Augmentation Lemma for Halin Conjecture

Augmen tation Lemma for Halin’s Conjecture Jerzy W ojciechowski W est Virginia Univ ersity Morgan town, WV, U.S.A. Marc h 27, 2026 Abstract The longstanding conjecture of Halin characterizing the existence of normal spanning trees in infinite graphs has been recently pro ved b y Max Pitz [3]. A critical step in the pro of in volv es the construction of dominated torsos, whose properties are essen tial to the ov erall proof. In this note, w e pro vide a correction to the proof of a k ey property of this construction. 1 In tro duction. W e follo w the notation in [1]. A ro oted spanning tree T of a graph G is normal if the ends of an y edge of G are comparable in the natural tree order of T . A graph G has c ountable c oloring numb er if there is a w ell-order ≤ ∗ on V ( G ) suc h that ev ery v ertex of G has only finitely many neigh b ors preceding it in ≤ ∗ . Halin [2] conjectured and Pitz [3] prov ed the follo wing result. Theorem 1. A c onne cte d gr aph has a normal sp anning tr e e if and only if every minor of it has c ountable c oloring numb er. A set U ⊆ V ( G ) of vertices of a graph G is disp erse d in G if every ra y in G can b e separated from U b e a finite set of v ertices. Given a subgraph H of G , a set of v ertices A ⊆ V ( H ) of the form A = N G ( D ) for some comp onen t D of G − H is an adhesion set . A subgraph H ⊆ G has finite adhesion if all adhesion sets in H are finite. The following Decomp osition Lemma (Lemma 3.3 in [3]) provides the back- b one of the proof of Theorem 1. Lemma 2 (Decomp osition Lemma) . L et G b e a c onne cte d gr aph of unc ountable size κ with the pr op erty that al l its minors have c ountable c oloring numb er. Then 1 G c an b e written as a c ontinuous incr e asing union S i ∈ σ G i of infinite, < κ -size d c onne cte d induc e d sub gr aphs G i of finite adhesion in G . T o use Lemma 2 in the pro of of Theorem 1, the key property is the following claim (Claim 4.1 in [3]) about the graphs G i from Lemma 2. Claim 3 . Ev ery V ( G i ) is a countable union of disp ersed sets in G . Let H b e a subgraph of a graph G . An augmentation of H in G is a con traction minor K of G that contains H as a subgraph. An augmentation K is faithful if every subset U ⊆ V H that is disp ersed in K is also disp ersed in G . It is c onservative if | V K | = | V H | . The pro of of Claim 3 in [3] inv olves the construction of a dominate d torso ˆ G i of G i that is a connected, faithful and conserv ativ e augmen tation of G i in G . Using the general notation, the dominated torso K of H in G is defined as follo ws. Let A b e the set of all adhesion sets of H in G . Th us A ∈ A if and only if A = N G ( D ) for some comp onen t D of G − H . F or each A ∈ A let D A b e the set of comp onen ts D of G − H with N G ( D ) = A . Let A ′ b e the set of all A ∈ A suc h that D A is finite and let A ′′ := A ∖ A ′ . Let D ′ := [ A ∈ A ′ D A and D ′′ := [ A ∈ A ′′ D A . Let η : D ′′ → V ( H ) b e a function suc h that for every A ∈ A ′′ the restriction η A of η to D A is a surjection onto A . Since each A ∈ A ′′ is finite and the corresp onding D A is infinite, such a surjection η A exists. Let { v D : D ∈ D ′ } b e a set disjoin t with V ( G ) such that v D  = v D ′ for D  = D ′ , and let V ( K ) := V ( H ) ∪ { v D : D ∈ D ′ } . Define ϱ : V ( G ) → V ( K ) by: ϱ ( u ) :=        u if u ∈ V ( H ); v D if u ∈ V ( D ) for some D ∈ D ′ ; η ( D ) if u ∈ V ( D ) for some D ∈ D ′′ . Let K b e the unique contraction minor of G witnessed b y ϱ . Define a tendril in G to b e a ray S that has infinitely many vertices in the subgraph H . T o prov e that K is a faithful augmen tation of H in G , given a tendril S in G , Pitz uses a canonical pro jection S ′ of S onto K . W e will call it a K -pro jection of S . It is defined as follows. 2 Giv en a tendril S = ( v n ) n ∈ ω in G , the H - masking of S is the sequence ( v ′ n ) n ∈ ω , where eac h v ′ n := v n if v n ∈ V ( H ) , and v ′ n := D if v n ∈ V ( D ) for some comp onen t D of G − H . The K - pr oje ction of the tendril S is the sequence S ′ := ( w m ) m ∈ ω of v ertices of K obtained from the H -masking ( v ′ n ) n ∈ ω of S by: 1. Replacing each maximal subsequence of identical consecutiv e terms D ∈ D ′ with the single vertex v D ∈ V K . 2. Remo ving all terms D ∈ D ′′ from the sequence entirely . It is clear from the construction that v D is adjacent in K to all v ertices in N G ( D ) for every D ∈ D ′ . Moreov er, for ev ery A ∈ A ′′ the induces subgraph K [ A ] of K is complete. Th us the K -pro jection S ′ = ( w m ) m ∈ ω of S is a lo cally finite tour in K (infinite walk visiting eac h vertex only finitely man y times). Let U ⊆ V ( H ) b e dispersed in K . T o sho w that U is disp ersed in G , we need to show, in particular, that if S is a tendril in G , then U is finitely separated from S . Let S ′ b e the K -pro jection of S . Since U is disp ersed in K there is a finite set F separating U from S ′ in K . Pitz in [3] claims that the set X := ( F ∩ V ( H )) ∪ [ { N G ( D ) : D ∈ D ′ , v D ∈ F } separates U from S in G . Consider the following example, which demonstrates that this claim ma y fail. Example 4. Let G b e a graph whose vertex set it the disjoint union X ∪ Y ∪ Z , where X , Y are countably infinite sets enumerated as X := { x j : j ∈ ω } , Y := { y j : j ∈ ω } and Z is enumerated as Z := { z α : α ∈ ω 1 } . The edge set of G consists of the edges x j x j +1 , x j y j , x j +1 y j , for each j ∈ ω and the edges x 1 z α , x 2 z α and x 3 z α for each α ∈ ω 1 . Let H := G [ X ] . The subgraph H has finite adhesion in G . Moreo ver, D ′ = { D ′ i : i ∈ ω } , where each comp onen t D ′ i has a single vertex y i and D ′′ = { D ′′ α : α ∈ ω 1 } , where each D ′′ α has a single vertex z α . The minor K has vertex set X ∪  v D ′ i : i ∈ ω  with edges x i x i +1 , x i v D ′ i and x i +1 v D ′ i for each i ∈ ω and the edge x 1 x 3 . Let S b e the ray ( x 2 , z 0 , x 3 , y 3 , x 4 , y 4 , x 5 , . . . , x n , y n , x n +1 , y n +1 , . . . ) in G . The K -pro jection of S is the sequence S ′ :=  x 2 , x 3 , v D ′ 3 , x 4 , v D ′ 4 , . . .  . Let U := { x 0 } and F := { x 2 , x 3 } . Then F separates U from S ′ (see Figure 3 x 0 x 1 x 2 x 3 x 4 x 5 . . . X v D ′ 0 v D ′ 1 v D ′ 2 v D ′ 3 v D ′ 4 . . . Y S ′ F U Figure 1: F separates U from S ′ in K x 0 x 1 x 2 x 3 x 4 x 5 . . . X y 0 y 1 y 2 y 3 y 4 . . . Y z 0 z 1 z 2 . . . Z tendril S X path from U to S U Figure 2: X does not separated U from S in G 1), but X = F do es not separate U from S since ( x 0 , x 1 , z 0 ) is a path joining x 0 ∈ U and z 0 ∈ S (see Figure 2). The construction of dominated torsos in [3] provides an implicit pro of of the follo wing Augmen tation Lemma. How ev er, the part of this pro of that demon- strates faithfulness is incorrect (see Example 4). The purp ose of this note is to pro vide a corrected pro of Lemma 5 (Augmen tation Lemma) . L et G b e an infinite c onne cte d gr aph and H b e an infinite c onne cte d induc e d sub gr aph with finite adhesion in G . Then ther e exists a c onne cte d, faithful, and c onservative augmentation K of H in G . 2 Pro of of the Augmen tation Lemma. Let G b e an infinite connected graph and H b e an infinite connected induced subgraph with finite adhesion in G . W e define K as the dominated torso of H in G as describ ed ab o ve. The minor K is clearly connected and as pro ved in [3] it is conserv ativ e. It remains to show that K is a faithful augmentation. Recall that A is the set of all adhesion sets of H in G , that A ′ is the set of 4 all A ∈ A suc h that D A is finite and A ′′ := A ∖ A ′ . Moreov er, D ′ = [ A ∈ A ′ D A and D ′′ = [ A ∈ A ′′ D A . Supp ose F ⊆ V K and S = ( v n ) n ∈ ω is a tendril in G . W e define the subset F S ⊆ V ( H ) as follo ws. Let ( v ′ n ) n ∈ ω b e the H -masking of S . Define the auxiliary sets of comp onen ts: ˆ D ′ := { D ∈ D ′ : v D ∈ F } and ˆ D ′′ := { D ∈ D ′′ : v ′ n = D and v n +1 ∈ F for some n ∈ ω } . Note that ˆ D ′ is indep enden t of S , but ˆ D ′′ dep ends on S . Let ˆ D := ˆ D ′ ∪ ˆ D ′′ and F S := ( F ∩ V ( H )) ∪ [ D ∈ ˆ D N G ( D ) . W e call F S the S - mo dific ation of F . It is clear that if F is finite, then F S is also finite. The difference b etw een the F S -mo dification and the set X used by Pitz in his pro of is the inclusion of the family ˆ D ′′ in the union used to define F S , while in the definition of X only ˆ D ′ w as used. Lemma 6. L et S b e a tendril in G with K -pr oje ction S ′ := ( w m ) m ∈ ω . If a set F ⊆ V K sep ar ates a subset U ⊆ V H fr om the set W := { w m : m ∈ ω } in K , then the mo dific ation F S sep ar ates U fr om V S in G . Pr o of. Let P = ( u 0 , u 1 , . . . , u k ) be a path in G joining from vertex u ∈ U to a v ertex v ∈ S . Our goal is to sho w that P has a vertex in F S . Let ( u ′ 0 , u ′ 1 , . . . , u ′ k ) be the H -masking of P and ( x 0 , x 1 , . . . , x q ) be its K - pro jection (defined similarly as for tendrils). Since u 0 ∈ V ( H ) , w e hav e x 0 = u ′ 0 = u 0 . First, consider the case where u k ∈ V H or u k ∈ V D for some D ∈ D ′ . If u k ∈ V H , then x q = u k ∈ W . Otherwise, since u k is a vertex of S , the definition of K -pro jection implies x q = v D ∈ W . Because F separates U from W in K , and ( x 0 , x 1 , . . . , x q ) is a walk in K connecting u ∈ U to x q ∈ W , there m ust b e some index i ∈ { 0 , 1 , . . . , q } such that x i ∈ F . ⋆ If x i ∈ V H , then by the definition of the S -mo dification, x i ∈ F S , whic h satisfies the requirement. ⋆ If x i = v D for some comp onent D ∈ D ′ , then v D ∈ F implies that D ∈ ˆ D ′ . Since x 0 ∈ V H , we kno w that i ≥ 1 and x i − 1 ∈ V H . Th us x i − 1 is a v ertex of P 5 that lies in N G ( D ) . Since D ∈ ˆ D ′ , the adhesion set N G ( D ) is con tained in F S . Therefore, x i − 1 ∈ F S , as required. No w assume that u k ∈ V ( D ) for some D ∈ D ′′ . Then A := N G ( D ) ∈ A ′′ . Since u 0 ∈ V ( H ) , there is the largest index j ∈ { 0 , 1 , . . . , k } with u j ∈ V ( H ) . Then u ′ j +1 = u ′ j +2 = · · · = u ′ k = D and x q = u j ∈ A . Let S = ( v n ) n ∈ ω and ( v ′ n ) n ∈ ω b e the H -masking of S . Since u k = v n for some n ∈ ω , it follows that v ′ n = D . Since S is a tendril, there is a smallest integer s > n with v s ∈ V ( H ) . Then v s − 1 ∈ V ( D ) so v s ∈ A . Note the graph K [ A ] is complete. Since both x q and v s b elong to A and A ∈ A ′′ , it follo ws that x q v s ∈ E ( K ) . Then ( x 0 , x 1 , . . . , x q , v s ) is a walk in K connecting u ∈ U to v s ∈ W . Because F separates U from W in K , if v s / ∈ F , then there is i ∈ { 0 , 1 , . . . , q } suc h that x i ∈ F . Then, arguing as ab o v e, w e find a vertex of the path P that b elongs to F S . Supp ose v s ∈ F . Then the definition of the auxiliary set ˆ D ′′ implies that D ∈ ˆ D ′′ and consequently x q ∈ F S . In either case w e find a vertex of P that b elongs to F S whic h implies that F S separates U from S in G . Pro of that K is a faithful augmen tation. Assume that U ⊆ V ( H ) is disp ersed in K . Let S = ( v i ) i ∈ ω b e a ra y in G . T o sho w that U is disp ersed in G , we must find a finite subset of V ( G ) that separates U from S . First, supp ose there exists n ∈ ω such that v i / ∈ V ( H ) for all i ≥ n . Then the tail { v i : i ≥ n } of S is contained in a single comp onent D of G − H . The union of the finite adhesion set N G ( D ) and the finite initial segmen t { v i : i < n } of S separates U from S . If no suc h n exists, then S is a tendril in G . Let ( w m ) m ∈ ω b e the K -pro jection of S . Since U is disp ersed in K , there is a finite set F ⊆ V ( G ) that separates U from { w m : m ∈ ω } in K . The S -mo dification F S is a finite subset of V ( G ) that separates U from S in G by Lemma 6. R emark. Note that if X := ( F ∩ V ( H )) ∪ [ D ∈ ˆ D ′ N G ( D ) is the separator defined as in [3], then there is n 0 ∈ ω is suc h that v n / ∈ X for every n > n 0 . Then the set X separates U from the tail { v n : n > n 0 } of S . Indeed, the proof of Lemma 6 implies that for any path P in G from a v ertex in U to a v ertex v n of S such that P av oids X there is n ′ > n with v n ′ ∈ F ∩ V ( H ) ⊆ X . 6 References [1] R. Diestel , Graph Theory , Springer, 5th edition, 2015 [2] R. Halin, Miscellaneous problems on infinite graphs, Journal of Graph Theory , 35(2):128–151, 2000 [3] M. Pitz, Pro of of Halin’s normal spanning tree conjecture, Israel Journal of Mathematics, 246:353–370, 2021 7