Subdivisions in digraphs of large out-degree or large dichromatic number *

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Introduction
For each positive integer k, we denote by [k] the subset of the natural numbers {1, . . . , k}.
Theorem 1 (Mader [19]). There exists an integer g(k) such that every graph with minimum degree at least g(k) contains a subdivision of K k .
Similarly, it would be interesting to find analogous results for digraphs. However, the obvious analogue that a digraph with sufficiently large minimum in-and out-degree contains a subdivision of the complete digraph of order n is false as shown by Mader [20].
Let γ be a digraph parameter. A digraph F is γ-maderian if there exists a least integer mader γ (F ) such that every digraph D with γ(D) ≥ mader γ (F ) contains a subdivision of F .
On the positive side, Mader conjectured that every acyclic digraphs is δ + -maderian. Since every acyclic digraph is the subdigraph of the transitive tournament on the same order, it is enough to prove that transitive tournaments are δ + -maderian.
In fact, Conjecture 3 is equivalent to Conjecture 2 because if transitive tournaments are δ 0 -maderian, then mader δ + (T T k ) ≤ mader δ 0 (T T 2k ) for all k. Indeed, let D be a digraph with minimum out-degree mader δ 0 (T T 2k ), and let D be the digraph obtained from disjoint copies of D and its converse (the digraph obtained by reversing all arcs) D by adding all arcs from D to D. Clearly, δ 0 (D ) ≥ mader δ 0 (T T 2k ). Therefore D contains a subdivision of T T 2k . Hence, either D or D (and so D) contains a subdivision of T T k .
Both conjectures are equivalent, but the above reasoning does not prove that a δ 0 -maderian digraph is also δ + -maderian. The case of oriented trees (i.e. orientations of undirected trees) is typical. Using a simple greedy procedure, one can easily find every oriented tree of order k in every digraph with minimum in-and out-degree k (so mader δ 0 (T ) = |T | − 1 for any oriented tree T ). On the other hand, it is still open whether oriented trees are δ + -maderian and a natural important step towards Conjecture 2 would be to prove the following weaker one.
We give evidences to this conjecture. First, in Subsection 2.1, we prove that every oriented path (i.e. orientation of an undirected path) P is δ + -maderian and that mader δ + (P ) = |V (P )| − 1. Next, in Subsection 2.2, we consider arborescences. An out-arborescence (resp. in-arborescence) is an oriented tree in which all arcs are directed away from (resp. towards) a vertex called the root. Trivially, the simple greedy procedure shows that mader δ + (T ) = |T | − 1 for every out-arborescence. In contrast, the fact that in-arborecences are δ + -maderian is not obvious since we have no control on the in-degree of each vertex in a digraph of out-degree at least k. We show in Theorem 23 that the in-arborescences are δ + -maderian.
In [21], Mader gave another partial result towards Conjecture 2. He proved the existence of a function f 1 (k) such that every digraph D with δ + (D) ≥ f 1 (k) contains a pair of vertices u and v with k disjoint directed paths between u and v. It is however not known if we can insist on these paths to be arbitrarily long. In Theorem 24, we do the first step toward this question by proving it in the case k = 2.
Conjecture 2 states that the acyclic digraphs are δ + -maderian. However, they are not the only ones. For example, it is folklore that every digraph with minimum out-degree at least 1 contains a directed cycle, which is a subdivision of C 2 , the directed 2-cycle. More generally, one can easily show, by considering a directed path of maximum length, that every digraph with minimum out-degree at least k − 1 contains a directed cycle of length at least k. In other words, mader δ + ( C k ) = k − 1, where C k denotes the directed k-cycle. Furthermore Alon [3] showed that every digraph with minimum out-degree at least 64k contains k disjoint directed cycles, which forms a subdivision of the disjoint union of k copies of C 2 . A celebrated conjecture of Bermond and Thomassen [4] states that the bound 64k can be decreased to 2k − 1. [4]). For every positive integer k, every digraph with minimum out-degree at least 2k − 1 contains k disjoint directed cycles.

Conjecture 5 (Bermond and Thomassen
In [3], Alon also conjectured the following. Conjecture 6 (Alon [3]). There exists a function h such that every digraph with minimum out-degree h(k) has a partition (V 1 , V 2 ) such that, for i = 1, 2, D V i has minimum out-degree k.
The difficulty of this question is remarkable, as the existence of h(2) still remains open. If true, this conjecture implies the following one. Conjecture 7. If F 1 and F 2 are δ + -maderian, then the disjoint union of F 1 and F 2 is also δ + -maderian.
Partial positive answers to this conjecture can be obtained via the Erdős-Pósa Property. For a digraph F and an integer k, we denote by k × F the disjoint union of k copies of F . A digraph F is said to have the Erdős-Pósa Property if for every positive integer k, there exists φ(k), such that every digraph D either contains a subdivision of k × F , or has a set S of size (at most) φ(k) such that D − S contains no subdivision of F . Theorem 8. If F is δ + -maderian and has the Erdős-Pósa Property, then k × F is also δ + -maderian for all positive integer k.
Proof. Let F be a maderian digraph having the Erdős-Pósa Property for some function φ. Let D be a digraph with δ + (D) ≥ φ(k)+mader δ + (F ). For every set S of size at most φ(k), δ + (D−S) ≥ mader δ + (F ), so D−S contains a subdivision of F . Thus, since F has the Erdős-Pósa Property, D contains a subdivision of k × F .
Reed et al. [23] proved that C 2 has the Erdös-Pósa Property, and using the Directed Grid Theorem [16], Akhoondian et al. [2] showed that many digraphs have the Erdös-Pósa Property, in particular all disjoint unions of directed cycles. Hence, Theorem 8 implies that disjoint unions of directed cycles are δ + -maderian.
On the negative side, Thomassen [27] showed a construction of digraphs with arbitrarily large inand out-degree and no directed cycles of even length (see also [10]). This gives a large class of digraphs that are not δ 0 -maderian, namely: digraphs such that all its subdivisions have a directed cycle of even length. In particular, K 3 , the complete digraph on three vertices, belongs to this class. All digraphs in this class have been characterized by Seymour and Thomassen [25]. Morever, Devos et al. [10] showed the existence of digraphs with large minimum out-degree and in-degree such that there exists no pair of vertices u and v with four internally disjoint directed paths between them, two from u to v and two from v to u. In Theorem 28, we show that if the minimum out-degree is large enough then we can find two vertices u and v with three internally disjoint directed paths between them, one from u to v and two from v to u.
Note that every graph with chromatic number at least p has a subgraph with minimum degree at least p − 1. This implies, by Theorem 1, that every graph with chromatic number at least g(k) + 1 contains a subdivision of K k . In the context of digraphs, there exist two natural analogues of the chromatic number. Given a digraph D, the chromatic number of D, denoted by χ(D), is simply the chromatic number of its underlying graph. The dichromatic number of D, denoted by χ(D), is the smallest integer k such that D admits a k-dicolouring. A k-dicolouring is a k-partition {V 1 , . . . , V k } of V (D) such that D V i is acyclic for every i ∈ [k]. Hence, it is natural to ask which digraphs are χ-maderian and which ones are χ-maderian.
Burr [7] proved that every (k − 1) 2 -chromatic digraph contains every oriented forest of order k. Later on, Addario-Berry et al. [1] slightly improved this value to k 2 /2 − k/2 + 1. This implies that every oriented forest is χ-maderian. Cohen et al. [9] showed that for any positive integer b, there are digraphs of arbitrarily high chromatic number that contain no oriented cycles with less than b blocks. (A block in an oriented path or cycle is a inclusion maximal directed subpath.) This directly implies that if a digraph is not an oriented forest, then it is not χ-maderian because it contains an oriented cycle, all subdivisions of which have the same number of blocks. The χ-maderian digraphs are known but determining mader χ for such digraphs is still open. Burr [7] made the following conjecture.
An interesting step towards Burr's conjecture is to prove the following consequence of it.
In Section 3, we prove that every digraph is χ-maderian. Again determining mader χ for every digraph is still open. Since every digraph D of order n is a subdigraph of K n , the complete digraph of order n, and so mader χ (D) ≤ mader χ ( K n ), it is natural to focus on K n . Problem 12. What is mader χ ( K n ) ?
To prove Theorem 1, Mader showed a stronger result about the average degree. Recall that the average degree of a graph G isd(G) = 2|E(G)|/|V (G)|. He proved that there exists a function g (k) such that every graph G with at leastd(G) ≥ g (k) contains a subdivision of K k . The average outdegree of a digraph D isd + (D) = |A(D)|/|V (D)|. (Note that this is equal to the average in-degree and half the average degree.) A digraph is − → P 3 -free if it does not contain − → P 3 as a subdigraph, where − → P 3 is the dipath on three vertices. There are bipartite graphs with arbitrarily large minimum degree and arbitrarily large girth (recall that the girth of a graph is the length of a smallest cycle or +∞ if it is a forest). Orienting edges of such graphs from one part to the other result in − → P 3 -free digraphs with arbitrarily large average out-degree and arbitrarily large girth (the girth of a digraph is the girth of its underlying graph). Consequently, a digraph isd + -maderian only if it is an antidirected forest, that is, an oriented forest containing no − → P 3 as a subdigraph. This simple necessary condition is also sufficient. Burr [8] showed that all antidirected forests ared + -maderian: for every antidirected forest F , The value k − 2 in this conjecture would be best possible, since the oriented star S + k , consisting of a vertex dominating k − 1 others, is not contained in any digraph in which every vertex has out-degree k − 2. It is also tight because the complete digraph K k−1 has (k − 2)(k − 1) arcs but trivially does not contain any oriented tree of order k.

Conjecture 15 (Erdős and Sós
Their conjecture has attracted a fair amount of attention over the last decades. Partial solutions are given in [6,15,24]. In the early 1990's, Ajtai, Komlós, Simonovits and Szemerédi announced a proof of this result for graphs of sufficiently large order. Since k-connected and k-edge-connected graphs have minimum degree at least k, Theorem 1 implies that every graph G with connectivity (resp. edge-connectivity) at least g(k) contains a subdivision of K k . Let κ(D), κ (D), be respectively the strong connectivity and the strong arc-connectivity of D.
The following conjecture due to Thomassen [28] implies that all digraphs are κ-maderian.
Conjecture 17 (Thomassen [28]). There exists f 1 (k) such that if κ(D) ≥ f 1 (k), and x 1 , . . . , x k and y 1 , . . . , y k are distinct vertices of D, then D contains k disjoint dipaths P 1 , . . . , P k such that P i goes from This conjecture would also imply the following one due to Lovász [18].
Conjecture 18 (Lovász [18]). There exists an integer p such that every p-strongly connected digraph has an even directed cycle.
2 Subdivision in digraphs with large minimum out-degree

Subdivisions of oriented paths
Let P = (x 1 , x 2 , · · · , x n ) be an oriented path. We say that P is an (x 1 , x n )-path. The vertex x 1 is the initial vertex of P and x n its terminal vertex. P is a directed path or simply a dipath, if Let k 1 be nonnegative integer and k 2 , . . . , k be positive integers. We denote by P (k 1 , k 2 , . . . , k ) the path obtained from an undirected path Proof. By induction on . If = 1, then the result holds trivially. Assume now ≥ 2, and suppose that, , let C be the connected component of D containing u, and let H be a sink strong component of C (i.e. a strong component without arcs leaving it) that is reachable by a directed path in C starting at u. We denote by As a consequence, H contains a directed cycle of length at least k 2 . Using this and the fact that H is strongly connected, we conclude that there exists a dipath P y, Therefore, the result follows immediately if = 2. Suppose now that ≥ 3. By the induction hypothesis, G contains a path W y := P (k 3 , . . . , k ) with initial vertex y such that k i ≥ k i if i is odd, and k i = k i otherwise. Therefore, Q v,y W y is the desired path P (k 1 , k 2 , . . . , k ) with initial vertex v.
. . , k ))| − 1, and that the complete digraph K k on k vertices has minimum out-degree k − 1 and contains no path on more than k vertices, we obtain the following corollary.

Subdivision of in-arborescences
The aim of this subsection is to prove that the in-arborescences are δ + -maderian. We need some preliminary results. The first one is the vertex and directed version of the celebrated Menger's theorem [22]. (See also [14] for a short proof.) Proof. Suppose to the contrary that there do not exist |S| vertex-disjoint (S, T )-dipaths in D. By Menger's theorem, there exists a set of vertices X ⊆ V (D) with cardinality |X| < |S| which is an (S, T )vertex-cut. Let C be the set of vertices in D − X that are reachable in D by a dipath with initial vertex in S \ X. Set k = |X ∩ S|.
Let us count the number a(C, X) of arcs with tail in C and head in X. Since the vertices in S have in-degree 0 and every vertex in C has out-degree at least ∆ − (D), Moreover, a(C, X) is at most the number of arcs with head in X which is at most This is a contradiction to |X| < |S|, since ∆ − (D) ≥ 1 because D has at least one arc.
Let k and be positive integers. The -branching in-arborescence of depth k, denoted by B(k, ), is defined by induction as follows.
• B(0, ) is the in-arborescence with a single vertex, which is the root and the leaf of B(0, ). Observe that every in-arborescence T is a subdigraph of B(k, ) with = ∆ − (T ) and k the maximum length of a dipath in T . Therefore to prove that in-arboresences are δ + -maderian, it suffices to show that B(k, ) is δ + -maderian for all k and .
We define a recursive function f : N → N as follows. For all positive integers k and such that ≥ 2, f (1, ) = and, for k ≥ 2, we define If D is a family of digraphs, a packing of elements of D is the disjoint union of copies of elements of D. Subproof. Let p = b(k − 1, ) · ( − 1) + 1. By the induction hypothesis, H contains a subdivision T of B(k − 1, p). Let R be the set of branching vertices of T , that is, R = {r ∈ T : d − T (r) = p}. We assume that each in-arborescence of F has at most b(k − 1, ) vertices, as any larger arborescence would yield the theorem. Thus, for each r ∈ R , there exists a vertex h r in the in-arborescence rooted at r such that h r is dominated in D by vertices of V (T ). Similarly, for each r ∈ V (T ) with in-degree 1, there exists a vertex h r in the in-arborescence rooted at r such that h r is dominated in D by a vertex of V (T ). Using these remarks, we next define a procedure to obtain a subdigraph of T that is a subdivision of B(k − 1, ).
For each r ∈ R , we remove from T all arcs with head r but exactly arcs from vertices in V (T ) that dominate h r in D. We denote by T the component of the subdigraph of T obtained by applying the above-described procedure and that contains the root of T . One may easily verify that T is a subdivision of B(k − 1, ). Let P r be the path from h r to r in the in-arborescence corresponding to r, for every r ∈ V (T ) such that either r ∈ R or r has in-degree 1.
Let Q be the in-arborescence obtained from T in the following way. For each r ∈ V (T ) such that either r ∈ R or r has in-degree 1, we add h r to T , and we add an arc from every in-neighbour of r in T to h r . Additionally, we remove all arcs with head r in T , and link h r to r by using the dipath P r . Finally, for each r ∈ V (T ) that is a leaf, we replace r by its corresponding in-arborescence belonging to F.
By this construction, we have that Q is a subdigraph of D such that every internal vertex has either in-degree or 1. Furthermore, it has depth at least k. Therefore, by possibly pruning some levels of Q, we obtain a subdivision of B(k, ). ♦ Let D be the digraph obtained from D[U ∪ X] by removing all arcs with head in X. From D , we construct a digraph G by replacing every vertex v ∈ X by t(k, ) new vertices v 1 , . . . , v t(k, ) , and adding, . In other words, we "redistribute" the out-neighbours of v in D among its t(k, ) copies in G so that every copy has out-degree at least ( − 1) · k, and the in-degrees of vertices belonging to U are not changed. Let S ⊆ V (G) be the set of vertices that replaced those of X, that is, S = v∈X {v 1 , . . . , v t(k, ) }. Let T be the set of vertices in U that have large out-degree outside U in the digraph D, more formally, forms a partition of the packing F. Additionally, observe that, due to the maximality of F, every vertex in U is dominated by at most −1 vertices belonging to U , and by at most −1 roots of in-arborescences in F i , for each i ∈ [k−1]. Thus, the in-degree in G of every vertex belonging to U is at most ( −1) Lemma 22, there exists a set P of |S| vertex-disjoint paths from S to T in G.
Note that, in D, every vertex belonging to T has at least t(k, ) + 1 out-neighbours in V (D) \ U . Therefore one can greedily extend each path of P with an out-neighbour of its terminal vertex in V (D)\U in order to obtain a set P of |S| vertex-disjoint paths from S to V (D) \ U such that for any v ∈ X all the paths in P with initial vertex v have distinct terminal vertices (and different from v).
We now construct the digraph M on the vertex set R where there exists an arc from v = r A to r B in M whenever either r A dominates some vertex of V (B) in D, or there is a dipath from some v i to V (B) in P .
Since, for each v ∈ X, all vertices in {v i } i∈[t(k, )] are the initial vertices of vertex-disjoint dipaths in P , we obtain δ + (M ) ≥ t(k, )/b(k − 1, ). Therefore, the result follows by Claim 23.1 with M playing the role of H.

Cycles with two blocks
We denote by C(k 1 , k 2 ) the digraph which is the union of two internally disjoint dipaths, one of length k 1 and one of length k 2 with the same initial vertex and same terminal vertex. C(k 1 , k 2 ) may also be seen as an oriented cycles with two blocks, one of length k 1 and one of length k 2 . Recall that the blocks of an oriented cycle are its maximal directed subpaths.
Since a (1, k 1 , k 2 )-fork has k 1 + k 2 + 1 vertices and δ + (D) ≥ k 1 + k 2 + 1, the digraph D contains a (1, k 1 , k 2 )-fork as a subdigraph. Let ≥ 1 be the largest integer such that D contains an ( , k 1 , k 2 )-fork as a subdigraph. Let F be such a fork. For convenience, we denote its subpaths and vertices by their labels in the above definition.
Suppose to the contrary that both c 1 i1 and c 2 i2 have no out-neighbour in . This contradicts the maximality of i 1 . Henceforth, we assume that c 1 i1 has an out-neighbour a j ∈ A \ {a } for some 0 ≤ j < . The case in which c 2 i2 has an out-neighbour in A \ {a } is similar.
For sake of contradiction, assume that both c 3 i3 and c 4 i4 have no out-neighbour in . This contradicts the maximality of i 3 .
So one of c 3 i3 , c 4 i4 has an out-neighbour in (A \ {a }) ∪ C 1 ∪ C 2 . We assume that it is c 3 i3 ; the case when it is c 4 i4 is similar. If c 3 i3 has an out-neighbour a q ∈ A \ {a } (for some q < ), then the union of either the dipaths (a , B 1 , C 1 , a j , . . . , a q ) and (a , B 2 , C 3 , a q ) (if q ≥ j), or the dipaths (a , B 1 , C 1 , a j ) and (a , B 2 , C 3 , a q , · · · , a j ) (if q < j), is a subdivision of C(k 1 , k 2 ).
If c 3 i3 has an out-neighbour c 1 h ∈ V (C 1 ) for some 1 ≤ h ≤ i 1 , then the union of the dipaths (a , B 1 , c 1 1 , · · · , c 1 h ) and (a , B 2 , C 3 , c 1 h ) is a subdivision of C(k 1 , k 2 ). Similarly, we find a subdivision of C(k 1 , k 2 ) if c 3 i3 has an out-neighbour in C 2 .
Theorem 24 shows that an oriented cycle with two blocks C is maderian and mader δ + (C) ≤ 2|V (C)| − 1. A natural question is to ask whether this upper bound is tight or not.
Consider now a digraph D with δ + (D) ≥ k. Let P be a longest dipath in D and let u be its terminal vertex. Necessarily N + (u) ⊆ V (P ). Let v (resp. w) be the first (resp. last) vertex of N + (u) along P . The path P [v, w] contains all vertices of N + (u), so it has length at least k − 1. Hence the union of (u, v) ∪ P [v, w] and (u, w) is a subdivision of C(k, 1).
A step further towards Conjecture 2 would be to prove that every oriented cycle is δ + -maderian. We even conjecture that for every oriented cycle C mader δ + (C) ≤ 2|V (C)| − 1.
Conjecture 27. Let D be a digraph with δ + (D) ≥ 2k − 1. Then D contains a subdivision of any oriented cycle of order k

Three dipaths between two vertices
A slight adaptation of the proof of Theorem 24 leads to a stronger result. Let k 1 , k 2 , k 3 be positive integers. Let P (k 1 , k 2 ; k 3 ) be the digraph formed by three internally disjoint paths between two vertices x, y, two (x, y)-dipaths, one of size at least k 1 , the other of size at least k 2 , and one (y, x)-dipath of size at least k 3 . When we want to insist on the vertices x and y, we denote it by P xy (k 1 , k 2 ; k 3 ).
If b 2 k2−1 has also an out-neighbour p j ∈ P , then F ∪ C 1 contains a P a k 3 pj (k 1 , k 2 ; k 3 ) if i ≤ j, and a P a k 3 pi (k 1 , So, we may assume that b 2 k2−1 has no out-neighbour in P . Hence, b 2 k2−1 has two distinct outneighbours, say c 3 1 and c 4 1 , not in F ∪ C 1 . Let i 3 ≥ 1 be the largest integer such that there exist two disjoint dipaths C 3 and C 4 in D − (F ∪ C 1 ) with initial vertex c 3 1 and c 4 1 respectively on i 3 and i 4 = min{k 2 − 1, i 3 } vertices. If i 3 ≥ k 1 , then i 4 ≥ k 2 − 1 and thus P ∪ A ∪ B 2 ∪ C 3 ∪ C 4 contains a fork that contradicts the maximality of F . Thus, we may assume that i 3 ≤ k 1 − 2. In particular Suppose to the contrary that both c 3 i3 and c 4 i4 have no out-neighbour in P , where c 3 i3 and c 4 i4 are the last vertices of C 3 and C 4 . Note that |V (A)∪V ( . This contradicts the maximality of i 3 . Therefore, one of c 3 i3 , c 4 i4 has an out-neighbour in p j in P . We assume that it is c 3 i3 ; the case when it is c 4 i4 is similar. We conclude that F ∪ C 1 ∪ C 3 contains a P a k 3 pj (k 1 , k 2 ; k 3 ) if i < j, and a P a k 3 pi (k 1 , k 2 ; k 3 ) if j < i.

Subdivisions in digraphs with large dichromatic number
Recall that a k-dicolouring is a k-partition {V 1 , . . . , V k } of V (D) such that D V i is acyclic for every i ∈ [k], and that the dichromatic number of D is the minimum k such that D admits a k-dicolouring. In this section, we first prove that every digraph is χ-maderian. We need some preliminaries. The first one is an easy lemma, whose proof is left to the reader.
Lemma 29. The dichromatic number of a digraph is the maximum of the dichromatic numbers of its strong components.
Our proof is based on levelling. Forthwith, we introduce the necessary definitions. Given a digraph D, the distance from a vertex x to another y, denoted by dist D (x, y) or simply dist(x, y) when D is clear from the context, is the minimum length of an (x, y)-dipath or +∞ if no such dipath exists. An out-generator in D is a vertex u such that, for every x ∈ V (D), there exists an (u, x)-dipath in D. Analogously, an in-generator in D is a vertex u such that, for every x ∈ V (D), there exists an (x, u)-dipath in D. For simplicity, we call a vertex generator if it is an in-or out-generator. Observe that every vertex in a strong digraph is an in-and out-generator.
Let D be a digraph. Let w, u be in-and out-generators of D, respectively. We remark that w and u are not necessarily different. For every nonnegative integer i, the ith out-level from u in D is the set An out-Breadth-First-Search Tree or out-BFS-tree T + with root u, is a subdigraph of D spanning V (D) such that T + is an oriented tree and, for every v ∈ V (D), dist T + (u, v) = dist D (u, v). Similarly, an in-Breadth-First-Search Tree or in-BFS-tree T − with root w, is a subdigraph of D spanning V (D) such that T − is an oriented tree and, for every It is well-known that if D has an out-generator, then there exists an out-BFS-tree rooted at this vertex. Likewise, if D has an in-generator, then there exists an in-BFS-tree rooted at this generator.
Let T denote an in-or out-BFS-tree rooted at u. For any vertex x of D, there is a single (u, x)-dipath in T if T is an out-BFS-tree, and a single (x, u)-dipath in T if T is an in-BFS-tree. The ancestors or successors of x in T are naturally defined. If y is an ancestor of x, we denote by T [y, x] the (y, x)-dipath in T . If y is a successor of x, we denote by T [x, y] the (x, y)-dipath in T .
Observe that Theorem 32 generalizes the consequence of Theorem 1, stating that every graph with sufficiently large chromatic number contains a subdivision of K k . In fact this statement corresponds to the case of symmetric digraphs of Theorem 32.

Better bounds on mader χ
The bound on mader χ given in Theorem 32 is not optimal. The aim of this subsection is to find better upper bounds.
A digraph is k-χ-critical if χ(D) = k and χ(D ) < k for every proper subdigraph D of D.

Proof.
Let v be a vertex of D. Since D is k-χ-critical, χ(D − v) ≤ k − 1, so D − v admits a (k − 1)dicolouring {V 1 , . . . , V k−1 }. Thus, for each i ∈ [k − 1], D V i ∪ {v} has a directed cycle that contains v. Therefore, v has an in-neighbour and an out-neighbour in each V i .
Corollary 34. mader χ (F ) ≤ mader δ 0 (F ) + 1 for every digraph F . Proof. The proof is identical to the one of Theorem 32, but instead of starting the induction with empty digraphs, we start it with a forest that is the union of spanning trees of the connected components.
Corollary 36 implies that mader χ ( K n ) ≤ 4 n(n−2)+1 (n − 1) + 1. On the other hand, we have mader χ ( K n ) ≥ Ω( n 2 log n ). Indeed, consider a tournament T on p vertices with a subdivision S of K n . For every two distinct vertices u, v of K n , at least one of the arcs (u, v), (v, u) is subdivided in S. Hence, S has at least n + n 2 = n+1 2 vertices, so p ≥ n+1 2 . Erdős and Moser [13] proved that for every integer p, there exists a tournament T p on p vertices with no transitive tournament of order 2 log p + 1. Thus χ(T p ) ≥ p 2 log p . Now set p = n+1 2 − 1. The tournament T p contains no subdivision of K n and χ(T p ) ≥ p 2 log p . Hence mader χ ( K n ) ≥ p 2 log p + 1 ≥ Ω( n 2 log n ). A k-source in a digraph is a vertex x with in-degree 0 and out-degree at most k; a k-sink in a digraph is a vertex x with out-degree 0 and in-degree at most k. A digraph is k-reducible if it can be reduced to the empty digraph by repeated deletion of k-sources or k-sinks. For instance, the 1-reducible digraphs are the oriented forests.
Proof. Suppose that F − x is χ-maderian and mader χ (F − x) = c. We shall prove that F is χ-maderian and mader χ (F ) ≤ 2c − 1. Note that the case c = 1 is trivial so we can assume c ≥ 2.
Let D be a digraph with χ(D) ≥ 2c − 1. By Lemma 29, and since c ≥ 2, we may assume that D is strong. Let u be a vertex in D, and let T be a BFS-tree with root u. By Lemma 30, there is a level such that χ(D L ) ≥ c. Consequently, D L contains a subdivision S of F − x. Let y 1 and y 2 be the vertices in S corresponding to the two out-neighours of x in F . Let v be the least common ancestor of y 1 and y 2 and, for i ∈ {1, 2}, let P i be the (v, y i )-dipath in T . Therefore, we conclude that the digraph S ∪ P 1 ∪ P 2 is a subdivision of F in D.
Corollary 38. The following statements hold.
Proof. Statement (a) follows by induction on |V (F )|. Observe that the result trivially holds when |V (F )| = 2.