On Partitioning the Edges of an Infinite Digraph into Directed Cycles

Nash-Williams proved in [1] that for an undirected graph G the set E¹Gº can be
partitioned into cycles if and only if there is no finite cut of odd size. Later C. Thomassen
gave a simpler proof for this in [2] and conjectured the following directed analogue of the
theorem: the edge set of a digraph can be partitioned into directed cycles if and only if for
each subset of the vertices the cardinality of the ingoing and the outgoing edges are equal. The aim of the paper is to prove this conjecture.


Introduction
Nash-Williams was one of the greatest researchers of the theory of nite and innite graphs in the 20th century.One of his famous result is that the edges of a 2k-edge-connected graph can be directed to obtain a k-edge-connected digraph.He proved it for nite graphs and claimed to be true for innite ones in [3].For the innite case there is still no published proof.The best partial result is due to C. Thomassen in [5], he showed that the edges of a 8k-edge-connected innite graph can be directed to obtain a k-edge-connected digraph.An other famous result of Nash-Williams is the following.
Theorem 1 (Nash-Williams, [2] (p.235 Theorem 3)).If G is an undirected graph, then E(G) can be partitioned into cycles if and only if every cut has either even or innite number of edges.
At the and of his article ( [2] page 237 Theorem 3') he claimed without proof the following directed analogue of his theorem: Theorem 2. If D = (V, A) is a directed graph, then A can be partitioned into directed cycles if and only if for all X ⊆ V the cardinalities of the ingoing and the outgoing edges of X are equal.
As far as we know there is no published (even partial) result about the directed version.
L. Soukup gave a new shorter proof to the undirected version (Theorem 5.1 of [4]) based on elementary submodels but nothing about the directed case.The main diculty in contrast to the undirected case, applying elementary submodel approach, is that in the undirected case one can nd a nite witness for the violation of the condition (an odd cut) but in the directed case we do not necessarily have a nite witness.Our main result is a proof of Theorem 2 by handling the additional diculties of the elementary submodel approach.

Notation
Let D = (V, A) be a digraph.We denote by out D (X) and by in D (X) the set of outgoing and ingoing edges of X in D respectively.For an X ⊆ V let D[X] the subgraph of D induced by X.The weakly connected components of a digraph are the connected components of its undirected underlying graph with the original orientations.We call a digraph weakly connected if its undirected underlying graph is connected.If x, y are vertices of the path P , then we denote by P [x, y] the segment of P between x and y.We will also use some basic standard notation from set theory and model theory.
We call X ⊆ V overloaded (with respect to D = (V, A)) if |out D (X)| < |in D (X)| and we call D balanced if there is no such an X.If M is an arbitrary set, then let D(M ) := (V ∩M, A∩M ) and let D M := (V, A \ M ).

An observation about overloaded sets
We need the following basic observation to nd overloaded sets in an unbalanced digraph in a special form.Proof: Let X ⊆ V be overloaded and let X i (i ∈ I) be the vertex sets of the weakly connected Denote by Z the vertex set of the weakly connected component of D that contains Y j0 then X := Z \ Y j0 is appropriate and Y j0 will be the desired Y .

Elementary submodels
We give here a quick survey about elementary submodel techniques that we use to prove the main result of this chapter.One can nd a more detailed survey with many combinatorial applications in [4].
All the formulas and models in this chapter are in the rst order language of set theory and the models are ∈-models i.e. the element of relation in them is the real The common practice by elementary submodel techniques is to x a large enough nite set Σ of formulas at the beginning and do not say explicitly what it is.After that, during the proof the author refers nitely many times that this and that formula is in Σ.If it is not satisfactory for someone, then he or she may consider Σ as the set of those formulas that have length at most 10 10 and contains at most the variables: v 1 , . . ., v 10 10 .Anyway, from now on Σ is a xed, large enough set of formulas.
Our next goal is to create Σ-elementary submodels.We will use the following two well-known theorems.One can nd them in [1] as well as in other textbooks in the topic.

Main result
Proof of Theorem 2. A directed cycle has the same number of ingoing and outgoing edges for an X ⊆ V thus if A can be partitioned into directed cycles, then D must be balanced.Next we deal with the nontrivial direction of the equivalence.
Observe that the weakly connected components of a balanced digraph are strongly connected thus each of their edges are in some directed cycle.Furthermore, a balanced digraph remains balanced after the deletion of the edges of a directed cycle.If a balanced digraph is at most countable and its edges are: e 1 , e 2 , . . ., then we can create a desired partition by the following recursion: in the n-th step delete the edges of a directed cycle which contains e n from the remaining digraph if it still contains e n , otherwise do nothing.
In the uncountable case the naive recursive method above does not work because in a transnite recursion one can not ensure that after the rst limit step the remaining digraph is still balanced.
If we prove Claim 9, then we are done with the proof of Lemma 8 as well.Indeed, by Claim 9, the digraphs D M α are balanced and therefore by using the 4th property of the recursion with D M α and with M α+1 we can partition A(D α ) into directed cycles for all β ≤ α < λ thus we get a desired partition of A ∩ M by uniting the partitions of the edge sets A(D α ).
Before the proof of Claim 9 we need some preparations.
Proposition 10.Let G be an undirected graph an let M be a Σ-elementary submodel such that Proof: We assume that Σ contains the formulas that expressing the followings: 2. E ⊆ E(G) separates the vertices u and v in graph G, f is a bijection between the sets X and Y .
separates the vertices u and v in graph G and f is a bijection between κ and E .Formulas 2 and 3 ensures that E ⊆ E(G) separates the vertices u and v in graph G and f is a bijection between κ and E .Since f ∈ M and κ ≤ |M | ⊆ M the range of f is a subset of M i.e.E ⊆ M therefore λ G M (u, v) = 0 since E separates v and u.
We need the following result of L. Soukup (see [4]  By using Proposition 11 to the undirected underlying graph of D with F and with arbitrary x ∈ X and y ∈ Y vertices we conclude that X and Y belongs to distinct weakly connected components of D F .Let us denote by X and Y the vertex set of these components.We claim that cut D M (X) = cut D (X ).Indeed, cut D (X ) might not have element that not in F by the denition of X and the elements of F goes between X and Y and therefore between X and Y .

Lemma 3 .
If D = (V, A) is an unbalanced digraph, then it has a weakly connected component with vertex set Z and an X ∪ Y partition of Z such that D[X] and D[Y ] are weakly connected and X is overloaded in D.

Theorem 5 (
Downward LöwenheimSkolem-Tarski Theorem).Let A be a rst order structure for language L with basic set A. Denote the set of L-formulas by Form(L).Assume that |Form(L)| ≤ |A|.Then for all B ⊆ A there exists an elementary submodel C of A with basic set C such that B ⊆ C and |C| = |Form(L)| + |B|.Remark 6.In the case of set theory |Form(L)| = ℵ 0 so if B is innite, then we may write |C| = |B| in Theorem 5. Now we can prove a fundamental fact about Σ-elementary submodels.Proposition 7.For all innite set B there is a Σ-elementary submodel M such that B ⊆ M and |M | = |B|.Proof: By Theorem 4 there is a β ≥ rank(B) such that V β is a Σ-elementary submodel.Then B ⊆ V β since β ≥ rank(B).Thus by using Theorem 5 with A = V β and with B we get an elementary submodel M of V β such that |M | = |B| and B ⊆ M .Finally M is a Σ-elementary submodel because it is an elementary submodel of a Σ-elementary submodel.
Lemma 5.3 on p. 16): Proposition 11.Let G be an undirected graph and let M be a Σ-elementary submodel such that G ∈ M and |M | ⊆ M .Assume that x = y ∈ V (G) are in the same component of G M and F ⊆ E(G M ) separates them where |F | ≤ |M |.Then F separates x and y in the whole G.Proof: Assume (reductio ad absurdum) that it is false and G, F, x, y, M witness it.Take a path P between x and y in G F .Denote by x and by y the rst and the last intersection of P with V ∩ M with respect to some direction of P .The vertices x and y are well-dened and distinct since P necessarily uses some edge from E(G) ∩ M .Fix also a path Q between x and y in G M .The paths P [x , x], Q, P [y, y ] shows that x and y are in the same component of G M .Thus by Proposition 10 λ G (x , y ) > |M |.We may x a path R between x and y in G F since λ G (x , y ) > |M | ≥ |F |.But then P [x, x ], R, P [y , y] shows that F does not separate x and y in G M which is a contradiction.Now we turn to the proof of Claim 9. Assume, seeking for contradiction, that D M is unbalanced.Then by Lemma 3 there is a weakly connected component of D M with vertex set Z and an X ∪Y partition of Z such that (D M )[X] and (D M )[Y ] are weakly connected and X is overloaded in D M .Let F = cut D M (X).We want to show that |F | ≤ |M |.We may suppose that F is innite and thus cut D (X) as well since F ⊆ cut D (X).Thus ℵ 0 ≤ |out D (X)| = |in D (X)|.The inequality out D M (X) < |out D (X)| holds because otherwise out D M (X) = |out D (X)| = |in D (X)| ≥ in D M (X)which contradicts to the choice of X. Hence M contains |out D (X)| elements of out D (X) and thus |out D (X)| ≤ |M |.Then |F | = in D M (X) + out D M (X) ≤ |in D (X)| + |out D (X)| = |out D (X)| ≤ |M | .
But then out D M (X) = |out D (X )| and in D M (X) = |in D (X )| thus |out D (X )| = out D M (X) < in D M (X) = |in D (X )| therefore X is overloaded in D which is a contradiction.