TTHE LOVÁSZ-CHERKASSKIY THEOREM IN COUNTABLE GRAPHS
ATTILA JOÓ
Abstract.
Lovász and Cherkasskiy discovered in the 1970s independently that if G is a finite graph with a given set T of terminal vertices such that G is inner Eulerien,then the maximal number of edge-disjoint paths connecting distinct vertices in T is P t ∈ T λ ( t, T − t ) where λ is the local edge-connectivity function.The maximal size of a system of edge-disjoint T -paths in the Lovász-Cherkasskiytheorem can be characterized by the existence of certain cuts applying Menger’s theorem.The infinite generalization of Menger’s theorem by Aharoni and Berger (earlier knownas the Erdős-Menger conjecture) together with the characterization of infinite Euleriangraphs due to Nash-Williams make possible to generalize the theorem for infinite graphsin a structural way. The aim of the paper is to prove this generalization when T iscountable. Introduction
There are several deep results and conjectures in infinite combinatorics whose restrictionto finite structures is a well-known classical theorem. For example the results [3], [2] and[1] by Aharoni, Nash-Williams and Shelah are known as Hall’s and Kőnig’s theorem whenonly finite graphs are considered. The finite case of the Aharoni-Berger theorem [5] (earlierknown as the Erdős-Menger Conjecture) is known as Menger’s theorem and the MatroidIntersection Conjecture [6] by Nash-Williams extends the Matroid Intersection Theorem[9] of Edmonds.There are several common aspects of the problems above. For example assuming thefiniteness of the involved structures simplifies the proof significantly. Induction on sizeor finitely many application of an “augmenting path” type of argument is applicable andsufficient. These tools are unavailable and insufficient respectively in the general casewhere more complex techniques and structural arguments are required. All these problemsexpress that a “primal-dual complementarity slackness” type of condition holds betweensuitable primal and dual objects: a matching M in G = ( A, B, E ) and vertex-cover C consisting of a single vertex from each e ∈ M ; disjoint path-system P between A and B in G = ( V, E ) with
A, B ⊆ V and AB -separation S ⊆ V consisting of choosing asingle vertex from each P ∈ P ; common independent set I of matroids M and M anda bipartition S = S ∪ S of their common ground set such that S i ∩ I spans S i in M i for i ∈ { , } . Alternative characterizations of “primal optimality” can be given throughthe concept of strong maximality. Let us call an element X of set family X stronglymaximal in X if | Y \ X | ≤ | X \ Y | for every Y ∈ X . Note that if X has only finite Mathematics Subject Classification.
Primary: 05C63, 05C38. Secondary: 05C40, 05C45.
Key words and phrases.
Lovász-Cherkasskiy theorem, infinite graph, packing paths.The author would like to thank the generous support of the Alexander von Humboldt Foundation andNKFIH OTKA-129211. a r X i v : . [ m a t h . C O ] F e b ATTILA JOÓ elements, then strongly maximal means maximum size but it is stronger than that ingeneral. It is known that in the three problems we mentioned that the strong maximality ofa matching/disjoint path system/common independent set is equivalent with the existenceof a vertex-cover/separation/bipartition such that the corresponding complementarityslackness conditions are satisfied.Our aim is to prove a theorem in this manner extending the following result obtainedby Lovász and Cherkasskiy independently in the 1970s:
Theorem 1.1 (Lovász-Cherkasskiy theorem, [16]) . Let G be a finite graph and let T ⊆ V ( G ) such that G is inner Eulerian (i.e. d G ( v ) is even for every v ∈ V ( G ) \ T ). Thenthe maximal number of pairwise edge-disjoint T -paths is X t ∈ T λ G ( t, T − t ) , where λ G ( t, T − t ) stands for the maximal number of pairwise edge-disjoint paths between t and T − t . The literal extension of Theorem 1.1 to infinite graphs fails. Indeed, consider the star K , and attach a one-way infinite path to its central vertex. Let T consists of the verticesof degree one. Then we have only even degrees out of T and the maximal number ofedge-disjoint T -paths is 1 although P t ∈ T λ ( t, T − t ) = . . . . Figure 1.
The failure of the literal infinite generalization the Lovász-Cherkasskiy theorem. Elements of T are black.The reason of this discrepancy is that after allowing G to be infinite the condition “ G isEulerian” (i.e. E ( G ) can be partitioned into edge-disjoint cycles) is no longer equivalentwith the property that G has only even degrees. Indeed, in the two-way infinite patheach degree is 2 but it is obviously not Eulerian. On the other hand, graphs with infinitedegrees can be easily Eulerian. The characterization of infinite Eulerian graphs due toNash-Williams is one of the fundamental theorems in infinite graph theory: Theorem 1.2 (Nash-Williams, [19] (p. 235 Theorem 3)) . A (possibly infinite) graph isEulerian if and only if it does not contain odd cut. Simpler proofs for Theorem 1.2 was given by L. Soukup (Theorem 5.1 of [22]) andThomassen [23] while its analogue for directed graphs (conjectured by Thomassen) wassettled affirmatively in [14]. Theorem 1.2 indicates that the condition “for every v ∈ V \ T : d ( v ) is even” should be replaced by “for every X ⊆ V \ T : d ( X ) is not odd” in order to paths connecting distinct vertices in T without having internal vertex in T . infinite cardinals considered neither odd nor even HE LOVÁSZ-CHERKASSKIY THEOREM IN COUNTABLE GRAPHS 3 allow infinite graphs. The latter means by Theorem 1.2 that the contraction of T resultsin an Eulerian graph while the former meant the same but restricted to finite graphs.The literal adaptation of the formula P t ∈ T λ ( t, T − t ) is also not really fruitful in thepresence of infinite quantities. Consider for example the graph ( { u, v } , E ) with T = { u, v } where E consists of ℵ parallel edges between u and v . Then any infinite P ⊆ E , consideredas a set of paths of length one, has the same size ℵ . It demonstrates that cardinality isan overly rough measure in the presence of infinite quantities and urges us to focus oncombinatorial instead of quantitative properties of an optimal path-system in Theorem 1.1.In a finite graph a system P of edge-disjoint T -paths has P t ∈ T λ ( t, T − t ) elements ifand only if P contains λ ( t, T − t ) paths between t and T − t for each t ∈ T . By Menger’stheorem it is equivalent with the fact that for every t ∈ T one can choose exactly one edgefrom each P ∈ P having t as an end-vertex such that the resulting edge set C is a cutseparating t from T − t . Now we are ready to state our main results: Theorem 1.3.
Let G be a graph and let T ⊆ V ( G ) be countable such that there is no X ⊆ V ( G ) \ T where d G ( X ) is an odd natural number. Then there exists a system P ofedge-disjoint T -paths such that for every t ∈ T : one can choose exactly one edge fromeach P ∈ P having t as an end-vertex in such a way that the resulting edge set C is a cutseparating t and T − t . We also prove the following closely related theorem.
Theorem 1.4.
Let G be a graph and let T ⊆ V ( G ) be countable such that there is no X ⊆ V ( G ) \ T where d G ( X ) is an odd natural number. Assume that for each t ∈ T thereis a system P t of edge-disjoint T -paths covering all the edges incident with t . Then thereexists a system P of edge-disjoint T -paths covering all the edges incident with any t ∈ T . We conjecture that the countability of T can be omitted in the theorems above. However,based on the experience with the similar problems mentioned earlier, we suspect that theproof is significantly harder.Mader generalized Theorem 1.1 for arbitrary finite graphs in [17]. The structural andalgorithmic aspects of the problem have been ever since a subject of interest (see forexample [21], [15], [7] and [12]) as well the analogous theorems considering vertex-disjoint[11] and internally vertex-disjoint [17] paths. Conjecture 1.5.
Let G be a graph and let T ⊆ V ( G ) . Then there exist a strongly maximalsystem P of edge-disjoint/vertex-disjoint/internally vertex-disjoint T -paths in G . We also conjecture that the path-system P in Conjecture 1.5 can be characterized inthe way that it extends the corresponding minimax theorem to infinite graphs based oncomplementarity slackness conditions. We discuss the details in Section 5. Before we turnto the proof of our main result in Section 4, we need to introduce some notation andrecall few results we are going to use in the proof. These are done in Sections 2 and 3respectively. ATTILA JOÓ Notation
In graphs we allow parallel edges but not loops. Technically we represent a graph as atriple G = ( V, E, I ) where the incidence function I : E → [ V ] defines the end-vertices ofthe edges. For X ⊆ V let δ G ( X ) := { e ∈ E : | I ( e ) ∩ X | = 1 } and we write d G ( X ) for | δ G ( X ) | . If graph G is obvious from the context, then we omit the subscript, furthermore,for a singleton { x } we write simply δ ( x ) and d ( x ). All the paths in the paper are finite.We refer sometimes the first vertex or last edge of a path. The context will always indicateaccording which direction we mean this. An AB -path for A, B ⊆ V is a path with firstvertex in A last vertex in B and no internal vertices in A ∪ B . A C ⊆ E is a cut if C = δ ( X )for some X ⊆ V . If G is connected then X is determined by C up to taking complementand the v -side of the cut C is the unique X with C = δ ( X ) and v ∈ X . We call δ ( X ) an AB -cut if A ⊆ X and B ∩ X = ∅ or the other way around. In a connected graph G , cut δ ( X ) is ⊆ -minimal if and only if the induced subgraphs G [ X ] and G [ V \ X ] are connected.We extend the definitions above for disconnected graphs G and cuts C living in a singleconnected component M by considering C as a cut in G [ M ]. For a U ⊆ V and a family F = { X u : u ∈ U } of pairwise disjoint subsets of V with X u ∩ U = { u } we define the graph G/ F obtained from G by contracting X u to u for u ∈ U and deleting the resulting loops.More formally V ( G/ F ) := ( V \ S F ) ∪ U, E ( G/ F ) := E \ { e ∈ E : ( ∃ u ∈ U ) I ( e ) ⊆ X u } and I ( G/ F )( e ) := { i F ( u ) , i F ( v ) } where I ( e ) = { u, v } and i F ( v ) = v if v / ∈ S F u if u ∈ X u . Preliminaries
Menger’s theorem and the other connectivity-related results that we recall in this sectionhave four versions depending on if the graph is directed and if we consider vertex-disjointor edge-disjoint paths. In all of these theorems the two directed variants are equivalentas well as the two undirected variants which can be shown by simple techniques likesplitting edges by a new vertex and blowing up vertices to a highly connected vertex sets.Furthermore, through replacing undirected edges by back and forth directed ones theundirected vertex-disjoint version can be reduced to the directed one.In this paper we deal only with undirected graphs and edge-disjoint paths so let usalways formulate immediately that variant even if historically other version was provedfirst.Let a connected graph G and distinct s, t ∈ V ( G ) be fixed. For st -cuts C and D wewrite C (cid:22) D if the s -side of cut C is a subset of the s -side of D . Note that the st -cutswith (cid:22) form a complete lattice. For a finite G the optimal (minimal-sized ) st -cuts form adistributive sublattice (see [10]) of it. In general graphs the size of the cut is an overlyrough measure for optimality. A structural infinite generalization of the class of “optimal” st -cuts is provided by the Aharoni-Berger theorem: Theorem 3.1 (Aharoni and Berger, [5]) . Let G be a (possibly infinite) graph and let s, t ∈ V ( G ) be distinct. Then there is a system P of edge-disjoint st -paths and an st -cut HE LOVÁSZ-CHERKASSKIY THEOREM IN COUNTABLE GRAPHS 5 C which is orthogonal to P , i.e. C consists of choosing exactly one edge from each pathin P . We say that the st -cut C in Theorem 3.1 is an Erdős-Menger st -cut and we let C ( s, t ) be the set of such cuts. Theorem 3.2 (J. [13]) . ( C ( s, t ) , (cid:22) ) is a complete lattice, although usually not a sublatticeof all the st -cuts. Finally we introduce two more classes C − ( s, t ) and C + ( s, t ) of st -cuts with C − ( s, t ) ∩ C + ( s, t ) = C ( s, t ) and C + ( s, t ) := C − ( t, s ). Let C − ( s, t ) consists of those st -cuts C forwhich there is a system W of pairwise edge-disjoint paths starting at s and having C asthe set of last edges (considering the paths directed away from s ). Such a W is called an st - wave and played an important role in the proof of Theorem 3.1. The cut defined as thelast edges of the paths in W is denoted by C W . The (cid:22) -smallest st -cut δ ( s ) is also a wave(considering the edges as paths of length one) that we call the trivial st -wave . Lemma 3.3. ( C − ( s, t ) , (cid:22) ) is a complete lattice and a sup-sublattice of all the st -cuts.After the contraction of the s -side of its largest element to s , there is no non-trivial st -wavein the resulting system. We call an st -wave W large if C W is the largest element of C − ( s, t ). Note that ifthe trivial st -wave is the only one, then δ ( s ) must be an Erdős-Menger st -cut because C ⊆ C − = { δ ( s ) } and the left side is nonempty by Theorem 3.1. This leads to the followingconclusion: Corollary 3.4.
If there is no non-trivial st -wave, then there is a system P of edge-disjoint st -paths covering δ ( s ) . Theorem 3.5 (Pym’s theorem, [20]) . Assume that G is a (possibly infinite) graph, s, t ∈ V ( G ) are distinct, moreover, P and Q are systems of edge-disjoint st -paths. Then thereexists a system R of edge-disjoint st -paths such that δ R ( s ) ⊇ δ P ( s ) and δ R ( t ) ⊇ δ Q ( t ) . Let P be a system of edge-disjoint st -paths and let W be a large st -wave. By contractingthe t -side of C W to t and applying Theorem 3.5 with the st -paths obtained from W andfrom the initial segments of the paths in P we conclude: Corollary 3.6.
Let P be a system of edge-disjoint st -paths. Then there is a large st -wave W with δ W ( s ) ⊇ δ P ( s ) . Finally, we will make use of the following classical lemma (see Lemma 3.3.2 and 3.3.3 in[8]):
Lemma 3.7 (Augmenting path lemma) . Assume that G is a (possibly infinite) graph, s, t ∈ V ( G ) are distinct and P is a system of edge-disjoint st -paths in G . Then either thereexists an st -cut C orthogonal to P or there is another system Q of edge-disjoint st -paths forwhich δ Q ( s ) ⊃ δ P ( s ) with | δ Q ( s ) \ δ P ( s ) | = 1 and δ Q ( t ) ⊃ δ P ( t ) with | δ Q ( t ) \ δ P ( t ) | = 1 . All the definitions and results in the section remain valid (but might sound less natural)if s and t are not vertices but disjoint vertex sets. ATTILA JOÓ The proof of the main result
We start by giving a short outline of the proof. In the first two subsections we applyrelatively simple techniques in order to reduce Theorem 1.3 to Theorem 1.4 and cut thelatter problem into countable sub-problems. The third subsection is devoted to the proofof the reduced problem, namely the countable case of Theorem 1.4. The core of thatproof is to show that for every given e ∈ S t ∈ T δ ( t ) there is a path P through e such that G − E ( P ) maintains the premisses of Theorem 1.4. Proof of Theorem 1.3.
We will use only that { t ∈ T : d ( t ) > } is countable instead ofthe countability of the whole T . As a first step we reduce Theorem 1.3 to the followingtheorem. Theorem 4.1.
Let G be a graph and let T ⊆ V ( G ) such that d ( t ) ≤ for all but countablymany t ∈ T and there is no X ⊆ V ( G ) \ T where d ( X ) is an odd natural number. Assumethat for each t ∈ T there is a system P t of edge-disjoint T -paths covering δ ( t ) . Then thereexists a system P of edge-disjoint T -paths covering S t ∈ T δ ( t ) . For s ∈ T we will write shortly s -wave instead of s ( T − s )-wave. Recall, it is a system W of pairwise edge-disjoint paths starting at s such that the set C W of the last edges ofthe paths is a cut separating s and T − s .4.1. Elimination of waves.
We will call shortly the condition about the existence ofpath-system P t in Theorem 4.1 the linkability condition for t (w.r.t. G and T ) and werefer to the conjunction of these for t ∈ T as linkability condition . First we define a processthat we call wave elimination . We may assume that G is connected otherwise we definethe elimination process component-wise. Let T ⊆ T be given where T = { t ξ : ξ < κ } andwe define by transfinite recursion G ξ for ξ ≤ κ . Let G := G . If G ξ is already defined thenlet W ξ be a large t ξ -wave with respect to G ξ and T (exists by Lemma 3.3). We obtain G ξ +1 by contracting the t ξ -side of the cut C W ξ in G ξ to t ξ (see Figure 2). If ξ is a limitordinal then we obtain G ξ by doing all the previous contractions simultaneously. Therecursion is done.The cardinal d G κ ( X ) for X ⊆ V ( G κ ) \ T cannot be an odd natural number because d G κ ( X ) = d G ( X ) and G was inner Eulerian w.r.t. T . Furthermore, Corollary 3.4 ensuresthat for ξ < κ there is no non-trivial t ξ -wave in G ξ +1 . Since any t ξ -wave in G κ is alsoa t ξ -wave in G ξ +1 , it follows that for each t ∈ T there is only the trivial t -wave in G κ .By taking T := T , this is (more than) enough to guarantee the linkability condition atTheorem 4.1 (see Corollary 3.4). Therefore G κ satisfies the premisses of Theorem 4.1 andhence assuming Theorem 4.1 we may conclude that there is a system P of T -paths in G κ covering S t ∈ T δ G κ ( t ). By using the waves W ξ , the system P can be extended to a system Q of T -paths in G where the t ξ ( T − t ξ )-cut C W ξ is orthogonal to Q t ξ := { Q ∈ Q : t ξ ∈ V ( Q ) } .Therefore Q satisfies the requirements of Theorem 1.3. HE LOVÁSZ-CHERKASSKIY THEOREM IN COUNTABLE GRAPHS 7 t t t W W W Figure 2.
The contracted vertex sets during the wave elimination process4.2.
Reduction to countable graphs.
In the next reduction we show that it is enoughto restrict our attention to countable graphs in the proof of Theorem 4.1. First of all, wemay assume without loss of generality that T does not span any edges. Indeed, otherwisewe delete those edges and a set of T -paths for the remaining system demanded by 4.1extended by the deleted edges as T -paths of length one suffices 4.1 for the original system.By applying some basic elementary submodel-type of arguments we cut E into countablepieces each of them satisfying both the inner Eulerian and the linkability condition w.r.t. T . The contraction of T to some t results in an Eulerian graph G/T by Theorem 1.2thus we can take a partition O of E ( G/T ) = E into (edge sets of) G/T -cycles. These arecycles and T -paths in G . Let T := { t ∈ T : d ( t ) > } . For t ∈ T let P t be a system of T -paths witnessing the linkability condition for t and let E := { E ( P ) : ( ∃ t ∈ T ) P ∈ P t } .We define a closure operation c on 2 E in the following way. Intuitively we want to closea set F ⊆ E under the property that if it shares an edge with some O or E ( P ), then itcontains it completely. Formally let c ( F ) := S n ∈ N F n where F n +1 := F n ∪ [ { O ∈ O : F n ∩ O = ∅} ∪ [ { E ( P ) ∈ E : F n ∩ E ( P ) = ∅} . We call an
F c -closed if c ( F ) = F . We claim that c satisfies the following properties:(1) The family of c -closed sets forms a complete Boolean algebra with respect to theusual ∪ and ∩ ;(2) If F is countable then so is c ( F );(3) If F is c -closed, then graph ( V, F, I ) and T satisfy the premisses of Theorem 4.1.Indeed, property (1) follows directly from the construction and (2) holds because of theassumption | T | ≤ ℵ . The inner Eulerian and linkability for t ∈ T parts of condition(3) ensured by F not subdividing any O and E ( P ) respectively. Recall that d ( t ) ≤ t ∈ T \ T by assumption. Preservation of the linkability for these t is “automatic”: Lemma 4.2. If H is an inner Eulerian graph w.r.t. T ⊆ V ( H ) , then the linkabilitycondition holds for all t ∈ T with d ( t ) ≤ .Proof. E ( H ) can be partitioned into the edge sets of cycles and T -paths. If d ( t ) = 1, thenthe unique edge incident with t cannot be in a cycle so must be in a T -path. (cid:3) ATTILA JOÓ
In order to reduce Theorem 4.1 to countable graphs, it is enough to partition E intocountable c -closed sets F ξ . Indeed, then G ξ := ( V, F ξ , I ) is countable (apart from isolatedvertices) and satisfies the premisses of Theorem 4.1 with T by property (3). Hence byapplying the countable case of Theorem 4.1, we can take a system P ξ of T -paths in G ξ covering the edges S t ∈ T δ G ξ ( t ). Finally, S ξ P ξ is as desired.Suppose that the pairwise disjoint countable c -closed sets { F ξ : ξ < α } are alreadydefined for some ordinal α . Then E \ S ξ<α F ξ is c -closed by property (1). If it is emptythen we are done. Otherwise let F α := c ( { e } ) for an arbitrary e ∈ E \ S ξ<α F ξ , which iscountable by property (2). The recursion is done.4.3. The proof of Theorem 4.1.
We will make use of the following simple observation.
Observation 4.3.
The deletion of the edges of a T -path preserves the condition that thereis no X ⊆ V \ T with d ( X ) odd. The core of our proof is the repeated application of the following claim:
Claim 4.4.
For every t ∈ T and e ∈ δ ( t ) there exists a T -path P through e such that G − E ( P ) satisfies the linkability condition. Indeed, we only need to prove Theorem 4.1 for countable G as discussed in the previoussubsection. Assuming Claim 4.4, a system of T -paths covering S t ∈ T δ ( t ) can be constructedby a straightforward recursion. Proof of Claim 4.4.
First we give a proof in the special case where there is some s ∈ T such that d ( t ) ≤ t ∈ T − s . Let us fix a system P s of edge-disjoint paths between s and T − s covering δ ( s ).For e ∈ δ ( s ), we simply take the unique P ∈ P s through e . By Observation 4.3 graph G − E ( P ) is still Eulerian w.r.t. T . By Lemma 4.2 it is enough to check that the linkabilitycondition is preserved for s but it is obviously true witnessed by P s \ { P } .Suppose now that e ∈ δ ( t ) for a t ∈ T − s . If t is an end-vertex of some P ∈ P s , then wetake P and argue as in the previous paragraph. If it is not the case, then either we replace P s by another P s where t is an end-vertex of some P ∈ P s or chose P to be edge-disjointfrom P s . To do so, let Q be an arbitrary path between t and T − t . If E ( Q ) ∩ E ( P s ) = ∅ ,then we take P := Q and the linkability condition holds for s since P lives in G − E ( P ).If E ( Q ) ∩ E ( P ) = ∅ , then let v ∈ V ( Q ) ∩ V ( P ) be the first common vertex while goingalong Q from t . Let P ∈ P s such that v ∈ V ( P ). We get P s by replacing P in P s withthe path P we obtain by uniting the initial segment of P from s to v with the initialsegment of Q from t to v .Applying this iteratively together with the technique discussed in Subsection 4.2 weconclude: Corollary 4.5.
Theorem 4.1 holds whenever there is an s ∈ T such that d ( t ) ≤ for t ∈ T − s . HE LOVÁSZ-CHERKASSKIY THEOREM IN COUNTABLE GRAPHS 9
Proposition 4.6.
Assume that G = ( V, E, I ) is an inner Eulerian graph w.r.t. T ⊆ V such that there is no non-trivial s -wave for some s ∈ T . Then for every f, h ∈ E , thelinkability condition holds for s in G − f − h .Proof. We may assume without loss of generality that G is connected, since only thecomponent containing s is relevant. Since deletion of edges in δ ( s ) make the linkability for s a weaker requirement, we can also assume that f, h ∈ E \ δ ( s ). If G is finite and X ⊆ V with X ∩ T = { s } , then d ( s ) and d ( X ) must have the same parity because d ( v ) is even for v ∈ X − s . This observation of Lovász led immediately to the justification of Proposition4.6 for finite graphs. Indeed, on the one hand, d ( s ) < d ( X ) if { s } (cid:40) X ⊆ V \ ( T − s ),since δ ( s ) is the only Erdős-Menger s ( T − s )-cut by assumption. On the other hand, thesame parity of d ( s ) and d ( X ) ensures d ( s ) + 2 ≤ d ( X ). The proof of Proposition 4.6 forinfinite graphs is more involved and we need some preparation.For a graph H and distinct s, t ∈ V ( H ), we call an Erdős-Menger st -cut C s -tight ifthere is system P of edge-disjoint paths in H between s and t covering δ H ( s ) but everysuch a path-system is orthogonal to C . Lemma 4.7.
Assume that H is a graph, s, t ∈ V ( H ) are distinct and there is a system P of edge-disjoint paths in H between s and t covering δ H ( s ) but there is an e ∈ E ( H ) \ δ H ( s ) such that e ∈ E ( P ) for every such a path-system. Then there exists an s -tight Erdős-Menger st -cut C containing e .Proof. We may assume that H is connected, since otherwise we consider the componentcontaining s and t . Let P and e as in the lemma. Then there is a unique P e ∈ P through e . If H − e is disconnected, then P = { P e } and C := { e } is as desired. Suppose that H − e is connected. Let D be the (cid:22) -smallest Erdős-Menger st -cut in H − e (see Lemma3.2). We are going to prove that C := D + e is as desired. To do so, it is enough to showthat Q := P \ { P e } is orthogonal to D . Indeed, if it is done, then e must connect the twoparts of cut D in H − e and therefore D + e is an st -cut in H and P is orthogonal to it.Suppose for a contradiction that Q is not orthogonal to D . s tD Q e Figure 3.
Graph H − e where D is not orthogonal to Q . The first edge of P e is e .Let H what we get by contracting the t -side of D to t in H . Then D = δ H − e ( t ) and itis the only element of C H − e ( s, t ) since it is the smallest but also the largest one. We applythe Augmenting path lemma 3.7 in H − e with s, t and the set Q of st -paths in H − e given by the initial segments of the paths in Q . The augmentation must be successful,since otherwise it would give a D ∈ C H − e ( s, t ) with D = D . Indeed, D \ E ( Q ) = ∅ by the indirect assumption but D ⊆ E ( Q ) according to Lemma 3.7. The successful augmentation provides a system Q of edge-disjoint st -paths in H − e covering δ H − e ( s ).Indeed, there is a unique e ∈ δ H − e ( s ) which is uncovered by Q , namely the first edge of P e , but Augmenting path lemma 3.7 ensures δ Q ( s ) ⊂ δ Q ( s ). Since D ∈ C H − e ( s, t ), thepaths in Q can be forward extended in H to obtain a system of edge-disjoint st -pathsin H − e covering δ H ( s ) contradicting the obligatory usage of e in the assumption of thelemma. (cid:3) Since the only Erdős-Menger s ( T − s )-cut is δ ( s ) (because there is no non-trivial s -wave)and f / ∈ δ ( s ), Lemma 4.7 ensures that there is a system P s of edge-disjoint paths in G − f between s and T − s covering δ ( s ). Suppose for a contradiction that such a path-systemcannot be found in G − f − h . By applying Lemma 4.7 again this time with G − f and h , we obtain an s -tight Erdős-Menger s ( T − s )-cut C in G − f containing h . Let S bethe s -side of cut C . Then δ G − f ( S ) = C and we must have f ∈ δ G ( S ) since otherwise theinitial segments of the paths in P s up to their unique edge in C would form a non-trivial s -wave with respect to G and T . Since Erdős-Menger cuts are ⊆ -minimal cuts, G [ S ] isconnected. We define G by extending G [ S ] with new vertices { t e : e ∈ δ G ( S ) } and withthe edges δ ( S ) where an e ∈ δ ( S ) keeps its original end-vertex in S and gets t e as theother end-vertex. We define T := { s } ∪ { t e : e ∈ δ G ( S ) } . st f t e t e t e Q f Q ¬ s S C + f Figure 4.
Graph G and path-system Q .For X ⊆ V ( G ) \ T , the cardinal d G ( X ) cannot be an odd number because d G ( X ) = d G ( X ) by construction. Moreover, the linkability condition with respect to G and T holds, since for s it is witnessed by the initial segments of the paths in P s while theconnectivity of G guarantees it for the vertices in T − s . Thus the premisses of Theorem4.1 are satisfied, furthermore, all the vertices in T except possibly s has degree 1. Byapplying Corollary 4.5 to G and T , we can take a system Q of T -paths in G coveringall the edges incident with some vertex in T .It cannot happen that all Q ∈ Q has s as an end-vertex because then Q would provide anon-trivial s -wave with respect to G and T (where f / ∈ δ ( s ) is used to ensure ‘non-trivial’).Let Q ¬ s ∈ Q be a path with s / ∈ V ( Q ¬ s ) and let us denote the end-vertices of Q ¬ s by t e and t e . Lemma 4.8.
Every system R of edge-disjoint T -paths in G covering δ ( s ) and avoiding t f must use all the vertices { t e : e ∈ C } . HE LOVÁSZ-CHERKASSKIY THEOREM IN COUNTABLE GRAPHS 11
Proof.
Suppose for a contradiction that R is a counterexample and let R be the path-system in G − f corresponding to R . Then R is a system of edge-disjoint paths startingat s and having exactly their last edges in C such that C \ E ( R ) = ∅ . Since C is anErdős-Menger s ( T − s )-cut in G − f , the paths in R can be forward extended to obtain asystem R + of edge-disjoint s ( T − s )-paths with C ∩ E ( R + ) = C ∩ E ( R ). Then R + alsocovers δ ( s ) and C \ E ( R + ) = ∅ contradicting the s -tightness of C in G − f . (cid:3) Let Q f ∈ Q the unique path with t f ∈ V ( Q f ). We claim that the other end-vertex of Q f must be s , thus in particular Q f = Q ¬ s and hence f / ∈ { e , e } . Indeed, since otherwisethe system Q s := { Q ∈ Q : s ∈ V ( Q ) } of edge-disjoint T -paths in G covers δ ( s ) usingneither t f nor the other end-vertex of Q f which contradicts Lemma 4.8. Now we considerthe path-system Q s \ { Q f } . It covers all but one edges in δ ( s ) and avoids e and e . Weapply the Augmenting path lemma 3.7 in G with Q s \ { Q f } , s and { t e : e ∈ C } . If theaugmentation is successful, then the resulting path-system covers δ ( s ) and at least oneof e and e is still unused contradicting Lemma 4.8. Thus the Augmenting path lemmaensures that we can pick a single edge from each path in Q s \ { Q f } such that the resultingedge set C separates s and { t e : e ∈ C } in G . We take the initial segments of the paths in P s until the first meeting with C and continue them forward using the terminal segmentsof the corresponding paths from Q s \ { Q f } to obtain a set of T -paths in G covering δ ( s )without using t e , t e and t f , which contradicts Lemma 4.8. (cid:3) Now we can finish the proof of Claim 4.4. Suppose for a contradiction that
G, T, s ∈ T and e ∈ δ ( s ) form a counterexample and P s = { P e : e ∈ δ ( s ) } is a system of edge-disjoint T -paths with e ∈ E ( P e ). We may assume that G, T, s, e and P s have been chosen tominimize | E ( P e ) | among the possible options. We know that | E ( P e ) | ≥ P e were consists of the single edge e , then P := P e would satisfy Claim 4.4. We proceed byapplying wave elimination with T := T − s (see Subsection 4.1) and denote the resultinggraph by G . The linkability condition for a t ∈ T w.r.t. G and T combined with Corollary3.6 allows us to choose wave W t during the elimination in such a way that δ W t ( t ) = δ G ( t ).Let us define W s to be the trivial s -wave in G . For any T -path Q in G with end-vertices t and t there is a unique W t ∈ W t and W t ∈ W t containing an extreme edge of Q . Unitingthese paths with Q results in a T -path ‘ ( Q ) between t and t in G . Furthermore, the imagesof pairwise edge-disjoint paths under this lifting operation ‘ are pairwise edge-disjoint.We claim that G , T, s and e must be also a counterexample for Claim 4.4. Suppose fora contradiction that P witnesses that it is not. We will conclude that then P := ‘ ( P )shows that G, T, s and e is also not a counterexample. Indeed, we need to check that thelinkability condition holds w.r.t. G − E ( P ) and T . For t ∈ T , we take a path-system Q t witnessing the linkability condition in G − E ( P ) for t . Since W t was chosen in the wayto cover δ G ( t ), the point-wise image ‘ [ Q t ] witnessing the linkability for t in G − E ( P ).Thus G , T, s and e form indeed a counterexample.Moreover, the set P s = { P e : e ∈ δ ( s ) } of the suitable initial segments of the paths in P s guarantee by (cid:12)(cid:12)(cid:12) P e (cid:12)(cid:12)(cid:12) ≤ | P e | that this newcounterexample also minimizing.We may assume (just to simplify the notation) that our original counterexample haschosen in such a way that there is no non-trivial t -wave for t ∈ T − s . Let f ∈ E ( P e ) be the edge right after e in P e . We replace e and f by a single new edge h connecting s and the end-vertex of f that is not shared with e (splitting technique by Lovász from[16]). Let P h be the path in the resulting graph G with E ( P h ) = E ( P e ) − e − f + h and let us define P s := P s − P e + P h . For X ⊆ V \ T the quantities d G ( X ) and d G ( X )are either both infinite or they have the same parity, thus G is also inner Eulerian w.r.t. T .The linkability condition for s holds in G witnessed by P s . Let t ∈ T − s be arbitrary. Thelinkability condition for t holds in G − e − f by Proposition 4.6, moreover, if h ∈ δ G ( t ),then h is an edge between s and t and hence a T -path itself. Note that G , T, s and h cannot be a counterexample for Claim 4.4 because | E ( P h ) | = | E ( P e ) | −
1. Therefore wecan pick some T -path P in G through h such that the linkability condition holds in G − E ( P ). Let us take then a T -path P in G through e with E ( P ) ⊆ E ( P ) − h + e + f .Since G − E ( P ) is a subgraph of G − E ( P ) and δ G − E ( P ) ( t ) = δ G − E ( P ) ( t ) holds for t ∈ T ,the linkability condition in G − E ( P ) implies the linkability in G − E ( P ). This contradictsthe assumption that G, T, s and e form a counterexample. (cid:3)(cid:3) Outlook
Edge-disjoint T-paths in non-Eulerian graphs.
Let G be a graph and let T ⊆ V ( G ). A T -subpartition is a family A = { X t : t ∈ T } of pairwise disjoint subsets of V ( G )such that X t ∩ T = { t } . If G is finite, then we call a component Y of G − S A obstructive if d ( Y ) is odd. Let o ( G, A ) be the number of the obstructive components. Theorem 5.1 (Mader, [17]) . Let G be a finite graph and let T ⊆ V ( G ) . Then the maximalnumber of pairwise edge-disjoint T -paths is min ( X t ∈ T d ( X t ) − o ( G, A ) ! : A is a T -subpartition ) . Let us define E ( A ) := S t ∈ T δ ( X t ). In Theorem 5.1, for a system P of edge-disjoint T -paths and a T -partition A we have equality if and only if the following conditions hold: Condition 5.2 (complementarity slackness) . (1) Each P ∈ P uses either only a single edge from E ( A ) (which must connect twovertex sets in A ) or two edges incident with a component of G − S A .(2) For each component Y of G − S A , path-system P uses all but at most one edgefrom δ ( Y ) . Conjecture 5.3.
Let G be a (possibly infinite) graph and let T ⊆ V ( G ) . Then there existsa system P of edge-disjoint T -paths and a T -subpartition A satisfying Condition 5.2. Although the Lovász-Cherkasskiy theorem 1.1 is a special case of Mader’s edge-disjoint T -path theorem 5.1, our Conjecture 5.3 does not seem to imply our main result Theorem1.3. It motivates to formulate a stronger conjecture based on the extension of the conceptof obstructive components. HE LOVÁSZ-CHERKASSKIY THEOREM IN COUNTABLE GRAPHS 13
For a possibly infinite graph G , we define a component Y of G − S A to be obstructive ifafter the contraction of V ( G ) \ Y to some vertex v the resulting graph H does not containa set of pairwise edge-disjoint cycles covering δ H ( v ). This extends our previous definitionof obstructive. Indeed, on the one hand, if d G ( Y ) is odd, then d H ( v ) is the same oddnumber and hence δ H ( v ) cannot be covered by edge-disjoint cycles. On the other hand,if d ( Y ) is even, then finding the desired cycles is equivalent to finding a J -join in theconnected graph G [ Y ] where J consists of those u ∈ Y for which there are odd number ofedges between u and v in H . Condition 5.4. (1) Each P ∈ P uses either only a single edge from E ( A ) (which must connect twovertex sets in A ) or two edges incident with a component of G − S A .(2) The path-system P uses all the edges E ( A ) except one from δ ( Y ) for each obstructivecomponent Y . Note that if G is inner Eulerian, then cannot be any obstructive components (regardless ofthe choice of A ) and therefore by replacing Condition 5.2 with Condition 5.4 in Conjecture5.3 it will imply Theorem 1.4 (even without the restriction | T | ≤ ℵ ). We also point outthat for finite graphs Conditions 5.2 and 5.4 are equivalent because if d ( Y ) is even, then P cannot misses exactly one edge from δ ( Y ).Recall a system P of edge-disjoint/vertex-disjoint/internally vertex-disjoint T -paths is strongly maximal if |Q \ P| ≤ |P \ Q| for every edge-disjoint/vertex-disjoint/internallyvertex-disjoint system Q of T -paths. Conjecture 5.5.
Let G be a (possibly infinite) graph and let T ⊆ V ( G ) . Then for asystem P of edge-disjoint T -paths the following statements are equivalent:(i) P is a strongly maximal system of edge-disjoint T -paths.(ii) There exists a T -subpartition A satisfying Condition 5.2 with P .(iii) There exists a T -subpartition A satisfying Condition 5.4 with P . Note that only the implication ( i ) = ⇒ ( iii ) is an open question. Indeed, the implication( iii ) = ⇒ ( ii ) is trivial. Assuming ( ii ), P must be an inclusion-wise maximal system ofedge-disjoint T -paths. If |P \ Q| = κ ≥ ℵ , then | E ( P \ Q ) | = κ and since each P ∈ Q \ P must contain an edge from E ( P \ Q ), we obtain |Q \ P| ≤ κ . If |P \ Q| = k ∈ N , then let G := G − E ( P ∩ Q ). Then d G ( Y ) is finite for every component of G − S A and for all ofbut finitely may Y it is 0, moreover,12 X t ∈ T d G ( X t ) − o ( G , A ) ! = k, from which |Q \ P| ≤ k follows. Thus P is strongly maximal.5.2. Vertex-disjoint T-paths.
We conjectured already in the Introduction (Conjecture1.5) the existence strongly maximal systems of T -paths with different concepts of dis-jointness. We believe that strong maximality can be characterized by the existence ofa certain dual object reflecting the corresponding classical theorems of Gallai [11] andMader [18]. If T = V ( G ), then a vertex-disjoint system of T -paths is a matching. Infinite matching theory was intensively investigated and is well-understood (see the survey [4]).The existence of a strongly maximal matching in a graph was proven by Aharoni (seeTheorem 5.3 of [4]) together with the following theorem: Theorem 5.6 (Aharoni, Theorem 5.2 of [4]) . In every (possibly infinite) graph G there isa matching M ⊆ E ( G ) and an X ⊆ V ( G ) such that:(1) For each component Y of G − X , the edges in M spanned by Y cover all but atmost one vertex of Y .(2) The vertices in X are covered by M in such a way that X does not span any edgein M .(3) G [ Y ] is factor-critical whenever Y is a component of G − X for which M doesnot contain a perfect matching of G [ Y ] .(4) Let Π( G, X ) be the bipartite graph whose vertex classes are X and the set Y ofthe factor-critical components of G − X , furthermore, an xY is an edge if x hasa neighbour in Y in G . Then for every Y ∈ Y there is a matching in Π( G, X ) covering X while avoiding vertex Y . Remark 5.7. • Properties (1) and (2) at Theorem 5.6 are already sufficient to ensure the strongmaximality of the matching M . • For every strongly maximal matching M there is an X satisfying (1)-(4). • Property (4) was originally not mentioned by Aharoni but it can be obtained easilyby applying for example Lemma 3.6 of [3]. • If there is a matching M for which V ( M ) is ⊆ -maximal (which is always the casein countable graphs), then the set X in Theorem 5.6 is unique.By omitting the assumption of T = V ( G ) we leave matchings theory and formulate aninfinite generalization of Gallai’s theorem [11]: Conjecture 5.8.
Let G be a (possibly infinite) graph and let T ⊆ V ( G ) . Then there existsa system P of vertex-disjoint T -paths and an X ⊆ V ( G ) such that:(1) For each component Y of G − X , the paths { P ∈ P : V ( P ) ⊆ Y } cover all but atmost one vertex of T ∩ Y .(2) X ⊆ V ( P ) where | V ( P ) ∩ X | ≤ for every P ∈ P . Assume now that there is a partition S of T and let us call S -path a T -path whoseend-vertices are in different members of S . A minimax formula for the maximal numberof S -paths was given by Mader in [18]. We expect the following generalization based onthe complementarity slackness conditions to be true: Conjecture 5.9.
Assume that G is a (possibly infinite) graph, T ⊆ V ( G ) and S is apartition of T . Then there exists a system P of vertex-disjoint S -paths, an X ⊆ V ( G ) anda partition Y of V ( G ) \ X such that:(0) After the deletion of the vertex set X and the edges of the subgraphs G [ Y ] for Y ∈ Y the resulting graph does not contain any S -path. does not have a perfect matching but after deleting any vertex the resulting graph has HE LOVÁSZ-CHERKASSKIY THEOREM IN COUNTABLE GRAPHS 15
For B Y := ( T ∩ Y ) ∪ { v ∈ Y : v has a neighbour in some Y ∈ Y \ { Y }} : (1) The paths in P cover X and all but at most one vertex of B Y for every Y ∈ Y .(2) For every P ∈ P either | V ( P ) ∩ X | = 1 and | V ( P ) ∩ B Y | ≤ for every Y ∈ Y or | V ( P ) ∩ X | = 0 and there is a unique Y P ∈ Y with | V ( P ) ∩ B Y P | = 2 while | V ( P ) ∩ B Y | ≤ for Y ∈ Y \ { Y P } .(3) For every Y ∈ Y there is at most one P ∈ P with | V ( P ) ∩ Y | = 1 . References [1] R Aharoni, C Nash-Williams, and S Shelah,
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Email address : [email protected] Attila Joó, Alfréd Rényi Institute of Mathematics, Set theory and general topologyresearch division, 13-15 Reáltanoda St., Budapest, Hungary
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