Bipartitions of highly connected tournaments
aa r X i v : . [ m a t h . C O ] F e b BIPARTITIONS OF HIGHLY CONNECTED TOURNAMENTS
JAEHOON KIM, DANIELA K ¨UHN, DERYK OSTHUS
Abstract.
We show that if T is a strongly 10 k log(2 k )-connected tournament, there exists apartition A, B of V ( T ) such that each of T [ A ], T [ B ] and T [ A, B ] is strongly k -connected. Thisprovides tournament analogues of two partition conjectures of Thomassen regarding highlyconnected graphs. Introduction
Partitions of highly connected tournaments.
The study of graph partitions wherethe resulting subgraphs inherit the properties of the original graph has a long history with somesurprises and numerous open problems, see e.g. the survey [7]. For example, a classical resultof Hajnal [1] and Thomassen [9] implies that for every k there exists an integer f ( k ) such thatevery f ( k )-connected graph has a vertex partition into sets A and B so that both A and B induce k -connected graphs. A related conjecture of Thomassen [12] states that for every k thereis an f ( k ) such that every f ( k )-connected graph G has a bipartition A, B so that the spanningbipartite graph G [ A, B ] is k -connected. It is not hard to show that one cannot achieve boththe above properties simultaneously in a highly connected graph. On the other hand, our mainresult states that for tournaments, we can find a single partition which achieves both the aboveproperties. Below we denote by T [ A, B ] the bipartite subdigraph of T which consists of all edgesbetween A and B but no others. Theorem 1.1.
Let T be a tournament and k ∈ N . If T is strongly k log(2 k ) -connected,there exists a partition V , V of V ( T ) such that each of T [ V ] , T [ V ] and T [ V , V ] is strongly k -connected. We have made no attempt to optimize the bound on the connectivity in Theorem 1.1. (Itwould be straightforward to obtain minor improvements at the expense of more careful calcula-tions.) On the other hand, it would be interesting to obtain the correct order of magnitude forthe connectivity bound.K¨uhn, Osthus and Townsend [4] earlier proved the weaker result that every strongly 10 k log(4 k )-connected tournament T has a vertex partition V , V such that T [ V ] and T [ V ] are both strongly k -connected (with some control over the sizes of V and V ). This proved a conjecture ofThomassen. [4] raised the question whether this can be extended to digraphs.As described later, our proof of Theorem 1.1 develops ideas in [4]. These in turn are basedon the concept of robust linkage structures which were introduced in [2] to prove a conjecture ofThomassen on edge-disjoint Hamilton cycles in highly connected tournaments. Further (asymp-totically optimal) results leading on from these approaches were obtained by Pokrovskiy [5, 6]. Date : March 19, 2018.The research leading to these results was partially supported by the European Research Council under theEuropean Union’s Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreements no. and 306349(J. Kim and D. Osthus) as well as 258345 (D. K¨uhn).
Subdivisions and linkages.
The famous Lov´asz path removal conjecture states that forevery k ∈ N there exists g ( k ) ∈ N such that for every pair x, y of vertices in a g ( k )-connectedgraph G we can find an induced path P joining x and y in G for which G \ V ( P ) is k -connected.In [11], Thomassen proved a tournament version of this conjecture. Here, we generalize hisargument to observe that highly connected tournaments contain a non-separating subdivisionof any given digraph H (with prescribed branch vertices). The case when d = 2 and m = 1corresponds to the result in [11]. Theorem 1.2.
Let k, d, m ∈ N . Suppose that T is a strongly ( k + m ( d + 2)) -connected tourna-ment, that D is a set of d vertices in T , that H is a digraph on d vertices and m edges and that φ is a bijection from V ( H ) to D . Then T contains a subdivision H ∗ of H such that (i) for each h ∈ V ( H ) the branch vertex of H ∗ corresponding to h is φ ( h ) , (ii) T \ V ( H ∗ ) is strongly k -connected, (iii) for every edge e of H , the path P e of H ∗ corresponding to e is backwards-transitive. Here a directed path P = x . . . x t in a tournament T is backwards-transitive if x i x j is an edgeof T whenever i ≥ j + 2. The graph version of Theorem 1.2 is still open and would follow fromthe following beautiful conjecture of Thomassen [10]. Conjecture 1.3.
For every k ∈ N there exists f ( k ) ∈ N such that if G is a f ( k ) -connectedgraph and M ⊆ V ( G ) consists of k vertices then there exists a partition V , V of V ( G ) suchthat M ⊆ V , both G [ V ] and G [ V ] are k -connected, and each vertex in V has at least k neighbours in V . The case | M | = 2 would already imply the path removal conjecture. The case M = ∅ wasproved in [3]. It implies the existence of non-separating subdivisions (without prescribed branchvertices) in highly connected graphs. Clearly, Theorem 1.1 implies a tournament version ofConjecture 1.3.The next theorem guarantees a spanning linkage in a highly connected tournament. It wasproved by Thomassen [11] with a super-exponential bound on the connectivity. He asked whethera linear bound suffices. Here we reduce the bound to a polynomial one. Pokrovskiy [5] showedthat a linear bound suffices to guarantee a linkage if we do not require it to be spanning. Theorem 1.4.
Let k ∈ N . Suppose that T is a strongly ( k + 3 k ) -connected tournament andthat x , . . . , x k , y , . . . , y k are vertices in T such that x i = y i for all i ∈ [ k ] and all the pairs ( x i , y i ) are distinct. Then T contains pairwise internally disjoint paths P i from x i to y i suchthat { x , . . . , x k , y , . . . , y k } ∩ V ( P i ) = { x i , y i } and V ( T ) = S ki =1 V ( P i ) . Both Theorem 1.2 and 1.4 can be deduced from Theorem 1.1 (but with weaker bounds).Instead, in Section 4 we adapt the argument from [11] to obtain a short direct proof of boththese results. 2.
Notation and tools
Given k ∈ N , we let [ k ] := { , . . . , k } , [ k, k + ℓ ] := { k, . . . , k + ℓ } and log k := log k . We write V ( G ) and E ( G ) for the set of vertices and the set of edges in a digraph G . We let | G | := | V ( G ) | .If u, v ∈ V ( G ) we write uv for the directed edge from u to v . We write d − G ( v ) and d + G ( v )for the in-degree and the out-degree of a vertex v in G . We write δ − ( G ) and δ + ( G ) for theminimum in-degree and the minimum out-degree of G and let δ ( G ) := min { δ − ( G ) , δ + ( G ) } . Aset A ⊆ V ( G ) in-dominates a set B ⊆ V ( G ) if for every vertex b ∈ B there exists a vertex a ∈ A IPARTITIONS OF HIGHLY CONNECTED TOURNAMENTS 3 such that ba ∈ E ( G ). Similarly, we say that A out-dominates B if for every vertex b ∈ B thereexists a vertex a ∈ A such that ab ∈ E ( G ). We say that a tournament T is transitive if wemay enumerate its vertices v , . . . , v m such that v i v j ∈ E ( T ) if and only if i < j . In this casewe call v the source of T and v m the sink of T . When referring to subpaths of tournaments,we always mean that these paths are directed (i.e. consistently oriented). The length of a pathis the number of its edges. We say that a path P is odd if its length is odd, and even if itslength is even. We say that two paths are disjoint if they are vertex-disjoint. A tournament T is strongly k -connected if | T | > k and for every set F ⊆ V ( T ) with | F | < k and every orderedpair x, y of vertices in V ( T ) \ F there exists a path from x to y in T − F . A tournament T iscalled k-linked if | T | ≥ k and whenever x , . . . , x k , y , . . . , y k are 2 k distinct vertices of T thereexist disjoint paths P , . . . , P k such that P i is a directed path from x i to y i for each i ∈ [ k ].We now collect the tools which we need in our proof of Theorem 1.1. The following propositionis a straightforward consequence of the definition of linkedness. Proposition 2.1.
Let k ∈ N . Then a tournament T is k -linked if and only if | T | ≥ k andwhenever ( x , y ) , . . . , ( x k , y k ) are ordered pairs of (not necessarily distinct) vertices of T , thereexist distinct internally disjoint paths P , . . . , P k such that for all i ∈ [ k ] we have that P i is adirected path from x i to y i and that { x , . . . , x k , y , . . . , y k } ∩ V ( P i ) = { x i , y i } . We will also use the following bound from [5] on the connectivity which forces a tournamentto be highly linked.
Theorem 2.2.
For each k ∈ N every strongly k -connected tournament is k -linked. The following two lemmas guarantee that every tournament contains almost out-dominatingand almost in-dominating sets which are not too large. (A similar observation was also usedin [2], see Lemmas 8.3 and 8.4.)
Lemma 2.3.
Let T be a tournament, let v ∈ V ( T ) and c ∈ N with c ≥ . Suppose that d − T ( v ) ≥ c − . Then there exist disjoint sets A, E ⊆ V ( T ) and a vertex a ∈ A such that thefollowing properties hold: (i) 2 ≤ | A | ≤ c and T [ A ] is a transitive tournament with source a and sink v , (ii) A \ { a } out-dominates V ( T ) \ ( A ∪ E ) , (iii) | E | ≤ (1 / c − d − T ( v ) . Proof.
Let v := v . Roughly speaking, we will find A by choosing vertices v , . . . , v i suchthat the size of their common in-neighbourhood (i.e. the intersection of their individual in-neighbourhoods) is minimised at each step. More precisely, suppose inductively that for some1 ≤ i < c we have already found a set A i = { v , . . . , v i } and a set W i such that the followingholds:(a) T [ A i ] is a transitive tournament with sink v ;(b) W i = ∅ or W i = { a } for some vertex a . Moreover, if W i = { a } then E i ∪ A i ⊆ N + ( a ),where E i := T ij =1 N − ( v j ) \ W i .(c) | E i | ≤ i − d − ( v ) . Moreover, | E i | > W i = ∅ .Note that (a)–(c) hold for i = 1 if we let A := { v } and W = ∅ .We first consider the case that | E i | ≤ c − d − ( v ). If W i = ∅ , choose any vertex a ∈ E i . Elselet a be the vertex in W i . In both cases let A := A i ∪ { a } and E := E i \ { a } . Then A and E satisfy (i)–(iii). JAEHOON KIM, DANIELA K ¨UHN, DERYK OSTHUS
So suppose next that | E i | > c − d − ( v ). (Note that in particular, this means that | E i | ≥ E i must contain a vertex x of in-degree at most | E i | / T [ E i ].If the in-degree of x in T [ E i ] is nonzero or W i = ∅ , let v i +1 := x . Else let v i +1 be a vertex ofin-degree at most | E i \ { x }| / T [ E i \ { x } ], and let W i +1 := { x } (note that we can find such a v i +1 as | E i | ≥ A i +1 := { v , . . . , v i +1 } and let E i +1 := ( E i ∩ N − ( v i +1 )) \ W i +1 . Then T [ A i +1 ] is a transitive tournament with sink v and | E i +1 | ≤ | E i | ≤ i d − ( v ) . So we have shown that (a)–(c) hold with i +1 playing the role of i . By repeating this construction,will eventually find A and E satisfying (i)–(iii). (Indeed, note that we must be in the first casefor some i < c , in particular this implies that | A | ≤ c .) (cid:3) The next lemma follows immediately from Lemma 2.3 by reversing the orientations of alledges.
Lemma 2.4.
Let T be a tournament, let v ∈ V ( T ) and c ∈ N with c ≥ . Suppose that d + T ( v ) ≥ c − . Then there exist disjoint sets B, E ⊆ V ( T ) and a vertex b ∈ B such that thefollowing properties hold: (i) 2 ≤ | B | ≤ c and T [ B ] is a transitive tournament with sink b and source v , (ii) B \ { b } in-dominates V ( T ) \ ( B ∪ E ) , (iii) | E | ≤ (1 / c − d + T ( v ) . We will also need the following observation, which guarantees a small set Z of vertices in atournament such that every vertex outside Z has many out- and in-neighbours in Z . Proposition 2.5.
Let k, n ∈ N and let T be a tournament on n ≥ vertices. Then there is a set Z ⊆ V ( T ) of size | Z | ≤ k log n such that each vertex in V ( T ) \ Z has at least k out-neighboursand at least k in-neighbours in Z . Proof.
We may assume that n ≥ k log n . We will use the fact that every tournament on n vertices contains an in-dominating set of size at most c := ⌈ log n ⌉ ≤ (3 log n ) /
2. (This canbe proved by choosing the vertices x , x , . . . in the in-dominating set one by one, similarly asin the proof of Lemma 2.3: at the i th step we let x i be a vertex with the smallest out-degreein T [ T j
Let X := { x , x , . . . , x k } ⊆ V ( T ) consist of 6 k vertices whose in-degree in T is as small aspossible, and let Y := { y , y , . . . , y k } be a set of 6 k vertices in V ( T ) \ X whose out-degree in T is as small as possible. Defineˆ δ − ( T ) := min v ∈ V ( T ) \ X d − T ( v ) and ˆ δ + ( T ) := min v ∈ V ( T ) \ Y d + T ( v ) . Let c := (cid:6) log (cid:0) k (cid:1)(cid:7) + 2 ≤ k . Apply Lemmas 2.3 and 2.4 with parameter c repeatedly(removing the dominating sets each time) to obtain disjoint sets of vertices A , A , . . . , A k , IPARTITIONS OF HIGHLY CONNECTED TOURNAMENTS 5 B , B , . . . , B k and sets of vertices E A , . . . , E B k satisfying the following properties for all i ∈ [6 k ], where we write D := S ki =1 ( A i ∪ B i ),(D1) 2 ≤ | A i | ≤ c and T [ A i ] is a transitive tournament with sink x i and source a i ,(D2) 2 ≤ | B i | ≤ c and T [ B i ] is a transitive tournament with source y i and sink b i ,(D3) A i \ { a i } out-dominates V ( T ) \ ( D ∪ E A i ) in T ,(D4) B i \ { b i } in-dominates V ( T ) \ ( D ∪ E B i ) in T ,(D5) | E A i | ≤ (1 / c − ˆ δ − ( T ),(D6) | E B i | ≤ (1 / c − ˆ δ + ( T ).Let E A := [ i ∈ [6 k ] E A i , E B := [ i ∈ [6 k ] E B i and E := E A ∪ E B . Note that(3.1) | E A | ≤ k (cid:18) (cid:19) c − ˆ δ − ( T ) ≤ ˆ δ − ( T )20 k and | E B | ≤ ˆ δ + ( T )20 k by our choice of c . Moreover, we may assume that | E A | ≤ | E B | . (The case | E A | > | E B | followsby a symmetric argument.) In particular, this implies that(3.2) | E | ≤ | E A | + | E B | ≤ | E B | ≤ ˆ δ + ( T )10 k . We will iteratively colour the vertices of T with colours α and β , and at each step V α willconsist of all vertices of colour α and V β is defined similarly. At the end of our argument, everyvertex of T will be coloured either with α or with β , i.e. V α , V β will form a partition of V ( T ).Our aim is to colour the vertices in such a way that we can take V := V α and V := V β .We say a path P is alternating if the colour of the vertices on P alternates as we move along P . P is monochromatic if all vertices of P have the same colour.At each step and for each γ ∈ { α, β } , we call a vertex v ∈ V γ forwards-safe if for any set F v of at most k − T [ V γ \ F ] from v to V ( T ) \ ( D ∪ E B ∪ F ). Similarly, we say that v ∈ V γ is backwards-safe iffor any set F v of at most k − T [ V γ \ F ] from V ( T ) \ ( D ∪ E A ∪ F ) to v .We call a vertex v ∈ V γ alternating-forwards-safe if for any set F v of at most k − T − F from v to V ( T ) \ ( D ∪ E B ∪ F ).Similarly, we say that v ∈ V γ is alternating-backwards-safe if for any set F v of at most k − T − F from V ( T ) \ ( D ∪ E A ∪ F ) to v .We say that a vertex v is safe if it is safe in all four respects.Note that the following properties are satisfied at every step (for each { γ, δ } = { α, β } ):(S1) all coloured vertices in V ( T ) \ ( D ∪ E ) are safe,(S2) all coloured vertices in V ( T ) \ ( D ∪ E B ) are forwards-safe as well as alternating-forwards-safe and all coloured vertices in V ( T ) \ ( D ∪ E A ) are backwards-safe as well as alternating-backwards-safe,(S3) if v ∈ V γ has at least k forwards-safe out-neighbours of colour γ then v itself is forwards-safe, the analogue holds if v has at least k backwards-safe in-neighbours of colour γ , JAEHOON KIM, DANIELA K ¨UHN, DERYK OSTHUS (S4) if v ∈ V γ has at least k alternating-forwards-safe out-neighbours of colour δ with δ = γ then v itself is alternating-forwards-safe, the analogue holds if v has at least k alternating-backwards-safe in-neighbours of colour δ ,(S5) if v ∈ V γ is safe and in the next step we colour some more (previously uncoloured)vertices then v is still safe.In what follows, by a (partial) colouring of the vertices of T we always mean a colouring withcolours α and β in which all the vertices in D : = [ i ∈ [ k ] ( A i ∪ B i ) ∪ [ i ∈ [3 k +1 , k ] ( A i \ { a i } ) ∪ [ i ∈ [3 k +1 , k ] ∪ [5 k +1 , k ] ( B i \ { b i } ) ∪ { a i | i ∈ [2 k + 1 , k ] ∪ [5 k + 1 , k ] } ∪ { b i | i ∈ [2 k + 1 , k ] ∪ [4 k + 1 , k ] } are coloured α , and all the vertices in D := D \ D are coloured β . Claim 0:
Suppose that there are paths P , . . . , P k of T satisfying the following properties: • for each i ∈ [6 k ] the path P i joins b i to a i , • the paths P , . . . , P k are disjoint from each other and meet D only in their endvertices.Suppose that we have coloured all vertices of T such that • every vertex in D ∪ V ( P ) ∪ · · · ∪ V ( P k ) is coloured α , • every vertex in D ∪ V ( P k +1 ) ∪ · · · ∪ V ( P k ) is coloured β , • P k +1 , . . . , P k are alternating, • every vertex is safe.Then the sets V := V α and V := V β form a partition of V ( T ) as required in Theorem 1.1. Note that the conditions of Claim 0 imply that P i must be an even path for i ∈ [2 k + 1 , k ]and an odd path for i ∈ [4 k + 1 , k ].To prove Claim 0, we first show that T [ V α ] is strongly k -connected. So consider any set F of at most k − x, y ∈ V α \ F . We need to check that T [ V α \ F ]contains a path from x to y . Since x is forwards-safe there exists a path Q x in T [ V α \ F ] from x to some vertex x ′ ∈ V α \ ( D ∪ E B ∪ F ). Similarly, since y is backwards-safe there exists apath Q y in T [ V α \ F ] from some vertex y ′ ∈ V α \ ( D ∪ E A ∪ F ) to y . Let i ∈ [ k ] be such that F avoids A i ∪ V ( P i ) ∪ B i . Since x ′ / ∈ D ∪ E B , (D4) implies that x ′ sends an edge to B i . Similarly,since y ′ / ∈ D ∪ E A , (D3) implies that y ′ receives an edge from A i . Altogether this implies that T [ V ( Q x ) ∪ V ( Q y ) ∪ A i ∪ V ( P i ) ∪ B i ] ⊆ T [ V α \ F ] contains path from x to y , as desired.A similar argument shows that V β is strongly k -connected too. It remains to show that T [ V α , V β ] is stongly k -connected. Consider any set F of at most k − x, y ∈ V ( T ) \ F . We will show that there is an alternating path between x and y avoiding F . Since x is alternating-forwards-safe there exists an alternating path Q x in T [ V α , V β ] − F from x to some vertex x ′ ∈ V ( T ) \ ( D ∪ E B ∪ F ). Similarly, since y is backwards-safe there exists apath Q y in T [ V α , V β ] − F from some vertex y ′ ∈ V [ T ] \ ( D ∪ E A ∪ F ) to y . We now choose anindex i as follows: • If x ′ , y ′ ∈ V α , let i ∈ [2 k + 1 , k ] be such that F avoids A i ∪ V ( P i ) ∪ B i . • If x ′ , y ′ ∈ V β , let i ∈ [3 k + 1 , k ] be such that F avoids A i ∪ V ( P i ) ∪ B i . • If x ′ ∈ V α and y ′ ∈ V β , let i ∈ [4 k + 1 , k ] be such that F avoids A i ∪ V ( P i ) ∪ B i . • If x ′ ∈ V β and y ′ ∈ V α , let i ∈ [5 k + 1 , k ] be such that F avoids A i ∪ V ( P i ) ∪ B i .Since x ′ / ∈ D ∪ E B , (D4) implies that x ′ sends an edge to B i \ { b i } . Similarly, since y ′ / ∈ D ∪ E A , (D3) implies that y ′ receives an edge from A i \ { a i } . Altogether this implies that IPARTITIONS OF HIGHLY CONNECTED TOURNAMENTS 7 T [ V ( Q x ) ∪ V ( Q y ) ∪ A i ∪ V ( P i ) ∪ B i ] ⊆ T − F contains an alternating path from x to y , asdesired. This completes the proof of Claim 0. Claim 1:
Consider a partial colouring of V ( T ) and let C denote the set of previously colouredvertices. (So D ⊆ C .) Let Z ⊆ V ( T ) \ ( X ∪ Y ) and N ⊆ V ( T ) \ Z and suppose that k | Z | + | C ∪ N | ≤ · k log(2 k ) . Then for every colouring of the vertices in Z \ C there is a set Z ′ ⊆ V ( T ) \ ( Z ∪ N ∪ C ) and a colouring of the vertices in Z ′ such that every vertex in Z ∪ Z ′ is safe and | Z ∪ Z ′ | ≤ k | Z | . To prove Claim 1, note that the strong 10 k log(2 k )-connectivity of T implies that δ ( T ) ≥ k log(2 k ). Hence(3.3) ˆ δ − ( T ) − k | E A | ( . ) ≥ ˆ δ − ( T )2 ≥ δ ( T )2 ≥ · k log(2 k ) , and similarly(3.4) ˆ δ + ( T ) − k | E | ( . ) ≥ ˆ δ + ( T )2 ≥ · k log(2 k ) . Consider any colouring of Z \ C . For each vertex z ∈ Z in turn we greedily choose 2 k uncoloured in-neighbours outside N ∪ E A , and colour k of them α and the remaining k by β .(We do not modify C in this process.) To see that we can choose all these vertices to be distinctfrom each other, note that the total number of vertices we wish to choose is 2 k | Z | and | C ∪ N ∪ Z | + 2 k | Z | ≤ · k log(2 k ) ( . ) ≤ ˆ δ − ( T ) − | E A | . For each vertex in Z as well as for each of the 2 k | Z | vertices that we coloured in the previousstep in turn, we greedily choose 2 k uncoloured out-neighbours outside N ∪ E , and colour k ofthem by α and the remaining k by β . To see that we can choose all these vertices to be distinctfrom each other, note that the total number of vertices we wish to choose is 2 k (1 + 2 k ) | Z | and | C ∪ N ∪ Z | + 2 k | Z | + 2 k (1 + 2 k ) | Z | ≤ | C ∪ N | + 9 k | Z | ≤ · k log(2 k ) ( . ) ≤ ˆ δ − ( T ) − | E | . Let Z ′ be the set of vertices outside C ∪ Z that we coloured. Then Z ′ ∩ N = ∅ . Moreover, using(S1)–(S4) it is easy to check that every vertex in Z ∪ Z ′ is safe. This completes the proof ofClaim 1.Recall that we have already coloured all the vertices in D by α and all the vertices in D by β . Step by step, we will now colour further vertices of T . Our final aim is to arrive at a colouringof V ( T ) which is as described in Claim 0. The first step is to colour some more vertices in orderto achieve that all the coloured vertices are safe. In what follows, when saying that we coloursome additional vertices we always mean that these vertices are uncoloured so far. Claim 2:
We can colour some additional vertices of T in such a way that every coloured vertexis safe and the set C consisting of all vertices coloured so far satisfies | C | ≤ k .To prove Claim 2, for every v ∈ { x , . . . , x k , y , . . . , y k } in turn, we greedily choose 2 k uncoloured in-neighbours and 2 k uncoloured out-neighbours, all distinct from each other, andcolour k of the in-neighbours and k of the out-neighbours by α and the remaining 2 k in/out-neighbours by β .Let Z ∗ denote the set of 4 k · k = 48 k new vertices we just coloured and let Z := Z ∗ ∪ ( D \ ( X ∪ Y )). Then | Z | ≤ | Z ∗ | + | D | ≤ k + c · k ≤ k . Apply Claim 1 with N := ∅ to find aset Z ′ of uncoloured vertices and a colouring of these vertices such that all the vertices in Z ∪ Z ′ JAEHOON KIM, DANIELA K ¨UHN, DERYK OSTHUS are safe and | Z ∪ Z ′ | ≤ k · | Z | ≤ k . Our choice of Z ∗ and (S3), (S4) together now implythat the vertices in X ∪ Y are safe as well. This completes the proof of Claim 2.Suppose that P is a path whose endvertices are already coloured, but whose internal verticesare still uncoloured. We say that we colour (the internal vertices of) P in an alternating mannerconsistent with its endvertices if the colouring results in an alternating path. (So for example,if the endvertices of P have the same colour, then P needs to be an even path.) Claim 3:
There are paths P , P , . . . , P k of T satisfying the following properties: (i) for each i ∈ [6 k ] , the path P i joins b i to a i , (ii) the paths P , . . . , P k are disjoint from each other and meet C only in their endvertices, (iii) we can colour the internal vertices of P , . . . , P k by α , the internal vertices of P k +1 , . . . , P k by β and the internal vertices of P k +1 , . . . , P k in an alternating manner consistent withtheir endvertices and can colour some additional vertices such that the set C of all colouredvertices satisfies the following properties: (a) all vertices in C are safe, (b) there is a set C ⊆ C such that the number of coloured vertices outside C is at most · k log(2 k ) , (c) every vertex outside C which has an in-neighbour in C has at least k in-neighboursof each colour, and every vertex outside C which has an out-neighbour in C has atleast k out-neighbours of each colour. We will prove Claim 3 via a sequence of subclaims. For i ∈ [6 k ] we define an i - path to be adirected path from the sink b i of B i to the source a i of A i whose internal vertices lie outside C .Ideally, we would like to find disjoint i -paths P i (one for each i ∈ [6 k ]) such that the followingproperties hold:(1) we can colour all the internal vertices of P , . . . , P k by α , the internal vertices of P k +1 , . . . , P k by β and the internal vertices of P k +1 , . . . , P k in an alternating manner consistent withtheir endvertices,(2) by colouring some additional vertices we can achieve that all coloured vertices are safe.For each i ∈ [6 k ] we will first try to find a short i -path P i such that all these i -paths are disjointand such that for each i ∈ [2 k + 1 , k ] the length of the path P i has the correct parity in order toensure that the internal vertices of P i can be coloured in an alternating manner consistent withthe endvertices of P i (so P i needs to be even for i ∈ [2 k + 1 , k ] and odd for i ∈ [4 k + 1 , k ]). Wewill then colour the vertices on these short i -paths as well as some additional vertices such that(1) and (2) are satisfied for the set I short of all indices i for which we have been able to choose ashort i -path (see Claim 3.1). This provides some of the paths required in Claim 3. To find theremaining paths, for all i / ∈ I short we will choose 10 k log(2 k ) i -paths Q i, , . . . , Q i, k log(2 k ) such that all these paths are internally disjoint from each other. For each i / ∈ I short with i ∈ [2 k ]there will be three distinct indices j i, , j i, , j i, ∈ [10 k log(2 k )] such that the path P i requiredin Claim 3 will consist of an initial segment of Q i,j i, , a middle segment of Q i,j i, , a final segmentof Q i,j i, as well as two edges joining these three segments. Similarly, for each i / ∈ I short with i ∈ [2 k + 1 , k ] the path P i required in Claim 3 will either be one of the Q i,j or will consist ofan initial segment of Q i,j i, and a final segment of Q i,j i, as well as an edge joining these twosegments.We will now choose the short i -paths. Let P correctshort be a collection of i -paths satisfying thefollowing properties:(P1) for each i ∈ [6 k ], P correctshort contains at most one i -path, IPARTITIONS OF HIGHLY CONNECTED TOURNAMENTS 9 (P2) all the paths in P correctshort are disjoint from each other,(P3) each path has length at most 10 k + 10,(P4) for each i ∈ [2 k + 1 , k ] for which P correctshort contains an i -path, this path P i has the correctparity, meaning that P i is even if i ∈ [2 k + 1 , k ] and odd if i ∈ [4 k + 1 , k ],(P5) subject to the above conditions, |P correctshort | is as large as possible.Let I correctshort be the set of all those indices i ∈ [6 k ] for which P correctshort contains an i -path, and let P i denote this i -path. Let V correctshort be the set of all internal vertices of the P i for all i ∈ I correctshort .Moreover, set I long := [6 k ] \ I correctshort . Recall that the definition of an i -path implies that all thevertices in V correctshort are uncoloured so far (i.e. V correctshort ∩ C = ∅ ). Claim 3.1:
We may colour all vertices in V correctshort as well as some additional vertices of T suchthat the following properties hold: (i) for each i ∈ I correctshort , all the vertices on P i are coloured α if i ∈ [ k ] and β if i ∈ [ k + 1 , k ] , (ii) for each i ∈ I correctshort \ [2 k ] , P i is coloured in an alternating manner consistent with itsendvertices, (iii) the set C consisting of all vertices coloured so far has size | C | ≤ k and all verticesin C are safe, (iv) for each i ∈ I long , any i -path whose internal vertices lie in V ( T ) \ C is either b i a i or haslength at least k + 10 . To prove Claim 3.1, consider any i ∈ I correctshort and colour all internal vertices of P i by α if i ∈ [ k ], by β if i ∈ [ k + 1 , k ], and in an alternating manner consistent with the endvertices of P i if i ∈ [2 k + 1 , k ] (this is possible by (P4)). Note that | V correctshort | ≤ k (10 k + 9) ≤ k . Togetherwith Claim 1 (applied with N := ∅ and Z := V correctshort ) and Claim 2 this implies Claim 3.1(i)–(iii),with room to spare in (iii). Indeed, the set C ′ of vertices coloured so far has size | C ′ | ≤ k .We will now colour some additional vertices to ensure that (iv) holds too. Consider any i ∈ I long . If there exists an i -path P whose internal vertices lie in V ( T ) \ C ′ and whose lengthis at most 10 k + 9, then P must have incorrect parity, i.e. P is odd if i ∈ [2 k + 1 , k ] and evenif i ∈ [4 k + 1 , k ]. Note that there cannot be two such i -paths of length at least two which areinternally disjoint from each other. Indeed, if P = v . . . v a and P ′ = v ′ . . . v ′ a ′ are two such i -paths which are internally disjoint, then we may assume that v v ′ ∈ E ( T ) and so v v v ′ v ′ . . . v ′ a ′ is an i -path of length at most 10 k + 10 with the correct parity which is disjoint from all theother paths in P correctshort , a contradiction to (P5).Let P incorrectshort be a collection of i -paths whose internal vertices lie in V ( T ) \ C ′ and whoselength is at least two and at most 10 k + 9, such that all these paths are disjoint from each otherand, subject to these properties, such that |P incorrectshort | is as large as possible. Let V incorrectshort bethe set of all internal vertices on these paths. Thus | V incorrectshort | ≤ k · (10 k + 8) ≤ k . Colourall vertices in V incorrectshort with α and apply Claim 1 again (with N := ∅ and Z := V incorrectshort ).Then the set C of all vertices coloured so far satisfies | C | ≤ k + 9 k · k ≤ k , so(iii) still holds. Moreover, now (iv) holds too. This completes the proof of Claim 3.1.Claim 3.1(iii) implies that all uncoloured vertices together with the a i and b i for all i ∈ I long induce a strongly (7 · · k log(2 k ))-connected subtournament T ′ of T (with some room tospare). Theorem 2.2 implies that T ′ is 7 · k log(2 k )-linked. Together with Proposition 2.1this implies that for each i ∈ I long we can find 10 k log(2 k ) i -paths in T ′ such that all these10 k log(2 k ) | I long | paths have length at least two and are internally disjoint from each otherand such that the internal vertices on all these paths lie outside C . We choose this collectionof 10 k log(2 k ) | I long | paths such that the set V long of all internal vertices on these paths is as small as possible. For all i ∈ I long and all j ∈ [10 k log(2 k )], let Q i,j denote the j th i -path wechose. Write Q i,j = q i,j q i,j . . . q | Q i,j | i,j , so that q i,j is b i and q | Q i,j | i,j is a i . Claim 3.1(iv) implies thateach Q i,j must have length at least 10 k + 10. Moreover, the minimality of | V long | implies thefollowing:(Q1) the interior of each Q i,j induces a backwards-transitive path,(Q2) if v ∈ V ( T ) \ ( C ∪ V long ) is an out-neighbour of q si,j , then v is also an out-neighbour of q s ′ i,j for all s ′ ≥ s + 3,(Q3) if v ∈ V ( T ) \ ( C ∪ V long ) is an in-neighbour of q si,j , then v is also an in-neighbour of q s ′ i,j for all s ′ ≤ s − Q i,j ) := q i,j . . . q | Q i,j |− i,j denote the interior of Q i,j . Let Q i,j , . . . , Q i,j be disjoint segmentsof int( Q i,j ) such that int( Q i,j ) = Q i,j . . . Q i,j , | Q i,j | = | Q i,j | = | Q i,j | = | Q i,j | = k , | Q i,j | = | Q i,j | = k + 2 and | Q i,j | = | Q i,j | = 2 k + 2. We let Q i,j := Q i,j ∪ Q i,j ∪ Q i,j ∪ Q i,j ∪ Q i,j ∪ Q i,j and write V long := [ ( i,j ) ∈ I long × [10 k log(2 k )] V ( Q i,j ) and V long := [ ( i,j ) ∈ I long × [10 k log(2 k )] V ( Q i,j ∪ Q i,j ∪ Q i,j ) . Thus V long ⊆ V long ⊆ V long and | V long | ≤ | V long | ≤ (10 k + 8) · k · k log(2 k ) ≤ · k log(2 k ) . Claim 3.2:
There exist disjoint index sets I R,α , I
R,β ⊆ I long × [10 k log(2 k )] such that, writing R α := [ ( i,j ) ∈ I R,α V ( Q i,j ) and R β := [ ( i,j ) ∈ I R,β V ( Q i,j ) , for each ( i, j ) ∈ I long × [10 k log(2 k )] every vertex in V ( Q i,j ) \ ( R α ∪ R β ) has at least k in-neighbours and at least k out-neighbours in each of R α and R β . Also | I R,α | , | I R,β | ≤ k log(2 k ) and | R α | , | R β | ≤ k log(2 k ) . To prove Claim 3.2, apply Proposition 2.5 to T [ V long ] to find a set Z α ⊆ V long with | Z α | ≤ k log | V long | ≤ k log(2 k ) and such that every vertex in V long \ Z α has at least k out-neighbours and k in-neighbours in Z α . Let I R,α := { ( i, j ) : V ( Q i,j ) ∩ Z α = ∅} and I ′ :=( I long × [10 k log(2 k )]) \ I R,α . We now consider W := S ( i,j ) ∈ I ′ V ( Q i,j ). By Proposition 2.5 ap-plied to T [ W ], there exists a set Z β ⊆ W with | Z β | ≤ k log | W | ≤ k log(2 k ) and suchthat every vertex in W \ Z β has at least k out-neighbours and in-neighbours in Z β . Let I R,β := { ( i, j ) ∈ I ′ : V ( Q i,j ) ∩ Z β = ∅} .Let R α and R β be as defined in the statement of Claim 3.2. Then by definition of I R,α and I R,β , for each ( i, j ) ∈ I long × [10 k log(2 k )] every vertex in V ( Q i,j ) \ ( R α ∪ R β ) has atleast k in-neighbours and at least k out-neighbours in each of R α and R β . Also | R α | , | R β | ≤ (6 k + 4) · k log(2 k ) ≤ k log(2 k ). This completes the proof of Claim 3.2.Let I R := I R,α ∪ I R,β , R := R α ∪ R β and R , := [ ( i,j ) ∈ I R V ( Q i,j ∪ Q i,j ) . IPARTITIONS OF HIGHLY CONNECTED TOURNAMENTS 11
Claim 3.3:
We may colour all vertices in R α ∪ R β ∪ R , as well as some additional verticeslying outside V long such that (i) all vertices in R α are coloured α , all vertices in R β are coloured β , (ii) for each ( i, j ) ∈ I R and each s ∈ { , } , Q si,j is an alternating path, (iii) all coloured vertices are safe, (iv) the set C consisting of all vertices coloured so far has size | C | ≤ · k log(2 k ) . To prove Claim 3.3, colour the vertices in R α ∪ R β ∪ R , such that (i) and (ii) hold. ApplyClaim 1 with C , R α ∪ R β ∪ R , , V long \ ( R α ∪ R β ∪ R , ) playing the roles of C , Z , N to obtaina set Z ′ ⊆ V ( T ) \ ( V long ∪ C ) and a colouring of the vertices in Z ′ such that every vertex in R α ∪ R β ∪ R , ∪ Z ′ is safe and | C | ≤ | C | + | R α ∪ R β ∪ R , ∪ Z ′ | ≤ k + 9 k · (2 · k log(2 k ) + (4 k + 4) · k log(2 k )) ≤ · k log(2 k ) . This completes the proof of Claim 3.3.
Claim 3.4:
For each i ∈ I long there is an i -path P i such that the following properties hold: (i) P i has no internal vertices in C , and P i and P i ′ are disjoint whenever i = i ′ , (ii) if i ∈ I long ∩ [2 k ] , then there exists three distinct indices j i, , j i, , j i, ∈ [10 k log(2 k )] suchthat P i = b i Q i,j i, Q i,j i, q k +1 i,j i, Q i,j i, . . . Q i,j i, q | Q i,ji, |− k − i,j i, Q i,j i, Q i,j i, a i , (iii) if i ∈ I long ∩ [2 k + 1 , k ] , then either P i = Q i,j i for some j i ∈ [10 k log(2 k )] or there existdistinct j i, , j i, ∈ [10 k log(2 k )] such that P i = b i Q i,j i, . . . Q i,j i, q k +5 i,j i, Q i,j i, . . . Q i,j i, a i , (iv) P i is even if i ∈ I long ∩ [2 k + 1 , k ] and odd if i ∈ I long ∩ [4 k + 1 , k ] . (Recall that q k +1 i,j i, is the first vertex of Q i,j i, , q | Q i,ji, |− k − i,j i, is the last vertex of Q i,j i, and q k +5 i,j i, is the first vertex of Q i,j i, .) To prove Claim 3.4, note that since | C | ≤ · k log(2 k ) < k log(2 k ) −
5, for each i ∈ I long there are at least five paths Q i,s i, , Q i,s i, , Q i,s i, , Q i,s i, , Q i,s i, whose internal vertices avoid C .Suppose first that i ∈ I long ∩ [2 k ]. Consider the subtournament T i of T spanned by q | Q i,si,t |− k − i,s i,t for t = 1 , , , , T i contains at least two vertices of out-degree at least two, assume they are q | Q i,si, |− k − i,s i, , q | Q i,si, |− k − i,s i, . We may also assume that q k +1 i,s i, sends an edge to q k +1 i,s i, . Finally,since q | Q i,si, |− k − i,s i, has at least two outneighbours in T i , we may assume q | Q i,si, |− k − i,s i, sends anedge to q | Q i,si, |− k − i,s i, . We set j i,t := s i,t and let P i be as described in Claim 3.4(ii).So suppose next that i ∈ I long ∩ [2 k + 1 , k ]. If Q i,s i,t is an even path for t = 1 or t = 2 wetake it to be P i . So suppose that these two paths are odd. We may assume that q k +5 i,s i, sends anedge to q k +5 i,s i, . We set j i, := s i, and j i, := s i, and let P i be as described in Claim 3.4(iii). If i ∈ I long ∩ [4 k + 1 , k ], we define P i similarly. This completes the proof of Claim 3.4.We are now ready to prove Claim 3. For each i ∈ I long , let P i be as given by Claim 3.4.We will colour all those vertices on the paths Q i,j with ( i, j ) ∈ I long × [10 k log(2 k )] which areuncoloured so far as follows.For each i ∈ I long ∩ [2 k ], we colour all internal vertices of P i by α if i ≤ k and by β if i > k .For each i ∈ I long ∩ [ k ], we also colour all vertices in ( Q i,j ∪ Q i,j ) \ ( V ( P i ) ∪ R ) by α and allvertices in ( Q i,j ∪ Q i,j ∪ Q i,j ∪ Q i,j ) \ ( V ( P i ) ∪ R ) by β (for all j ∈ [10 k log(2 k )]). Similarly, for each i ∈ I long ∩ [ k + 1 , k ], we colour all vertices in ( Q i,j ∪ Q i,j ) \ ( V ( P i ) ∪ R ) by β and allvertices in ( Q i,j ∪ Q i,j ∪ Q i,j ∪ Q i,j ) \ ( V ( P i ) ∪ R ) by α . For each i ∈ I long ∩ [2 k ], we colour allvertices in ( Q i,j ∪ Q i,j ) \ ( V ( P i ) ∪ R , ) by α .For each i ∈ I long ∩ [2 k + 1 , k ], we colour all internal vertices of P i in an alternating mannerconsistent with the endvertices of P i . (Claim 3.4(iv) ensures that this is possible.) For all j ∈ [10 k log(2 k )] we also colour all vertices in Q i,j ∪ Q i,j ∪ Q i,j \ ( V ( P i ) ∪ R ∪ R , ) in analternating manner. (That is, if b i = q i,j is coloured α , we colour q si,j by α for all even numbers s ≤ k + 4, and colour q si,j by β for all odd numbers s ≤ k + 4. We colour of each vertex x in( Q i,j ∪ · · · ∪ Q i,j ) \ ( V ( P i ) ∪ R ∪ R , ) in a similar way, depending on the colour of a i and thedistance of x to a i in Q i,j .)Now all uncoloured vertices of V long belong to Q i,j for some i, j . We let C be the union of V ( Q i,j ) over all ( i, j ) ∈ I long × [10 k log(2 k )]. We colour all uncoloured vertices in C by α ,and let C denote the set consisting of all the vertices coloured so far. Note that | C \ C | ≤| C | + | V long | ≤ · k log(2 k ) + 2 · k log(2 k ) ≤ · k log(2 k ). Together with Claim 3.1this implies that parts (i), (ii) and (iii)(b) of Claim 3 hold.We now show that all the vertices on the paths Q i,j are safe. Together with Claim 3.3(iii) thiswill imply that all vertices in C are safe, i.e. Claim 3(iii)(a) will hold. Consider first any vertex v ∈ V long . If v ∈ R , then v is safe by Claim 3.3(iii). If v / ∈ R , then by Claim 3.2 v has at least k out-neighbors and at least k in-neighbours in each of R α and R β , so it has k safe out-neighboursand k safe in-neighbours of each colour. Thus v is safe by (S3) and (S4). So all the the verticesin V long are safe.Note that if ( i, j ) / ∈ I R , then V ( Q i,j ∪ Q i,j ∪ Q i,j ) \ { q k +2 i,j } contains at least k vertices of eachcolour and so does V ( Q i,j ∪ Q i,j ∪ Q i,j ) \ { q | Q i,j |− k − i,j } . (Recall that q k +2 i,j is the final vertex of Q i,j and that q | Q i,j |− k − i,j is the initial vertex of Q i,j .)Now consider a vertex v ∈ V long \ V long , and let i, j be such that v ∈ V ( Q i,j ∪ Q i,j ). If v ∈ R , , then v is safe by Claim 3.3(iii). If v / ∈ R , , then ( i, j ) / ∈ I R . But by (Q1) all vertices in Q i,j ∪ Q i,j ∪ Q i,j (apart from possibly the final vertex of Q i,j ) are out-neighbours of v , so v has k safe out-neighbours coloured α and k safe out-neighbours coloured β . Similarly, all verticesin Q i,j ∪ Q i,j ∪ Q i,j (apart from possibly the initial vertex of Q i,j ) are in-neighbours of v , so v has k safe in-neighbours coloured α and k safe in-neighbours coloured β . Hence v is safe.Now consider any vertex v ∈ V ( Q i,j ). If ( i, j ) / ∈ I R , a similar argument as above shows that v is safe. If ( i, j ) ∈ I R , then by (Q1) all vertices in Q i,j (apart from possibly its final vertex)are out-neighbours of v , and all vertices in Q i,j (apart from possibly its initial vertex) are in-neighbours of v . Together with Claim 3.3(ii),(iii) this shows that v has k safe out-neighboursand k safe in-neighbours of each colour. So v is safe. This completes the proof of Claim 3(iii)(a).To check Claim 3(iii)(c), note that if a vertex v outside C has an out-neighbour in C , thenby (Q3) all vertices in Q i,j ∪ Q i,j ∪ Q i,j ∪ Q i,j (apart from possibly the last two vertices of Q i,j )are out-neighbours of v . Thus v has at least k out-neighbours of each colour. In a similar wayone can use (Q2) to show that v also has k in-neighbours of each colour. This completes theproof of Claim 3. Claim 4:
We can colour all uncoloured vertices in such a way that every vertex is safe.
To prove Claim 4, we colour all the vertices outside C one by one. We first deal with allvertices in E A \ C (see STEP 1), then we move to the vertices in E B \ C (see STEP 2). Finally, IPARTITIONS OF HIGHLY CONNECTED TOURNAMENTS 13 we colour all the remaining vertices (see STEP 3). We let Z A := ∅ . While dealing with eachvertex in E A \ C in turn (i.e. during STEP 1), we will update Z A . At each substep, Z A willsatisfy the following properties:(a) Z A consists of coloured vertices and Z A ∩ ( C ∪ E A ) = ∅ ,(b) every coloured vertex lies in C ∪ E A ∪ Z A ,(c) | Z A | ≤ k | E A | .STEP 1. We can colour all vertices in E A \ C as well as some set Z A of additional verticesin such a way that all the vertices in E A \ C are backwards-safe and alternating-backwards-safeand Z A satisfies (a)–(c). Consider each vertex v ∈ E A \ C in turn. Suppose first that v has 2 k uncoloured in-neighbours v , v , . . . , v k outside E A . We colour k of them by α and k of them by β and replace Z A by Z A ∪ { v , v , . . . , v k } . We also colour v with α . Note that (S2) implies that v , v , . . . , v k arebackwards-safe and alternating-backwards-safe. Together with (S3) and (S4) this shows that v is backwards-safe and alternating-backwards-safe.So suppose that v has less than 2 k uncoloured in-neighbours outside E A . Recall fromClaim 3(iii)(b) that at most 3 · k log(2 k ) vertices in C lie outside the set C . Togetherwith (3.3) and (c) this shows thatˆ δ − ( T ) − | E A ∪ Z A | ≥ ˆ δ − ( T ) − k | E A | ≥ · k log(2 k ) ≥ k + | C \ C | . Since all coloured vertices lie in C ∪ E A ∪ Z A , this implies that v has an in-neighbour in C .But now Claim 3(iii)(c) implies that v has k in-neighbours of colour α and k in-neighbours ofcolour β in C . Since all the vertices in C are safe, this implies that v becomes backwards-safeand alternating-backwards-safe by colouring v with α .Note that we add at most 2 k vertices to Z A for each vertex v ∈ E A \ C . So at the endof STEP 1, we will still have that | Z A | ≤ k | E A | . Since by (S2) every vertex outside E B isforwards-safe and alternating-forwards-safe, after STEP 1, all vertices in E A \ E B will be safe,while the vertices in E A ∩ E B might only be backwards-safe and alternating-backwards-safe.Let Z B := ∅ . While dealing with each vertex in E B \ C in turn during STEP 2, we willupdate Z B . At each substep, Z B will satisfy the following properties (where Z := Z A ∪ Z B ):(a ′ ) Z B consist of coloured vertices and Z B ∩ ( C ∪ E ∪ Z A ) = ∅ ,(b ′ ) every coloured vertex lies in C ∪ E ∪ Z ,(c ′ ) | Z B | ≤ k | E B | and so | Z | ≤ k | E | .STEP 2. We can colour all uncoloured vertices in E B \ C as well as some set Z B of additionalvertices in such a way that all the vertices in E B \ C are safe and Z B satisfies (a ′ )–(c ′ ). Consider each vertex v ∈ E B \ C in turn. If v / ∈ E A , then v is backwards-safe and alternating-backwards-safe by (S2). If v ∈ E A , then by STEP 1 v is also backwards-safe and alternating-backwards-safe.Suppose first that v has 2 k uncoloured out-neighbours v , v , . . . , v k outside E . We colour k of them by α and k of them by β . We replace Z B by Z B ∪ { v , v , . . . , v k } . If v is uncoloured,we colour v with α . Then (S2)–(S4) together imply that v becomes safe.So suppose that v has less than 2 k uncoloured out-neighbours outside E . Note thatˆ δ + ( T ) − | E ∪ Z | ≥ ˆ δ + ( T ) − k | E | ≥ · k log(2 k ) ≥ k + | C \ C | by (3.4), (c ′ ) and Claim 3(iii)(b). Since all coloured vertices lie in C ∪ E ∪ Z , this implies that v has an out-neighbour in C . But now Claim 3(iii)(c) implies that v has k out-neighbours of colour α and k out-neighbours of colour β in C . Since all the vertices in C are safe, this impliesthat v becomes safe by colouring v with α (in case v is still uncoloured).Note that we add at most 2 k vertices to Z B for each vertex in E B \ C . Thus we always havethat | Z B | ≤ k | E B | and so | Z | ≤ k | E | . After STEP 2, all vertices in C ∪ E are safe.STEP 3. By colouring all the remaining uncoloured vertices with α , every vertex becomes safe. This follows immediately from (S2).This completes the proof of Claim 4 and thus of Theorem 1.1. (cid:3) Spanning linkedness and non-separating subdivisions
The following lemma generalizes a result of Thomassen [11]. Theorems 1.2 and 1.4 then bothfollow easily by an inductive application of Lemma 4.1.
Lemma 4.1.
Let k, d be nonnegative integers. Let x, y, z , . . . , z d be any distinct vertices in astrongly ( k + d + 4) -connected tournament T and let P be a shortest xy -path in T − { z , . . . , z d } .Then T − ( V ( P ) \ X ) is strongly k -connected for any (possibly empty) subset X ⊆ { x, y } . Proof.
Write P := x x . . . x m with x = x and y = x m . Note that P must be a backwards-transitive path. If P has length at most two, the result trivially holds. So suppose that P haslength more than two. Note that in this case it suffices to show that T − V ( P ) is strongly k -connected (otherwise we consider x ′ ∈ { x, x } , y ′ ∈ { y, x m − } and proceed through the argumentwith x ′ , y ′ playing the role of x, y ). So suppose T − V ( P ) is not strongly k -connected. Thenthere exist a partition of V ( T ) \ V ( P ) into nonempty sets S, S , S such that | S | ≤ k − S sends an edge to S . Since T − ( S ∪ { z , . . . , z d } ) is strongly 5-connected,there are five paths P , . . . , P from S to S which are internally disjoint and do not intersect S ∪ { z , . . . , z d } . We may assume that the P i are backwards-transitive. Moreover, the interiorof each P i is nonempty and is contained in V ( P ). Altogether, this means that the intersectionof P i and T [ V ( P )] is either a segment of P or a path of the form x j x ℓ with j ≥ ℓ + 2 or of theform x j x j +1 x j − or x j x j − x j − . We let p be the largest number such that some P i containsan edge ux p from S to x p and we let q be the smallest number such that some P i containsan edge x q v from x i to S . Note that p ≥ q + 4. Then the path obtained from P by deleting x q +1 x q +2 . . . x p − and adding x q vux p is shorter than P , a contradiction. (cid:3) Proof of Theorem 1.2.
Write D = { w , . . . , w d } . We proceed by induction on m . For m = 1, the assertion holds by Lemma 4.1 applied with d − d . Supposethat m ≥ m −
1. Consider any edge uv ∈ E ( H ). Withoutloss of generality, we may assume that φ ( u ) = w and φ ( v ) = w . Then we apply Lemma 4.1(with d − d ) to find a w w -path P whose interior does not intersect D andso that T ′ := T − V (int( P )) is strongly ( k + ( m − d + 2))-connected. Now by the inductionhypothesis, we can find a subdivision H ∗ of H \ { uv } in T ′ which satisfies (i)–(iii) (with T ′ playing the role of T ). Finally, let H ∗ := H ∗ ∪ int( P ). Then H ∗ satisfies all requirements. (cid:3) IPARTITIONS OF HIGHLY CONNECTED TOURNAMENTS 15
Proof of Theorem 1.4.
We proceed by induction on k . For k = 1, the assertion was provenby Thomassen [8]. Assume that k ≥ k −
1. Let Z := { x , . . . , x k − , y , . . . , y k − } and let X := { x k , y k } ∩ Z . We can now apply Lemma 4.1 with d = | Z \ X | to a find a x k y k -path P avoiding Z \ X so that T [ W ] is strongly (( k − + 3( k − W := V ( T ) \ ( V ( P ) \ X ). Now by the induction hypothesis, we can find P , . . . , P k − in T [ W ] so that P i is a path from x i to y i and W = S k − i =1 V ( P i ). Let P k := P .Then P , . . . , P k are as desired. (cid:3) Acknowledgements
We are grateful to Jørgen Bang-Jensen for helpful comments on an earlier version of thispaper.
References [1] P. Hajnal, Partition of graphs with condition on the connectivity and minimum degree,
Combinatorica (1983), 95–99.[2] D. K¨uhn, J. Lapinskas, D. Osthus and V. Patel, Proof of a conjecture of Thomassen on Hamilton cycles inhighly connected tournaments, Proc. London Math. Soc. (2014), 733–762.[3] D. K¨uhn and D. Osthus, Partitions of graphs with high minimum degree or connectivity,
J. CombinatorialTheory B (2003), 29–43.[4] D. K¨uhn, D. Osthus and T. Townsend, Proof of a tournament partition conjecture and an application to1-factors with prescribed cycle lengths, Combinatorica , to appear.[5] A. Pokrovskiy, Highly linked tournaments, preprint, arXiv:1406.7552.[6] A. Pokrovskiy, Edge disjoint Hamiltonian cycles in highly connected tournaments, preprint, arXiv:1406.7556.[7] A. Scott, Judicious partitions and related problems,
Surveys in Combinatorics 2005, London Math. Soc.Lecture Note Ser. , Cambridge Univ. Press, Cambridge, (2005) 95–117.[8] C. Thomassen, Hamiltonian-connected tournaments,
J. Combinatorial Theory B (1980), 142–163.[9] C. Thomassen, Graph decomposition with constraints on the connectivity and minimum degree, J. GraphTheory (1983), 165–167.[10] C. Thomassen, Graph decomposition with applications to subdivisions and path systems modulo k , J. GraphTheory (1983), 261–271.[11] C. Thomassen, Connectivity in tournaments in, Graph Theory and Combinatorics, A volume in honour ofPaul Erd˝os (B. Bollob´as, ed.), Academic Press, London (1984), 305–313.[12] C. Thomassen Configurations in graphs of large minimum degree, connectivity, or chromatic number,
Com-binatorial Mathematics: Proceedings of the Third International Conference (New York, 1985), vol. 555 ofAnn. New York Acad. Sci. , New York, 1989, 402–412.Jaehoon Kim, Daniela K¨uhn, Deryk OsthusSchool of MathematicsUniversity of BirminghamEdgbastonBirminghamB15 2TTUK
E-mail addresses: { kimJS, d.kuhn, d.osthus }}