aa r X i v : . [ m a t h . C O ] J un QUICKLY PROVING DIESTEL’S NORMAL SPANNING TREE CRITERION
MAX PITZ
Abstract.
We present two short proofs for Diestel’s criterion that a connected graph has anormal spanning tree provided it contains no subdivision of a countable clique in which everyedge has been replaced by uncountably many parallel edges. § Overview
This paper continues a line of inquiry started in [7] with the aim to find efficient algorithms forconstructing normal spanning trees in infinite graphs. A rooted spanning tree T of a graph G iscalled normal if the end vertices of any edge of G are comparable in the natural tree order of T .Intuitively, all the edges of G run ‘parallel’ to branches of T , but never ‘across’.Every countable connected graph has a normal spanning tree, but uncountable graphs might not,as demonstrated by complete graphs on uncountably many vertices. While exact characterisationsof graphs with normal spanning trees exist, see e.g. [5, 6], these may be hard to verify in practice.The most applied sufficient condition for normal spanning trees is the following criterion due toHalin [4], and its strengthening due to Diestel [2], see also [6, §6] for an updated proof. Theorem 1 (Halin, 1978) . Every connected graph without a
T K ℵ has a normal spanning tree. Theorem 2 (Diestel, 2016) . Every connected graph without fat
T K ℵ has a normal spanning tree. Here, a
T K ℵ is any subdivision of the countable clique K ℵ , and a fat T K ℵ is any subdivisionof the multigraph obtained from a K ℵ by replacing every edge with ℵ parallel edges.Until recently, only fairly involved proofs of these results were available: Halin’s original proofemploying his theory of simplicial decompositions [4], and Diestel’s proof strategy building on theforbidden minor characterisation for normal spanning trees [2, 6].In [7], however, the present author found a simple greedy algorithm which constructs the desirednormal spanning tree in Halin’s Theorem 1 in just ω many steps. The purpose of this note is toprovide two simple proofs also for Theorem 2, one of them again an ω -length algorithm.Notably, this algorithm also yields a new, local version of Theorem 2: Given a set of vertices U of a connected graph G , there exists a normal tree of G containing U if and only if every fat T K ℵ in G can be separated from U by a finite set of vertices, see Theorem 3 below. § Tree orders and normal trees
We follow the notation in [1]. The tree-order T of a tree T with root r is defined by setting u T v if u lies on the unique path from r to v in T . For a vertex v of T , let ⌈ v ⌉ := { t ∈ T : t T v } .For rooted trees that are not necessarily spanning, one generalises the notion of normality asfollows: A rooted tree T ⊆ G is normal (in G ) if the end vertices of any T -path in G (a path in G with end vertices in T but all edges and inner vertices outside of T ) are comparable in the treeorder of T . If T is spanning, this clearly reduces to the definition given in the introduction. If T ⊆ G is normal, then the set of neighbours N ( D ) of any component D of G − T forms a chainin T , i.e. all vertices of N ( D ) are comparable in T . Moreover, incomparable nodes v, w of anynormal tree T ⊆ G are separated in G by ⌈ v ⌉ ∩ ⌈ w ⌉ . Fact 1 (Jung [5, Satz 6]) . Let G be a graph with a normal spanning tree. Then for every connectedsubgraph C ⊆ G and every r ∈ C there is a normal spanning tree of C with root r . For distinct vertices v, w of G we denote by κ ( v, w ) = κ G ( v, w ) the connectivity between v and w in G , i.e. the largest size of a family of independent v − w paths. If v and w are non-adjacent,this is by Menger’s theorem equivalent to the minimal size of a v − w separator in G . Fact 2 (Halin, [3, (15)]) . Let U be a countable set of vertices in G . There is a fat T K ℵ withbranch vertices U if and only if κ ( u, v ) is uncountable for all u = v ∈ U . § The first proof
First proof of Theorem 2.
By induction on | G | . We may assume that | G | is uncountable. Supposewe have a continuous increasing ordinal-indexed sequence ( G i : i < σ ) of induced subgraphs all ofsize less than | G | with G = S i<σ G i such that ( i ) the end vertices of any G i -path in G have infinite connectivity in G i , and ( ii ) the end vertices of any G i -path in G have uncountable connectivity in G .Then we can construct normal spanning trees T i of G i extending each other all with the sameroot by (transfinite) recursion on i . If ℓ < σ is a limit, we may simply define T ℓ = S i<ℓ T i . For thesuccessor case, suppose that T i is already defined. By ( ii ) , the neighbourhood N ( C ) is finite forevery component C of G i +1 − G i (otherwise we get a fat T K ℵ by Fact 2), and by ( i ) , N ( C ) lies ona chain of T i (as incomparable vertices in T i are separated in G i by the intersection of their finitedown-closures). Let t C ∈ N ( C ) be maximal in the tree order of T i , and let r C be a neighbour of t C in C . By the induction hypothesis and Fact 1, C has a normal spanning tree T C with root r C .Then T i together with all T C and edges t C r C is a normal spanning tree T i +1 of G i +1 . Once therecursion is complete, T = S i<σ T i is the desired normal spanning tree of G .It remains to construct a sequence ( G i : i < σ ) with ( i ) and ( ii ) . This can be done, for example,by taking a continuous increasing chain ( M i : i < σ ) with σ = cf ( | G | ) of < | G | -sized elementary UICKLY PROVING DIESTEL’S NORMAL SPANNING TREE CRITERION 3 submodels M i of a large enough fragment of ZFC with G ∈ M i , such that G ⊆ S i<σ M i , see [8].Then G i = G ∩ M i is as required.Alternatively, use a countable closure argument to construct G i such that for every pair v, w ∈ V ( G i ) with κ G ( v, w ) ℵ , the graph G i contains a maximal family of independent v − w paths in G (this will guarantee ( ii ) ), and for all other pairs, G i contains at least countably many independent v − w paths (this will guarantee ( i ) ); and note that properties ( i ) and ( ii ) are preserved underincreasing unions. (cid:3) § The second proof
Our second proof extracts the closure properties ( i ) and ( ii ) in the previous construction, andcombines them into a single algorithm constructing the normal spanning tree in ω many steps,avoiding ordinals and transfinite constructions altogether. Second proof of Theorem 2.
For every pair of distinct vertices v and w of G with κ ( v, w ) at mostcountable, fix a maximal collection P v,w = { P v,w , P v,w , . . . } of independent v − w paths in G .Construct a countable chain T ⊆ T ⊆ T ⊆ · · · of rayless normal trees in G with the sameroot r ∈ V ( G ) as follows: Put T = { r } , and suppose T n has already been defined. Since T n isa rayless normal tree, any component D of G − T n has a finite neighbourhood N ( D ) in T . Foreach pair v = w ∈ N ( D ) with countable connectivity select the path P Dv,w with least index P v,w intersecting D . By [1, Proposition 1.5.6], we may extend T n finitely into every such component D as to cover P Dv,w ∩ D for all v = w ∈ N ( D ) (or at least one abitrarily chosen vertex making theextension into D is non-trivial), so that the extension T n +1 ⊇ T n is a rayless normal tree with root r . This completes the construction.The union T = S n ∈ N T n with root r is a normal tree in G . We claim that T is spanning unless G contains a fat T K ℵ . If T is not spanning, consider a component C of G − T . Then N ( C ) ⊆ T is infinite: otherwise, N ( C ) ⊆ T n for some n ∈ N but then we would have extended T n into C , acontradiction. For every n , let D n be the unique component of G − T n containing C .By Fact 2, it suffices to show that κ ( v, w ) is uncountable for every v = w ∈ N ( C ) . Consider a T -path P from v to w with ˚ P ⊆ C . If κ ( v, w ) was countable, then by maximality of P v,w thereis P kv,w ∈ P v,w with say P kv,w ∩ ˚ P ∋ x . Let m be minimal with v, w ∈ T m . Since the P D n v,w arepairwise distinct, the path P kv,w was selected as P D n v,w for some n with m n m + k . But then x ∈ P kv,w ∩ ˚ P ⊆ P D n v,w ∩ D n ⊆ T n +1 ⊆ T contradicts that P is a T -path. (cid:3) § Local versions of Diestel’s criterion
By a slight modification of this ω -length algorithm, one readily obtains a proof of the followingresults, which answer [6, Problem 3]. MAX PITZ
Theorem 3.
A set of vertices U in a connected graph G is contained in a normal tree of G providedevery fat T K ℵ in G can be separated from U by a finite set of vertices.Proof. Let U be a set of vertices such that every fat T K ℵ in G can be separated from U by afinite set of vertices. Use the algorithm from Section 4, but only extend T n into a component D of G − T n with U ∩ D = ∅ . Additionally, make sure to cover at least one vertex from U ∩ D .It remains to argue that U is contained in T = S T n . Otherwise, there is a component C of G − T containing a vertex from U . As in Section 4, this gives us a fat T K ℵ in G which furthermorecannot be separated from U by a finite set of vertices, cf. [7]. (cid:3) Theorem 4.
A connected graph has a normal spanning tree if and only if its vertex set is acountable union of sets each separated from any fat
T K ℵ by a finite set of vertices.Proof. For the forward implication, recall that the levels of any normal spanning tree can beseparated by a finite set of vertices from any ray, and hence in particular from any fat
T K ℵ .Conversely, let { V n : n ∈ N } be a collection of fat T K ℵ -dispersed sets in G with V ( G ) = S n ∈ N V n .Adapt the algorithm from Section 4, so that when extending T n into a component D of G − T n ,we additionally cover a vertex v D ∈ D ∩ V n D where n D minimal such that V n D ∩ D = ∅ . The proofthen proceeds as in [7]. (cid:3) References [1] Reinhard Diestel.
Graph Theory . Springer, 5th edition, 2015.[2] Reinhard Diestel. A simple existence criterion for normal spanning trees.
The Electronic Journal of Combina-torics , 2016. P2:33.[3] Rudolf Halin. Unterteilungen vollständiger Graphen in Graphen mit unendlicher chromatischer Zahl. In
Ab-handlungen aus dem Mathematischen Seminar der Universität Hamburg , volume 31, pages 156–165. Springer,1967.[4] Rudolf Halin. Simplicial decompositions of infinite graphs. In
Annals of Discrete Mathematics , volume 3, pages93–109. Elsevier, 1978.[5] Heinz A. Jung. Wurzelbäume und unendliche Wege in Graphen.
Mathematische Nachrichten , 41(1-3):1–22, 1969.[6] Max Pitz. Proof of Halin’s normal spanning tree conjecture. https://arxiv.org/abs/2005.02833, 2020.[7] Max Pitz. A unified existence theorem for normal spanning trees. https://arxiv.org/abs/2003.11575, 2020.[8] Lajos Soukup. Elementary submodels in infinite combinatorics.
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