − ( e n ) : n ∈ N } inherits an order from the order on I . So toevery path f we can associate a countable linear order L f , which we informally callthe shape of f .To illustrate this, consider L = L f where f is the Eulerian loop in the Cantorbouquet of circles diagrammed above. Then f traverses the top edge from left toright, covers the right-hand copy of CBC, traverses the bottom edge from rightto left, and then covers the left-hand copy of CBC. So L satisfies the equation L = 1 + L + 1 + L . It follows that L is an infinite ordinal. Thus L = 1 + L , andwe see L = L + L . The first infinite ordinal which is a fixed point under additionof linear orders is the ordinal ω ω . Hence L = ω ω .We now see how to construct for each countable linear order L a graph-likecontinuum X L with an Eulerian loop f so that L f = L . To do so recall: everycountable linearly ordered set L can be realized (is order isomorphic to) a countablefamily of disjoint open subintervals of I , with dense union. For further material onthe interaction of linear orders and graph-like compacta, see [2, § S , in the plane the ‘circle with diameter S ’ is the circlewith center the midpoint of the line segment, and radius half the length of thesegment. Example 4.
Let L be a countable linear order. Fix a family U of pairwise disjointopen subintervals of I , with dense union, which is order isomorphic to L . Define X L to be the subspace of the plane obtained by starting with X = I × { } , and foreach U in U , remove U × { } from X and add the circle with diameter U × { } .The Eulerian loops on X L are naturally bijective with all functions ρ : L → {± } .To see this take any ρ : L → {± } . Since U and L are isomorphic we can think that RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 7 the domain of ρ is actually U . Define g ρ : [0 , → X L by requiring (i) g ( t ) = t on I \ S U , and (ii) on U in U the path g traverses the top (resp. bottom) semi-circlein X L corresponding to U if ρ ( U ) = +1 (resp. ρ ( U ) = − f ρ –the desired Eulerian loop – by f ρ ( t ) = g ρ (2 t ) on [0 , /
2] and f ρ ( t ) = g − ρ (2 − t )on [1 / , , /
2] the path f ρ travels from left to right along X L crossing the circles by either taking the upper or lower semi-circles depending on ρ ; and then on [1 / ,
1] it travels across X L from right to left taking the oppositeupper/lower semi-circles than before. Every Eulerian loop arises in this way, andobserve that they all have the same shape, L .The following diagram depicts X Q where Q is the linearly ordered set of dyadicrationals in (0 , Q is order isomorphic to the rationals, Q . QX Q The graph-like continuum X Q provides an example of the difficulties involved inna¨ıvely trying to lift arguments for graphs to graph-like continua. In the standardproof of Theorem (B) for graphs one moves from (iv) ‘the edges of the graph can bedecomposed into disjoint cycles’ to (i) ‘there is an Eulerian circuit’ by amalgamatingthe cycles, one after another to form the circuit. Notice that in X Q there is acanonical decomposition of X Q into edge disjoint circles – namely the circles in thedefinition of X Q . But these circles are pairwise disjoint . Hence there is no naturalmethod of amalgamating them into an Eulerian loop for X Q . Example 5.
The Hawaiian earring, H , is also Eulerian. Unlike the X L examplesabove, every countable linear order can be realized as the L f of an Eulerian loop.Write H as = (0 ,
0) (the sole vertex) and the unionof circles in the plane C n , for n ∈ N , where C n hasradius 1 /n and is tangential at to the x -axis.We can identify the Eulerian loops in the Hawaiianearring as follows. For any countable linear order L andfunction ρ : L → N × {± } such that π ◦ ρ : L → N isa bijection, there is a naturally corresponding Eulerianloop f ρ of H . Indeed, given L and ρ , let U be a familyof pairwise disjoint open subintervals of I , with denseunion, which is order isomorphic to L (and identifythem). Define f ρ to have value on the complementof S U , and on U in U , writing ρ ( U ) = ( n, i ), it shouldtraverse C n clockwise (respectively, anticlockwise) if i = +1 (respectively, i = − BENJAMIN ESPINOZA, PAUL GARTSIDE, AND MAX PITZ Properties and Characterizations of Graph-like continua
Basic Properties.
Most of the following basic properties of graph-like spacesare well-known, see e.g. [22]. Nonetheless, it might be helpful to give a self-containedoutline of the most important properties we use.Let (
X, V, E ) be a compact graph-like space. We often identify the label, e , ofan edge, with the subspace e × (0 ,
1) of X . Note that since V is zero-dimensional,for every edge e , the closure, e , of e adds at most two vertices – the ends of theedge – and e is homeomorphic to the circle, S , or I = [0 , separation ( A, B ) of a graph like space X is a partition of V ( X ) into twodisjoint clopen subsets. The cut induced by the separation ( A, B ) is set of edges withone end vertex in A and the other in B , denoted by E ( A, B ). More generally, we calla subset F ⊂ E a cut if there is a separation ( A, B ) of X such that F = E ( A, B ). A multi-cut is a partition U = { U , U , . . . , U n } of V ( X ) into pairwise disjoint clopensets. For each two U i , U j , not necessarily different, E ( U i , U j ) denotes the set ofedges with one endpoint in U i and the other endpoint in U j . By X [ U i ] we denotethe induced subspace of X , i.e. the closed graph-like subspace with vertex set U i and edge set E ( U i , U i ). Finally, a clopen subset U ⊂ V ( X ) is called a region if theinduced subspace X [ U ] is connected. Lemma 1.
In a compact graph-like space, all cuts are finite.Proof.
Suppose there is an infinite cut F = { f n : n ∈ N } induced by a separation( A, B ) of a graph-like space X . Then A and B are disjoint closed subsets of X , soby normality there are disjoint open subsets U ⊇ A and V ⊇ B . Since edges areconnected, there exist x n ∈ f n \ ( U ∪ V ) for all n . It follows that { x n : n ∈ N } isan infinite closed discrete subset, contradicting compactness. (cid:3) Lemma 2.
Let X be a compact graph-like space. For every vertex v of X andany open neighborhood U of v , there is a clopen C ⊂ V ( X ) such that v ∈ C and X [ C ] ⊂ U . Moreover, if X is connected, then C can be chosen to be a region.Proof. Since V ( X ) is totally disconnected we have { v } = \ { X [ A ] : ( A, B ) a separation of
X, v ∈ A } . Now T X [ A ] ⊂ U and compactness implies that there is a finite subcollection A , . . . , A n such that for the clopen set B = A ∩ · · · ∩ A n we have v ∈ X [ B ] = X [ A ] ∩ · · · ∩ X [ A n ] ⊂ U. For the moreover part, since E ( B, V \ B ) is finite by Lemma 1, it follows fromconnectedness of X that X [ B ] consists of finitely many connected components, say X [ B ] = X [ C ] ⊕ · · · ⊕ X [ C k ], one of which contains the vertex v . This is our desiredregion C . (cid:3) Definition 3.
Let X be a graph-like space and U be a multi-cut of X . The multi-graph associated with U is the quotient space G ( U ) = X/ { X [ U ]: U ∈ U} . The map π U : X → G ( U ) denotes the corresponding quotient map.We remark that G ( U ) is indeed a finite multi-graph. The identified X [ U ] forma finite collection of vertices, which are connected by finitely many edges (seeLemma 1). The degree of π U ( U i ) in G ( U ) is given by | E ( U i , V \ U i ) | < ∞ . Ournext proposition gathers properties of graphs associated with multi-cuts.
RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 9
Proposition 4.
Let X be a graph-like compact space. Then(1) X is connected if and only if G ( U ) is connected for all multi-cuts U of X .(2) All cuts of X are even if and only if all vertices in G ( U ) have even degreesfor all multi-cuts U of X .Proof. (1) If X is connected, then connectedness of G ( U ) follows from the fact thatit is the continuous image of X . Conversely, a disconnection of X gives rise to a G ( U ) which is the empty graph on two vertices.(2) If every cut of X is even, then the above degree considerations show thatevery vertex in G ( U ) has even degree. And conversely, any odd cut of X gives riseto a graph G ( U ) on two vertices of odd degree. (cid:3) Recall that a standard subspace Y of a graph-like space X is a closed subspacethat contains all edges it intersects (i.e. whenever e ∩ Y = ∅ then e ⊂ Y ). Standardsubspaces of graph-like spaces are again graph-like. Write E ( Y ) for the collectionof edges of Y . Lemma 5.
Let X be a graph-like space and C ⊆ X a copy of a topological circle.Then C is a standard subspace.Proof. Assume, by contradiction, that there exists e ∈ E ( X ) such that e ∩ C = ∅ and that e C . Let y ∈ e \ C . Then there exist x ∈ C with the properties thatthe arc [ x , y ] is a subset of e and [ x , y ] ∩ C = { x } . Observe that x V . Let U be an open set containing x such that U ∩ V = ∅ . Let α be the component in[ x , y ] of x contained in U and β be the component in C of x contained in U .Then α ∪ β contains a triod and α ∪ β ⊂ X \ V which is a contradiction to the factthat X \ V ∼ = E × (0 ,
1) contains no triods. (cid:3)
Lemma 6.
Let X be a graph-like space and C ⊆ X a copy of a topological circle.Then E ( C ) ∩ F is finite and even for all cuts F = E ( A, B ) of X .Proof. By Lemma 5 we may assume X = C . Let F = E ( A, B ). That F is finite isimmediate from Lemma 1, so we only need to prove that | F | is even.Let C [ A ] (resp. C [ B ]) be the standard subspace containing A (resp. B ) and alledges with both endpoints in A (resp. B ). Observe that(a) C = C [ A ] ∪ F ∪ C [ B ], and(b) C [ A ] and C [ B ] have finitely many components.Let A , . . . , A r be the components of C [ A ] and B , . . . , B s be the components of C [ B ]. These components induce a multi-cut, U = { U A , . . . , U A r , U B , . . . , U B s } , ofthe vertices of C where U A i (resp. U B i ) consists of all vertices contained in A i (resp. B i ). Then G ( U ), the multi-graph associated with U , is a cycle whose edges are theelements of F and whose vertices are the equivalence classes containing the sets U A , . . . , U B s . Observe that the sets A = { U A , . . . , U A r } and B = { U B , . . . , U B s } give a 2-coloring of the vertices of G ( U ). Hence G ( U ) has an even number of edges,i.e. | F | is even. (cid:3) Characterizations and Representations.
In this section we prove Theo-rem (A). The equivalence of (i) and (iii) is given by Proposition 10, the equivalenceof (i) and (ii) is Theorem 11, while the equivalence of (i), (iv) and (iv)’ follows fromTheorems 13 and 14. Compact graph-like spaces were explicitly defined to includestandard subspaces of the Freudenthal compactification of locally finite graphs.Theorem 15 provides the converse, establishing equivalence of (i) and (v).
Recall that a continuum X is regular if it has a basis of open sets, each with finiteboundary, and it is called completely regular if each non-degenerate subcontinuumof X has non-empty interior in X , see [4, Page 1176]. A continuum is hereditar-ily locally connected (hlc) if every subcontinuum is locally connected, and finitelySouslian if each sequence of pairwise disjoint subcontinua forms a null-sequence, i.e.the diameters of the subcontinua converge to zero. It is known that for continua( ‡ ) completely regular ⇒ regular ⇒ finitely Souslian ⇒ hlc ⇒ arc-connected.For the first three implications, see [16, Proposition 1.1]. Lemma 7.
Every compact graph-like space is regular.Proof.
Let X be a compact graph-like space, p ∈ X , and U be an open of X setsuch that p ∈ U . We will show that there is an open set O with finite boundarysuch that p ∈ O ⊆ U .The case when p is in the interior of an edge follows from the fact that the set ofedges is discrete. So we may assume p ∈ V . For this case let X [ B ] as in the proof ofLemma 2, then p ∈ X [ B ] ⊆ U . Now for each e ∈ E ( B, V \ B ), let ( v, x e ) be a subarcof e such that ( v, x e ) ⊆ U and such that v ∈ B . Since cuts are finite, then thereare only finitely many of these arcs. The desire open set O is then X [ B ] ∪ { ( v, x e ) : e ∈ E ( B, V \ B ) } as its boundary is the set { x e : e ∈ E ( B, V \ B ) } . (cid:3) Corollary 8.
Every graph-like continuum is finitely Souslian, hereditarily locallyconnected and arc-connected.Proof.
By Lemma 7 and ( ‡ ), this is a consequence of regular.For a direct proof that graph-like continua are finitely Souslian, suppose fora contradiction that { A i : i ∈ N } forms a sequence of disjoint subcontinua of X with non-vanishing diameter. It follows from the sequential compactness of thehyperspace of subcontinua, [19, Corollary 4.18], that there is a subsequence A i j such that A = lim j →∞ A i j = S j A i j \ S j A i j is a non-trivial subcontinuum of X .But since edges are open, we also have that A ⊂ V ( X ), so is totally disconnected,a contradiction.For a direct proof that graph-like continua are hlc, see Lemma 2. (cid:3) In particular, noting that a compact graph-like space has at most countablymany edges (as they form a collection of disjoint open subsets), it follows that theedges of X form a null-sequence, i.e. lim n →∞ diam( e n ) = 0. Here, for a subset A of a metric space, we denote by diam( A ) the diameter of A .In the next theorem we use the following notation. For a subspace A ⊂ X wedenote by Bd( A ) its boundary. A subarc A ⊂ X is called free if A removed itsendpoints is open in X . Theorem 9 ([16, Theorem 1.3]) . A continuum X is completely regular if and onlyif there exists a -dimensional compact subset F of X and a finite or countable nullsequence of free arcs A , A , . . . in X such that X = F ∪ (cid:16)[ { A n : n ≥ } (cid:17) and A j ∩ F = Bd( A j ) for each j ≥ X is a graph-like continuum, then the set of RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 11 vertices V is zero-dimensional. Also by Corollary 8, E ( X ) forms a null sequence.By Theorem 9, X is a completely regular continuum. Proposition 10.
Let X be a continuum. Then X is completely regular if and onlyif X is a graph-like space. Recall that a graph can be characterized in terms of order: a continuum is agraph if and only if every point has finite order, and all but finitely many pointshave order 2, [19, Theorem 9.10 & 9.13].
Theorem 11 (Graph-like Characterization) . A continuum is graph-like if and onlyif it is regular and has a closed zero-dimensional subset V such that all points outsideof V have order .Proof. Sufficiency follows from the definition of graph-like and Lemma 7.For the necessity, first observe that regular implies local connectedness. Let V ⊂ X be a closed zero-dimensional collection of points in X such that all pointsoutside of V have order 2. By local connectedness, all components of X \ V areopen subsets of X . In particular, we have at most countably many components,and each component is non-trivial, non-compact, and consists exclusively of pointsof order 2. So each component is homeomorphic to an open interval. So all thatremains to show for graph-like is that the closure of each edge is compact, which isautomatic. (cid:3) Corollary 12 (Canonical Representation of Graph-like Spaces) . Let X = S be agraph-like continuum. Then there is a unique minimal set V ⊂ X which witnessesthat X is a graph-like space. We call ( X, V, E ) the standard representation of X .Proof. Let { V s : s ∈ S } be the collection of all subsets of X which witness that X is graph-like. We claim that V = T s ∈ S V s is also a vertex set.Clearly, V is closed and zero-dimensional. Further, if x / ∈ V , then x / ∈ V s forsome s ∈ S , so x has order 2. So either V is empty, in which case X ∼ = S ; or V is non-empty, in which case every component of X \ V is non-compact, open, andconsists of points of order 2, so is homeomorphic to an open interval. (cid:3) Our next theorem has been proved, for completely regular continua, by Nikiel,[20, 3.8]. We reprove his theorem here (and extend it to graph-like compacta),phrased for convenience in the language of graph-like continua.
Theorem 13 (Inverse Limit Representation) . Every graph-like compact space X can be represented as an inverse limit of multi-graphs G n ( n ∈ N ) with onto sim-plicial bonding maps that have non-trivial fibres at vertices only, such that(1) X connected if and only if G n is connected for all n , and(2) all cuts E ( A, B ) in X are even ⇔ all vertices in G n are even for all n .Moreover, if X is connected, then the bonding maps can additionally be chosenmonotone.Proof. Let X be a graph-like continuum with vertex set V and edge set E . Withoutloss of generality, X contains no loops, as otherwise we can subdivide each edgeonce (this does not change the homeomorphism type of X , and the new edge setis still a dense open subset, so the new vertex set is a compact, zero-dimensionalsubspace as required).Since V is a compact, zero-dimensional metrizable space, we can find, as inLemma 2, a sequence of multi-cuts {U n : n ∈ N } such that (a) U n +1 is a refinement of U n ,(b) S n ∈ N U n forms a basis for V ( X ), andWriting U n = { U n , U n , . . . , U ni ( n ) } we observe that every v ∈ V has a unique descrip-tion in terms of { v } = T n ∈ N U nl ( v ) and that conversely, for every nested sequence ofcut elements, there is precisely one vertex in T n ∈ N U nl n by compactness and ( b ). The inverse system:
Let {U n : n ∈ N } be as above. To simplify notation, let q n stand for π U n . For each n ∈ N let f n : G ( U n +1 ) → G ( U n ) be defined as f n ( x ) = q n (cid:0) q − n +1 ( x ) (cid:1) for all x ∈ G ( U n +1 ) . Observe that if U n +1 i , U n +1 j ⊂ U ns ,(i) then f (cid:0) q n +1 ( U n +1 i ) (cid:1) = f (cid:0) q n +1 ( U n +1 j ) (cid:1) = q n ( U ns );(ii) and if e ∈ E ( U n +1 i , U n +1 j ); in particular e ∈ E ( U ns , U ns ), then f n ( e ) = q n ( U ns ).In particular, each f n is an onto simplicial map with non-trivial fibres only atvertices of G ( U n ). Then { G ( U n ) , f n } n ∈ N is an inverse sequence of multi-graphs.Hence, its inverse limit is compact and nonempty. We will show that there is acontinuous bijection f : X → lim ←−−− n ∈ N G ( U n ) . For x ∈ X , we define f ( x ) = ( q ( x ) , q ( x ) , . . . ). By the product topology, thisis a continuous map into the product Q n G ( U n ), as all coordinate maps q n arecontinuous. Moreover, it is straightforward from the definition of f n to check that f ( x ) ∈ lim ←− G ( U n ). That the map f is surjective follows from the fact that each q n is continuous and X is compact (see [19, 2.22]). Finally, f is injective because ofthe neighborhood bases requirement (b) on U n . Since X is compact and lim ←− G ( U n )is Hausdorff, it follows that f is a homeomorphism as desired, and properties (1)and (2) now follow from Proposition 4.For the moreover part, simply require that besides (a) and (b), our sequence ofmulti-cuts {U n : n ∈ N } also satisfies(c) every multi-cut U n partitions V ( X ) into regions.That this is possible follows from Lemma 2; and clearly, property (c) implies thateach f n as defined above will be a monotone map. (cid:3) In fact, a converse of the above theorem holds. This has been mentioned, forcompletely regular continua, by Nikiel, [20, 3.10(i)], though without proof. Weprovide the proof in the language of graph-like continua.
Theorem 14.
Let X be a countable inverse limit of connected multi-graphs X n with finite vertex sets V ( X n ) and onto monotone bonding maps f n : X n +1 → X n satisfying: (+) f n ( V ( X n +1 )) ⊆ V ( X n ) . Then X is a graph-like continuum.Proof. By Theorem 11, every regular continuum with the property that all but aclosed zero-dimensional subset of points are of order 2 is a graph-like continuum.That X is regular follows from [20, 3.6]. For sake of completeness, we providethe argument. Let π n : X → X n denote the projection maps, and for m ≥ n write f m,n = f n ◦ f n +1 ◦ · · · ◦ f m − ◦ f m : X m +1 → X n . Claim: For every n ∈ N , the set P n = (cid:8) y ∈ X n : (cid:12)(cid:12) π − n ( y ) (cid:12)(cid:12) > (cid:9) is countable. RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 13
This holds, because for every m ≥ n , the set Q m = (cid:8) y ∈ X n : (cid:12)(cid:12) f − m,n ( y ) (cid:12)(cid:12) > (cid:9) is countable: By assumption, all bonding maps f m are monotone, and hence so is f m,n . Thus, the collection of non-degenerate f − m,n ( y ) from a disjoint collections ofsubcontinua of X m , all with non-empty interior. It follows that P n = S m ≥ n Q m iscountable, completing the proof of the claim.To conclude that X is regular, let x ∈ X and let U be an open neighborhood of x ∈ X . Then there is k ∈ N and an open subset W ⊂ X k with x ∈ π − k ( W ) ⊂ U .Note that since X k is a graph, and P k is countable by the claim, we may choose W with finite boundary such that Bd( W ) ∩ P k = ∅ . It follows that π − k ( W ) has finiteboundary, as well.Our candidate for the vertex set of X is V ( X ) = T n ∈ N π − n ( V ( X n )). By (+), thefamily { ( V ( X n ) , f n ) : n ∈ N } gives a well-defined inverse limit, which is identicalwith our vertex set, i.e. V ( X ) = lim ← { V ( X n ) , f n } . Since all V ( X n ) are finitediscrete sets, it follows that V ( X ) is a compact zero-dimensional metric space, asdesired.To see that elements y ∈ X \ V ( X ) have order 2, note that y / ∈ V ( X ) meansthere is an index N ∈ N such that π n ( y ) is an interior point of an edge of X n forall n ≥ N . Consider an open neighborhood U with y ∈ U ⊂ X . As before, thereis an index k > N and an open subset W ⊂ X k with y ∈ π − k ( W ) ⊂ U . Since π k ( y ) ∈ X k is an interior point of an edge, and P k is countable by the claim, wemay assume that W has 2-point boundary with Bd( W ) ∩ P k = ∅ . It follows that π − k ( W ) has a 2-point boundary, as well. (cid:3) In fact, the class of continua, which can be represented as countable monotoneinverse limits of finite connected multi-graphs are precisely the so-called totallyregular continua , [5] – for each countable P ⊂ X , there is a basis B of open setsfor X so that for each B ∈ B , P ∩ Bd( B ) = ∅ and B has finite boundary. Thesecontinua have also been studied under the name continua of finite degree . The classof totally regular continua is strictly larger than the class of completely regularcontinua. In particular, the condition in Theorem 14 on f n having nontrivial fibersonly at vertices cannot be omitted. For example, the universal dendrite D n of order n can be obtained as the inverse limit of finite connected graphs, see [7, Section 3],and D n has a dense set of points of order = 2.In [8] the graph-like continuum depicted on the left side of the diagram belowserved to show that graph-like continua form a wider class than Freudenthal com-pactifications of locally finite graphs. Note that the two black nodes simultaneouslyact as ends for the blue double ladder, and as vertices for the red edge. ֒ → However, after subdividing the red edge appropriately – turning it into a doubleray – we see from the right side that it can be realized as a standard subspaceof the Freudenthal compactification of the triple ladder. We now show that everygraph-like continuum has the same property.
Theorem 15.
Every graph-like continuum can be embedded as a standard subspaceof a Freudenthal compactification of a locally finite graph.
We remark that Theorem 15 can be rephrased as saying that every graph-likecontinuum has a subdivision, turning each edge into a double ray, which is a stan-dard subspace of a Freudenthal compactification of a locally finite graph.
In the proof of Theorem 15, we use the following notation. Let G be a finite,connected graph with vertex set V , and let L ( G ) be its (connected) line graph,both considered as 1-complexes. For every edge e ⊂ G , let m e ∈ e be the mid-pointof that edge. Then by G ⊛ we denote the graph G ⊛ = ( G ⊕ L ( G )) / ∼ , where m e ∼ e for m e ∈ G and e ∈ V ( L ) . Geometrically, we subdivide each edge of G in its mid-point, and connect two newsuch vertices if and only if their underlying edges share a common vertex. Proof of Theorem 15.
Let X be a graph-like continuum. Represent X as a mono-tone inverse limit of finite multi-graphs G n with onto, monotone simplicial bondingmaps f n : G n +1 → G n having non-trivial fibres at vertices only.Recall first that the Freudenthal compactification of a locally finite graph canbe realized as an inverse limit: Let L be a locally finite graph with vertex set V ( L ) = { v k : k ∈ N } say. Let k n be an increasing sequence of integers, and considerfor each n the induced subgraph L n = L [ v , . . . , v k n ]. Let L n denote the multi-graph quotient of L where we contract every connected component of the inducedsubgraph L [ V ( L ) \ V ( L n )], deleting all arising loops. Since L was locally finite, itis easy to check that L n is a finite multi-graph. Then { L n : n ∈ N } forms an inversesystem under that natural projection maps g n : L n +1 → L n , such that the resultinginverse limit lim ← L n ∼ = γL is the Freudenthal compactification of L ; moreover,this holds independently of the sequence k n .Now our proof strategy is as follows. We plan to find a locally finite graph L asabove such that there are subgraphs T n ⊂ L n such that(i) ˆ g n = g n ↾ V ( T n +1 ) → V ( T n ) restricts to a surjection (so that the T n forma subsystem of the inverse limit with bonding maps ˆ g n ), and(ii) for each n ∈ N , the graph T n witnesses that G n is a topological minor of L n , meaning there are homeomorphisms h n : G n → T n of the underlying1-complexes which map V ( G n ) ֒ → V ( T n ), and map distinct edges vw and xy of G n to independent h n ( v ) h n ( w )- and h n ( x ) h n ( y )-paths in T n , and(iii) we have ˆ g n ◦ h n +1 = h n ◦ f n for all n , i.e. the following diagram commutes: T n T n +1 G n G n +1 h n ˆ g n f n h n +1 Under these assumptions, it follows that X = lim ← G n is homeomorphic to theinverse limit lim ← T n , which in turn, as it was constructed as a subsystem, embeds RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 15 into the inverse limit lim ← L n = γL , which equals the Freudenthal compactificationof L by the foregoing discussion. Thus, it remains to find a locally finite graph L subject to requirements (i)–(iii).We will build this locally finite graph L by geometric considerations as a directlimit of finite connected graphs F n , so that F n = L [ V ( F n )] = L n . More precisely,we will define finite connected 1-complexes F n such that(1) F ֒ → F ֒ → F ֒ → · · · forms a direct limit such that for all n >
0, no vertexof F n +1 \ F n is incident with a vertex of F n − , and(2) F n is embedded together with G n in some ambient 1-complex H n = F n ∪ G n such that(a) no vertex of G n lies in F n ,(b) every vertex of F n lies on an edge of G n ,(c) every open edge of F n is either disjoint from G n , or completely con-tained in an edge of G n , and(d) every edge of G n intersects with F n in a non-trivial path P ⊂ F n suchthat the end-vertices of P are vertices of V ( F n ) \ V ( F n − ).To begin, put H = G ⊛ , and let F denote the subgraph L ( G ) ⊂ G ⊛ . Then (2) issatisfied since vertices of F are mid-points of edges of G , and every open edge of F is disjoint from G ; and (1) is trivially true. x yz v f ←− x yz v v v Figure 1.
Depicts the first bonding map f between graphs G and G in black, where f ( { v , v , v } ) = { v } . Further, the figureon the left shows F ⊂ G ⊛ in red, and on the right F ⊂ H as theunion of ˜ F in red, S v L v in blue, and edges induced by ˜ F and S v L v in green.Now inductively, suppose we have already defined H n = G n ∪ F n for some n ∈ N according to (1) and (2). First, consider the natural pull-back ˜ F n ⊂ G n +1 of F n under f n . More precisely, by (2), the preimage f − n ( F n ) ⊂ G n +1 is isomorphic to asubgraph of F n . Let ˜ F n be an isomorphic copy of F n on the vertex set f − n ( V ( F n ))obtained by adding all edges missing from f − n ( F n ) so that they are disjoint from G n +1 .For every component C v of the topological subspace H n \ F n (which by (2)(a) and(b) will be a vertex v of G n incident with finitely many half-open edges), considerthe subcontinuum K v = f − n ( C v ) ⊂ G n +1 . Then K v is a finite connected graph.For each v , consider K ⊛ v , and L v = L ( K v ) ⊂ K ⊛ v , and define F n +1 to be theinduced subgraph F n +1 = ˜ F n ∪ S { L v : v ∈ V ( G n ) } . Claim 1: F n +1 is a connected graph. By induction on n . If F n is connected, then so is its isomorphic copy ˜ F n . As linegraphs of connected graphs, every L v is connected. Since by construction, every L v is connected via an induced edge to ˜ F n , it follows that F n +1 is connected. Claim 2:
Property (2) holds for F n +1 and G n +1 . (a) No vertex v of G n +1 lies on ˜ F n , as otherwise f n ( v ) would be a vertex of G n on F n . Also, since all L v are partial line graphs of G n +1 , we see that (a) holds atstep n + 1.(b) Similar.(c) By construction, this holds for all edges of ˜ F n . Further, all edges of L v are disjoint from G n +1 , and all edges of F n +1 induced ˜ F n and L v are completelycontained in one edge of G n +1 be definition.(d) Let e = vw be an edge of G n +1 . If e / ∈ E ( G n ) then F n +1 ∩ e = L v ∩ e is a trivial path consisting of one new vertex. Otherwise, if e ∈ E ( G n ), then byconstruction and induction assumption, ˜ F n intersects e in a non-trivial path P ⊂ ˜ F n such that the end-vertices of P have been added only at the previous step. Butnow, we see that F n +1 ∩ G is a path P ′ which is one edge longer on either sidethan P , because we added two edges induced by L v and L w . In particular, the endvertices of P ′ are vertices of L v and K v , and so have only been added at this step. Claim 3:
Property (1) holds.
Since F n ∼ = ˜ F n ⊂ F n +1 it is clear how to choose the embedding F n ֒ → F n +1 .The second part of the claim now follows from (2)(d) as follows: Every vertex of F n +1 \ F n is a vertex of some L v . By construction, any such vertex is connectedat most to one of the end vertices on some path P , which is, by (2)(d), a vertex of F n \ F n − .This completes the recursive construction. As indicated above, the graphs F ֒ → F ֒ → F ֒ → · · · give rise to a direct limit, which we call L . Let V ( L ) = { v k : k ∈ N } be an enumeration of the vertices of L , first listing all vertices of F , then all(remaining) vertices of F etc. It is clear that there is an increasing sequence ofintegers k n such that L n = L [ { v , . . . , v k n } ] = F n . Claim 4: L is a locally finite connected graph. To see that L is locally finite, note that any vertex v ∈ L is contained in some F n for some n , and then (1) implies that deg L ( v ) = deg F n +1 ( v ) < ∞ . And sinceevery L n is connected, so is L . Claim 5:
There are isomorphisms ϕ n : H n → L n . It suffices to show that L n ∼ = F n +1 / { L v : v ∈ V ( G n ) } . Indeed, (1) implies that the connected componentsof L \ F n correspond bijectively to the connected components of F n +1 \ F n , whichare, by construction, precisely the L v indexed by the different v ∈ V ( G n ). Inparticular, ϕ n is a bijection between V ( G n ) and the dummy vertices of L n thatcommutes with the respective bonding maps, i.e.( † ) g n − ◦ ϕ n ( v ) ↾ V ( G n ) = ϕ n − ◦ f n − ( v ) ↾ V ( G n ) . Claim 6:
For T n = ϕ n ( G n ) ⊂ L N the subgraph of L n which is the image of1-complex G n ⊂ H n subdivided by the vertices of L n , satisfies (i)–(iii). Everythingis essentially set up by construction; (iii) follows by ( † ) with h n = ϕ n ↾ G n . (cid:3) Note that our embedding of X into γL has the property that every vertex ofthe graph-like continuum X is represented by a compactification point (an end) of γL . By exercising some extra care in the above construction, one could arrange forisolated vertices of V ( X ) to be mapped to vertices of L . RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 17
Remark.
Theorem 15 has the following notable consequence. Diestel asked in [9]whether every connected subspace of the Freudenthal compactification of a locallyfinite graph is automatically arc-connected. In 2007, Georgakopoulos gave a neg-ative answer, [15]. However, the analogous problem for arbitrary continua is awell-studied problem. Indeed, a continuum is said to be in class A if every con-nected subset is arc-wise connected. Continua in class A have been characterizedby Tymchatyn in 1976, [23]. Even earlier, in 1933, Whyburn gave an example of acompletely regular continuum which is not in class A, [24, Example 4]. ApplyingTheorem 15, Whyburn’s example shows at once that Freudenthal compactificationsof locally finite graphs are not necessarily in class A.3.
Eulerian graph-like continua
Characterizing Eulerian Graph-Like Continua.
We now prove the equiv-alence of (i), (ii), (iii) and (iv) of Theorem (B) in the case of closed paths, and thendeduce the same equivalences in Theorem (B) for open paths. To start note that(iv) ⇒ (iii) and (i) ⇒ (iii) of Theorem (B) follow from Lemma 6. The next lemmatakes care of (iii) ⇒ (iv). Lemma 16.
A graph-like continuum such that every topological cut is even can bedecomposed into edge-disjoint topological cycles.Proof.
We adapt the proof from [21] as follows. Let G be a graph-like continuum,and E ( G ) = { e , e , . . . } an enumeration of its edges. Note that G − e is notdisconnected: If G − e = A ⊕ B then ( A, B ) would be a separation in G with E ( A, B ) = { e } , so odd, a contradiction. Since G is arc-connected by Corollary 8,there is an arc in G − e connecting x and y . Together with e that gives a topologicalcircle C .Now let e i = x i y i be the first edge not on C . We claim that there is a pathconnecting x i to y i in G \ ( E ( C ) ∪ { e i } ). Otherwise, there is a cut ( A, B ) of G ′ = G \ E ( C ) such that E G ′ ( A, B ) = { e i } . But then the same cut viewed in G would be odd by Lemma 6. A contradiction.It is clear that we can continue in this fashion until all edges are covered. (cid:3) To establish the equivalence of clauses (i), (iii) and (iv) of Theorem (B), itremains to show (iii) implies (i), which is established by the next result.
Proposition 17.
Let X be a graph-like continuum. If all topological cuts of X have even size then X has an Eulerian loop.Proof. By Theorem 13 (2), X can be written as an inverse limit of graphs G n ,which are all closed Eulerian. Let f n denote the bonding map f n : G n +1 → G n .For each n , let E n be the collection of all Euler cycles of G n . Since G n is finite,so is E n . For each n ∈ N , let ˆ f n : E n +1 → E n be the map induced by f n . That is, if E = ( v e v e v e · · · v k e k v ), thenˆ f n ( E ) = ( f n ( v ) f n ( e ) f n ( v ) f n ( e ) · · · f n ( e k ) f n ( v )) . Observe that from the proof of Theorem 13 some of the edges in E get contractedto a vertex. So ˆ f n ( E ) is an Eulerian circuit in G n . Now, {E n , ˆ f n } n ∈ N forms aninverse system, and since each E n is compact, we see lim ← E n = ∅ .Let ( E n ) ∈ lim ← E n . For each n ∈ N , fix an Eulerian loop φ n : S → G n followingthe pattern given by E n . Now observe that since the ( E n ) n ∈ N are compatible, thereare monotone continuous maps g n : S → S ( n ∈ N ) such that the diagram G f ←− G f ←− G f ←− · · ·↑ φ ↑ φ ↑ φ S g ←− S g ←− S g ←− · · · commutes. As an inverse limit of circles under monotone bonding maps, we havelim ←− S ∼ = S , [6, 4.11], and so the map g : lim ←− S → lim ←− G n , ( x n ) n ∈ N ( φ n ( x n )) n ∈ N is our desired Eulerian loop. (cid:3) The proof of the equivalence of (i), (ii), (iii) and (iv) in Theorem (B), for closed loops, is completed by Lemma 19 showing the equivalence of (ii) and (iii). Apreliminary lemma is needed.
Lemma 18.
Let X be a graph-like continuum, ( A, B ) be a separation of V, and U = { A , . . . , A n } be a multi-cut of A . If the cut E ( A, B ) is odd, then E ( A j , V \ A j ) is odd for some ≤ j ≤ n .Proof. Consider the contraction graph induced by the multi-cut (
B, A , . . . , A n ).By assumption, the vertex { B } has odd degree. Since by the HandshakingLemma, the number of odd-degree vertices in a finite graph is even, there must besome further vertex { A j } with odd degree, so E ( A j , V \ A j ) is odd. (cid:3) Lemma 19.
Let X be a graph-like continuum. All topological cuts of X are evenif and only if every vertex of X is even.Proof. If all cuts are even, then from the definition every vertex is even. We provethe converse by contrapositive. Assume there exists a separation ( A , B ) of V such that E ( A , B ) is odd. Let U = { U , . . . , U n } be a separation of A intosets with diameter < diam ( A ). By Lemma 18 there exists 1 ≤ j ≤ n such that E ( U j , V \ U j ) is odd. Denote U j by A and V \ U j by B . Let U = { U , . . . , U n } be a separation of A into sets with diameter < diam ( A ). Again by Lemma 18there exists 1 ≤ j ≤ n such that E ( U j , V \ U j ) is odd. Denote U j by A and V \ U j by B . Continuing with this procedure we obtain a nested sequenceof nonempty cut elements { A i } i ∈ N . By construction T i ∈ N A i = { v } ∈ V and E ( A i , B i ) is odd for every i ∈ N , hence v is not even. (cid:3) It remains to deduce the equivalence of (i), (ii), (iii) and (iv) in Theorem (B)for the case of open paths from that of closed paths. This can be achieved with asimple trick.Suppose, to start, that item (i) for open paths of Theorem (B), holds for a graph-like continuum X . So in X there is an open Eulerian path starting at a vertex v and ending at another vertex w . Create a new graph-like continuum Z by addingone edge to X with endpoints at v and w . Then Z is a graph-like continuum withan Eulerian loop. So, by Theorem (B) applied to Z , each of (ii)-(iv) (for closedpaths) of that theorem hold for Z . But now it easily follows from the definitionsthat each of (ii)-(iv) (for open paths) of Theorem (B) hold for X .Now let X be a graph-like continuum for which one of items (ii)-(iv) for openpaths in Theorem (B) holds. To complete the deduction we show (i) holds for openpaths. Each of these items highlights two distinct vertices (the two odd verticesin (ii) and the ends of the arc in (iv)). Call them v and w . Create a new graph-like continuum Z by adding one edge to X with endpoints at v and w . Then RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 19 Z is a graph-like continuum and it is easily verified from the definitions that itsatisfies one of (ii)-(iv) for closed paths in Theorem (B). Hence (i) for closed pathsof Theorem (B) holds, and there is a closed Eulerian path in Z . Removing theadded edge yields an open Eulerian path in X .3.2. Counting All Eulerian Loops and Paths.
In this section we aim to countthe number of distinct Eulerian loops and paths in a given graph-like continuum. Todo so we must decide what it means for two paths to be equivalent. This is a well-studied problem in combinatorial group theory, and we adopt the approach takenthere. Two maps f, g : I → X are equivalent if v = f (0) = g (0), w = f (1) = g (1), v and w are vertices, and f is homotopy equivalent to g relative to v, w . As notedin the Introduction, every map f : I → X with vertices for endpoints is equivalentto a standard path.Let X be a graph-like continuum. By Theorem 13 (2), X can be written as aninverse limit of graphs G n , via bonding maps f n : G n +1 → G n . As in Proposi-tion 17, for each n , let E n be the collection of all Eulerian cycles in G n , and letˆ f n : E n +1 → E n be the map induced by f n . Recall, ( E n , ˆ f n ) n forms an inverse sys-tem, and set E = E ( X ) = lim ← E n . As in Proposition 17, every ( E n ) n in E ( X )gives rise to an Eulerian loop in X . It is straightforward to check that distinctmembers of E ( X ) gives rise to inequivalent Eulerian loops. The converse is alsotrue, although we do not need that for our counting result. In any case we consider E ( X ) to be the space of Eulerian loops in X . Theorem 20.
A closed Eulerian graph-like continuum has either finitely many dis-tinct Eulerian loops in which case it is a graph, or it has continuum many Eulerianloops.Proof.
Since every E ( G n ) is finite discrete, the inverse limit is a compact subspaceof a Cantor set. As compact subspaces of a Cantor set without isolated points havesize continuum, the result follows from the next claim. Claim: If E ( X ) contains an isolated point, then X is homeomorphic to a graph.Fix an isolated element ( E n ) n in E ( X ). Fix an Eulerian loop f : I → X of X corresponding to ( E n ) n (as in Proposition 17). To witness that f is isolated, findcoordinate graph G n induced by a multi-cut U = ( U , . . . , U n ) of X such that the thequotient map q : X → G acting on the set of (distinct) Euler cycles E ( X ) → E ( G n )satisfies q − ( q ( f )) = { f } . We claim that every X [ U i ] (the subspace of X inducedby the vertex set U i ) is a graph. This would show that X itself is also a graph.Without loss of generality, f (0) / ∈ X [ U i ]. The map f induces a linear order on E ( U i , V \ U i ), say ( e , . . . , e k − ). For all 0 ≤ l < k write x l for the end vertex of e l in U i (of course, the x l need not be distinct). Let f m be the arc between x m and x m +1 induced by f . We claim that the arcs { f m : 0 ≤ m < k } witness that X [ U i ] is a graph.First of all, X [ U i ] = S m 1. Otherwise, suppose that y = z aretwo vertices lying in the interior of both arcs. Denote by e m = f m ↾ [ y, z ] and e p = f p ↾ [ y, z ] (or e m = f m ↾ [ z, y ] depending on which vertex comes first). Since f m , f p are edge disjoint, e m = e p . Then replace • f m by f m ↾ [ x m , y ] ∪ e p ∪ f m ↾ [ y, x m +1 ], and • f p by f p ↾ [ x p , y ] ∪ e m ∪ f p ↾ [ y, x p +1 ].This change gives rise to an Eulerian loop f ′ of X distinct from f , with q − ( q ( f )) ⊃{ f, f ′ } , a contradiction. (cid:3) We can deduce the analogous result for the number of open Eulerian paths bythe same trick used to derive the open version of Theorem (B) from the closedversion. Let X be an open Eulerian graph-like continuum, and let v, w be thetwo odd vertices of X . Add an edge connecting v and w , to get a closed Euleriangraph-like continuum Z . Apply the preceding result to deduce Z has either finitelymany distinct Eulerian loops in which case it is a graph, or it has continuum manyEulerian loops. Removing the added edge yields either that X is a graph or hascontinuum many open Eulerian paths. Theorem 21. An open Eulerian graph-like continuum has either finitely manydistinct open Eulerian paths in which case it is a graph, or it has continuum manyopen Eulerian paths. Bruhn & Stein Parity Let X be a graph-like continuum with vertex set V . Let v be a vertex of X .Then we say that v has strongly even degree (respectively, strongly odd degree ) ifthere is a clopen neighborhood C of v such that for every clopen neighborhood A of v contained in C the maximal number of edge-disjoint arcs from V \ A to v is even(respectively, odd). By Lemma 1, this is well-defined. We further say that v has weakly even degree (resp., weakly odd degree ) if v does not have strongly odd (resp.even) degree. Equivalently, v has weakly even degree if v has a neighborhood baseof clopen sets, C , so that the maximal number of edge-disjoint arcs from V \ C to v is even. And similarly for weakly odd degree. Bruhn & Stein [3] use the sameterminology for ‘strongly odd’ and ‘weakly even’ degrees, but use ‘even’ for our‘strongly even’ and ‘odd’ for our ‘weakly odd’.Note that isolated vertices have finite degree by Lemma 1, so for them beingeven and having strongly even degree coincide (and similarly for odd). In general,our notion of ‘even’ and ‘odd’ vertices implies those of Bruhn & Stein. To see this,we shall need a version of Menger’s theorem in the edge-disjoint version. ThatMenger-like theorems hold for graph-like continua is not surprising, and vertex-disjoint versions of Menger have been proved in [22]. We complement their resultsby the following theorem. Note that in finite graph theory, the edge disjoint versionfollows from the vertex disjoint version by applying the latter theorem to the linegraph. As it is unclear, what a line-graph for graph-like spaces should be, we needa different proof. Theorem 22 (Menger for Graph-like Continua—Edge Disjoint Version) . Let X be a graph-like continuum. For disjoint closed sets A and B of vertices of X , themaximum number of edge-disjoint A − B paths equals the minimum cut separating A from B .Proof. Let k be the size of a smallest cut separating A from B . Note that since A and B are closed disjoint, it follows from compactness that such a cut exists, andhence k is finite by Lemma 1. It is clear that the maximum number of edge-disjoint A − B paths is bounded by k . RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 21 Conversely, write X as an inverse limit X = lim ← G n with simplicial bondingmaps f n : G n +1 → G n and simplicial projection maps π n : X → G n . Without lossof generality, π n ( A ) ∩ π n ( B ) = ∅ for all n . Let T n be the (finite) space of all k -tuples of edge-disjoint connected subgraphs of G n that intersect both π n ( A ) and π n ( B ). By Menger’s theorem for finite graphs, T n = ∅ for all n , so T n with naturalbonding maps ˆ f n form their own inverse system, which is non-empty. Taking theinverse limit in each coordinate, we obtain k edge-disjoint subcontinua of X eachintersecting both A and B . By Corollary 8, we can find A − B paths inside eachsubcontinuum, which are then edge-disjoint by construction. (cid:3) Lemma 23. Let X be a graph-like continuum and v an even (resp. odd) vertex in X . Then v has strongly even (resp. odd) degree.Proof. Let v be an even vertex and let C be a clopen neighborhood of v such thatif A is a clopen neighborhood of v contained in C , then E ( V ( X ) \ A, A ) is even.Observe that E ( V ( X ) \ A, A ) is the minimum cut separating V ( X ) \ A from v .Hence by Theorem 22, the maximum number of edge-disjoint paths from V ( X ) \ A to v is equal | E ( V ( X ) \ A, A ) | which is even. This shows that v is strongly even. (cid:3) However, in general, strongly even degree vertices need not be even. Example. The right hand vertex in the graph-like continuum illustrated below isneither even nor odd but has strongly even degree.If each simple circle, , in the above example is replaced with a copy of , then inthe resulting graph-like continuum the right hand vertex has strongly odd degree.Our aim is to prove the following theorem, generalizing corresponding resultsof Bruhn & Stein [3] and Berger & Bruhn [1] for Freudenthal compactifications ofgraphs, and their Eulerian subspaces. Observe that this theorem can be rephrasedas saying that although not every vertex of strongly even degree must be even, if all vertices of a graph-like continuum have strongly even degree then they are all even. Theorem 24. A graph-like continuum is closed Eulerian if and only if all itsvertices have strongly even degree. It is an interesting open problem, whether the same conclusion holds under theassumption that all vertices have weakly even degree. The forward implication ofTheorem 24 follows from Lemma 23, Lemma 19 and Proposition 17. Theorem 33establishes the converse. The plan for the proof of Theorem 33 is to establishthe contrapositive: if X is a graph-like continuum which is not closed Eulerianthen it contains a vertex without strongly even degree (i.e. of weakly odd degree).Lemma 26 shows how a certain sequence of regions leads to such a vertex. Now if X is a graph-like continuum which is not closed Eulerian, then by Theorem (B) (iii) = ⇒ (i), there must be an odd cut in X . This provides the starting point for thesequence needed to apply Lemma 26. Theorem 27 then provides the ‘ContractionMachine’ required to create the remaining elements of the sequence.If v and w are distinct vertices in a graph-like continuum X , and they bothhave strongly odd degree, then after connecting them with a new edge they willboth have strongly even degree. Conversely if they both have strongly even degree,then after removing an edge connecting them, they will have strongly odd degree.Hence, as we deduced the open version of Theorem (B) from the closed version, wenow derive the following characterization of open Eulerian graph-like continua. Theorem 25. A graph-like continuum is open Eulerian if and only if it has exactlytwo strongly odd degree vertices, and the rest have strongly even degree. The odd-end lemma. For a clopen subset U ⊂ V ( X ), consider the inducedgraph-like space X [ U ]. We say that a clopen subset U ⊂ V ( X ) is a region if X [ U ]is connected. By ∂U ⊂ E ( X ) we denote the set of edges between the separation( U, V \ U ). This set is finite for regions U . Let us call a region U of X a k -regionif | ∂U | = k , and an even or an odd region depending on whether k is even or odd.The following lemma generalizes the corresponding lemma of Bruhn & Stein forlocally finite graphs, [3, p.7f], to graph-like continua. Lemma 26. Let X be a graph-like continuum, and let E ( X ) = { e , e , . . . } be anenumeration of its edges. Assume there exists a sequence of regions U , U , . . . of X with the following properties:(1) | ∂U n | is odd for all n ∈ N ,(2) U n ⊃ U n +1 ,(3) if D is a region of X with U n ⊃ D ⊃ U n +1 then | ∂U n | ≤ | ∂D | for all n ∈ N ,and(4) e n / ∈ E [ U n +1 ] .Then X has a vertex which has weakly odd degree.Proof. Since A = T n ∈ N X [ U n ] is a nested intersection of continua by (2), it is non-empty and connected. It follows from (4) that A ⊂ V ( X ), so A = { v } for somevertex v , since V ( X ) is totally disconnected. Furthermore, compactness impliesthat { U n : n ∈ N } is a neighborhood base for v in V ( X ).Property (3) together with Theorem 22 shows that for all U n the maximal numberof edge disjoint arcs from V \ U n to v equals | ∂U n | , so is odd by (1). Since the U n form a neighborhood base, it follows that v has weakly odd degree. (cid:3) The contraction machine. Suppose we have an odd region U . We want toconstruct a sequence as in Lemma 26. If we recursively choose an odd region U n +1 of minimal | ∂U n +1 | amongst all odd regions contained in U n , then (1) and (2) arefine, and property (3) is satisfied at least for all odd regions D nested between U n and U n +1 . Following Bruhn & Stein’s idea [3], our plan for evading all even regions D with | ∂D | < | ∂U n | nested between U n and U n +1 is roughly as follows: first, wecontract all even regions D ⊂ U n with boundary smaller than | ∂U n | to single points.Only then do we pick our region U n +1 . After uncontracting, this means that everysmall even region lies either behind U n +1 , or is completely disjoint from U n +1 .The next result formalizes this idea for contracting regions. RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 23 Theorem 27 (Contraction Theorem) . Let X be a graph-like continuum such thatall isolated vertices are even. Suppose further that U ⊂ X is an odd region of X such that for some even m > , there is no infinite k -region with k < m of X contained in U .Then there is a collection M of disjoint regions of U such that after contract-ing every element of M to a single point, the graph-like continuum X/ M , withassociated (monotone) quotient map π : X → X/ M , has the property that(i) all isolated vertices of X/ M are even,(ii) there are no infinite k regions with k ≤ m contained in the region π ( U ) ⊂ X/ M , and(iii) if D ⊂ U is an ℓ -region of X , then there is an ≤ ℓ -region D ′ ⊂ π ( U ) suchthat | π ( D ) \ D ′ | < ∞ . We divide the proof into a sequence of lemmas. For two subsets A, B ⊂ X , saythat A splits B , or B is split by A , if A ∩ B = ∅ 6 = B \ A . Lemma 28. Let X be a graph-like continuum, and U ⊂ X a region. Let R, S , . . . , S n be infinite m -regions contained in U , where S , . . . , S n are pairwise disjoint and | R \ S i ≤ n S i | = ∞ .If there is no infinite k -region with k < m of X contained in U , then there is an m -region ˜ R which doesn’t split any S i such that | R \ ( S i ≤ n S i ∪ ˜ R ) | < ∞ . For the proof we need the following lemma, which can be proven, as for graphs,by a simple double-counting argument. Lemma 29. Let X be a graph-like space, and Y, Z ⊂ V ( X ) clopen subsets. Then | ∂Y | + | ∂Z | ≥ max {| ∂ ( Y ∩ Z ) | + | ∂ ( Y ∪ Z ) | , | ∂ ( Y \ Z ) | + | ∂ ( Z \ Y ) |} . Proof of Lemma 28. Without loss of generality, assume that S is split by R , i.e.that R ∩ S = ∅ 6 = S \ R . We claim that one of S ∪ R or R \ S is an m -region.They are clearly clopen subsets of vertices of X .Otherwise, since S ∪ R and R \ S are infinite, we have | ∂S ∪ R | > m and | ∂R \ S | > m . Thus, Lemma 29 implies that | S \ R | < m and | S ∩ R | < m , soboth regions are are finite, contradicting that S is infinite.Hence, one of S ∪ R or R \ S is has a boundary of size m , and they can’tbe disconnected, as otherwise their components had to be finite. Now put R ′ tobe either one of them, whichever was the m -region. Then R ′ splits strictly fewer S i than R , but covers the same set together with the S i . Thus, we may pick ˜ R to be such that it splits the fewest number of S i , subject to the condition that | R \ ( S i ≤ n S i ∪ ˜ R ) | < ∞ . By the preceding argument, it follows that ˜ R does notsplit any of the S i . (cid:3) Let X be a graph-like continuum, and U ⊂ X a region. Assume there is noinfinite k -region with k < m of G contained in U . Let R = { R n : n ∈ N } be anenumeration of all infinite m -regions of G contained in U . Since each R i is faithfullyrepresented by the finite cut ∂R i ⊂ E , and E is countable, there are indeed at mostcountably many such regions. Below we write S (cid:22) S ′ if S is a refinement of S ′ , i.e.for all S ∈ S there is S ′ ∈ S ′ such that S ⊆ S ′ . Lemma 30. For every n ∈ N there are finite collections S n ⊂ R of disjoint m -regions of U such that(1) for all R j with j ≤ n we have | R j \ S S n | < ∞ , and (2) S n (cid:22) S n +1 . Construction. We begin with S = { R } . Suppose S n ⊆ R has been found satis-fying the above properties. Applying Lemma 28 with R n +1 and the collection S n ,we obtain an infinite m -region ˜ R n +1 . We claim that S n +1 = { ˜ R n +1 } ∪ { S ∈ S n : S ∩ ˜ R n +1 = ∅} is as desired. Indeed, by construction, S n +1 covers R n +1 up tofinitely many vertices; and S S n ⊆ S S n +1 , so we preserved the covering propertiesof earlier stages. (cid:3) We would like to contract the ‘maximal’ m -regions (with respect to inclusion)contained in S = S S n . However, for graph-like continua, there can be infinite non-trivial chains in S . Still, for any such chain S ( S ( S ( · · · of m -regions, wecan contract a suitable collection of disjoint even regions such that after contraction,all S n are finite. Our plan is to contract S , and each component of S n +1 \ S n , to asingle point for all n ∈ N . Our next lemma provides the details for the second case. Lemma 31. Let X be a graph-like continuum, and U ⊂ X a region. Assume thereis no infinite k -region with k < m of G contained in U .If S ( R are infinite m -regions contained in U , then X [ R \ S ] has at most m connected components, and every such component is an even region of X .Proof. Note that since X [ R ] is path-connected, it follows that every component of X [ R \ S ] has to limit onto an end vertex of some e ∈ ∂S . Thus, X [ R \ S ] has atmost | ∂S | = m components. In particular, every component is clopen in X [ R \ S ],and hence a region of X .To see that ∂ ( R \ S ) is even, consider the graph induced by the multi-cut ( S, R \ S, V \ R ). This graph has two even vertices, namely { S } and { V \ R } . So by theHandshaking Lemma, also the last vertex is even, i.e. R \ S induces an even cut.Moreover, since in the contraction graph, both { S } and { V \ R } have degree m , itfollows that the third vertex has the same number of edges to { S } and to { V \ R } .In other words, we have | ∂ ( R \ S ) ∩ ∂R | = | ∂ ( R \ S ) ∩ ∂S | .Let C denote the vertex set of one such component. It follows that in order toestablish that C is an even region, it suffices to show that | ∂C ∩ ∂R | ≥ | ∂C ∩ ∂S | . (1)Indeed, once we know that (1) holds for every component C , then | ∂R | = m = | ∂S | gives equality in (1). To see that (1) holds, note that if | ∂C ∩ ∂R | < | ∂C ∩ ∂S | ,then we see that | ∂ ( S ∪ C ) | < m , so this is a finite region, contradicting that S wasinfinite. (cid:3) We now collapse all maximal m -regions in S = S S n , and for every infinite properchain in S we perform the above contractions. Write M for the disjoint collectionof even regions we contract. Write q M : V ( X ) → V ( X/ M ), which extends to acontinuous (monotone) quotient map on X → X/ M (where we also contract allpotential loops), which we also call q M . Note that since we contracted regions of acompact space, the map q M : X → X/ M is a closed, monotone map. In particular,this implies that preimages of regions are regions, see Theorem 9 of [17]. Proof of Theorem 27. First, to see that X/ M is still a graph-like continuum, notethat our countable family M forms a null-sequence of clopen sets by Corollary 8.It follows from the fact that if X is separable metrizable, and A = { A n : n ∈ N } anull-sequence of non-empty compact subsets of X , then X/ A is separable metrizable ,[18, A.11.6], that X/ M is a continuum. Further, it is graph-like, because its vertex RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 25 set V ( X ) / M is totally disconnected: If there was any non-trivial connected set C ⊂ V ( X/ M ), then C cannot contain contracted vertices (they are isolated), so C ⊂ V ( X ) is non-trivial connected, contradiction.Item (i), that every isolated vertex of X/ M is even, follows from Lemma 31, aswe only contracted even regions.For (ii), that all m -regions of X/ M contained in π ( U ) are finite, note that forany such m-region D of X/ M , the clopen vertex set D ′ = π − ( D ) is an m -regionof X . If D ′ was infinite, then D ′ appears in our list, so is covered by some finite S n . Consider S ∈ S n . Note that S either gets contracted to a single point, or S appears in an infinite chain with at most n predecessors, in which case we contract S to at most ( m · n + 1)-many points. It follows that D ′ gets contracted to finitelymany points, i.e. D is finite.For (iii), let D be an ℓ -region of X . There are at most ℓ many elements M , . . . , M ℓ ∈ M such that ∂D ∩ E [ M i ] = ∅ . Now if D ⊂ M i for some i thenit is clear that π ( D ) is finite. Otherwise, choose disjoint m -regions S i ⊃ M i in S .We claim that either ˜ D = D ∪ S or ˜ D = D \ S is an ≤ ℓ -region. Otherwise, itfollows from Lemma 29 that | ∂ ( D ∩ S ) | < m and | ∂ ( S \ D ) | < m . So S is finite,a contradiction. Continue with the other S i . This gives us an ≤ ℓ -region D ′ , whichdiffers from D by finitely many S ∈ S . (cid:3) Chasing odd regions. After having established Theorem 27, the proof nowproceeds essentially as in [3]. We need one more simple lemma. Lemma 32. A graph-like continuum in which all isolated vertices are even doesnot contain finite odd regions.Proof. If A = { v , . . . , v n } ⊂ V ( X ) is a finite region, consider the finite graphinduced by the multi-cut ( V \ A, { v } , . . . , { v n } ). Since all vertices v i are even, itfollows from the Handshaking Lemma that also { V \ A } must be even. (cid:3) Theorem 33. A graph-like continuum is Eulerian if all its vertices have stronglyeven degree.Proof. Assume X is not Eulerian. To prove the contrapositive we show X con-tains a vertex without strongly even degree. If some isolated vertex does not have(strongly) even degree then we are done. So assume all isolated vertices of X areeven. We construct a sequence of graph-like continua X = X , X , . . . such that(a) X π −→ X π −→ X π −→ · · · are successive quotients with monotone openquotient maps π n , and write f n = π n ◦ π n − ◦ · · · ◦ π ,(b) all X n have the property that all isolated vertices are even,(c) there are regions V n ⊂ X n such that(1) ′ | ∂V n | is odd for all n ∈ N ,(2) ′ π n +1 ( V n ) ⊃ V n +1 ,(3) ′ any ℓ -region of X gets contracted to a ≤ ℓ -region of X n modulo finitelymany isolated vertices; and any k -region of X n contained in V n with k < | ∂V n | gets contracted to finitely many vertices in X n +1 ,(4) ′ e n / ∈ E [ V n +1 ].Before describing the construction, let us see that that U n = f − n ( V n ) defines regionssatisfying the requirements of Lemma 26, and so X has a vertex which does nothave strongly even degree, as desired. Indeed, as inverse images under monotone closed maps, they are connected, andhence regions in X . Next, it is easy to check that (1) ′ ⇒ (1), (2) ′ ⇒ (2) and (4) ′ ⇒ (4). Finally, to see (3), i.e. that U n +1 does not lie behind some region D of U n withsmall | ∂D | , note that by (3) ′ , this region D would have been contracted to finitelymany points in X n +1 , and hence V n +1 would be finite, which is a contradiction by(b) and Lemma 32.Now towards the construction of our sequence X , X , . . . with (a)–(c). First,since X = X is not Eulerian, it has an odd cut. By choosing V = U to be an oddregion of X such that | ∂U | is minimal, we see that V is as desired. Now supposewe have constructed V n ⊂ X n according to (a)–(c). Put m n +1 = | ∂V n | − X ( k ) and region q k ◦· · · ◦ q ( U n ) to obtain graph-like continua X n = X ( m n ) ≻ X ( m n +1) ≻ · · · ≻ X ( m n +1 ) = X n +1 with corresponding monotone quotient maps q k : X ( k − → X ( k ) for all even 0 < k ≤ m . Define π n +1 = q m n +1 ◦ · · · ◦ q m n +1 : X n → X n +1 .Note that Theorem 27(i) implies (b), and (ii) and (iii) imply (c)(3) ′ . We nowwant to find an odd cut V ⊂ π n +1 ( V n ) such that e n / ∈ E ( V ). Towards this, notethat f n +1 ( e n ) is either an isolated vertex v of X n +1 , or f n +1 ( e n ) is an edge withend vertices say x and y in X n +1 . Find a multi-cut V of π n +1 ( V n ) into regionswhich either displays v as a singleton, or contains x and y in different partitionelements. By Lemma 18, there is an odd region V ∈ V . Since isolated vertices of X n +1 are even, V is not the singleton { v } . In the other situation, note that in theinduced graph G ( V ), the edge f n +1 ( e n ) is displayed as cross edge. In either case,we have e n / ∈ E ( V ).Finally, amongst all odd regions of X n contained in V pick any odd region V n +1 ⊂ V such that | ∂V n +1 | is minimal. This choice satisfies items (1) ′ , (2) ′ and (4) ′ . (cid:3) References [1] E. Berger and H. Bruhn, Eulerian edge sets in locally finite graphs , Combinatorica (2011),21–38.[2] N. Bowler, J. Carmesin and R. Christian, Infinite graphic matroids, Part I , http://arxiv.org/abs/1309.3735 .[3] H. Bruhn and M. Stein, On end degrees and infinite cycles in locally finite graphs , Combi-natorica (2007), 269–291.[4] W. Bula, J. Nikiel, and E. D. Tymchatyn, The K¨onigsberg bridge problem , Can. J. Math. (1994), 1175–1187.[5] R.D. Buskirk, J. Nikiel, E.D. Tymchatyn, Totally regular curves as inverse limits , Houston.J. Math. (3) (1992), 319–327.[6] C.E. Capel, Inverse limit spaces , Duke Math. J. (1954), 233–245.[7] J. J. Charatonik Monotone mappings of universal dendrites , Topology App. (1991), 163–187.[8] R. Christian, R. B. Richter, B. Rooney, The Planarity Theorems of MacLane and Whitneyfor Graph-like Continua , Electron. J. Combin. , Research Paper 12, 2010.[9] R. Diestel, The cycle space of an infinite graph , Comb. Probab. Comput. 14 (2005), 59–79.[10] R. Diestel, Locally finite graphs with ends: a topological approach I-III , Discrete Math 311–312 (2010–11).[11] R. Diestel & D. K¨uhn, On infinite cycles I , Combinatorica (2004), 68–89.[12] R. Diestel & D. K¨uhn, On infinite cycles II , Combinatorica (2004), 91–116.[13] B. Espinoza and E. Matsuhashi, Arcwise increasing maps , Topology Appl. (2015) 74–92.[14] A. Georgakopoulos, Topological circles and Euler tours in locally finite graphs , Electronic J.Comb. : RAPH-LIKE COMPACTA: CHARACTERIZATIONS AND EULERIAN LOOPS 27 [15] A. Georgakopoulos, Connected but not path-connected subspaces of infinite graphs , Combi-natorica (6) (2007), 683–698.[16] J. Krasinkiewicz, On two theorems of Dyer , Colloq. Math., (1986) 201–208.[17] K. Kuratowski, Topology Vol. 2, Academic Press, New York and London, PWN–Polish Sci-entific Publishers, Warszawa, 1968.[18] J. van Mill, The Infinite-Dimensional Topology of Function Spaces , Elsevier, 2001.[19] S. B. Nadler, Jr, Continuum Theory: An Introduction , Pure and Applied Mathematics Series,Vol. 158, Marcel Dekker, Inc., New York and Basel, 1992.[20] J. Nikiel, Locally connected curves viewed as inverse limits , Fund. Math. (2) (1989) 125–134.[21] C. St. J. A. Nash-Williams, Decompositions of graphs into closed and endless chains , Proc.London Math. Soc. (3) (1960), 221–238.[22] C. Thomassen, A. Vella, Graph-like continua, augmenting arcs, and Menger’s Theorem ,Combinatorica (5) (2008) 595–623.[23] E.D. Tymchatyn, Characterizations of continua in which connected subsets are arcwise con-nected , Trans. Amer. Math. Soc. (1976) 377–387.[24] G.T. Whyburn, Sets of local separating points of a continuum , Bull. Amer. Math. Soc. (1933), 97–100. 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