Complete directed minors and chromatic number

The dichromatic number χ→(D) $\overrightarrow{\chi }(D)$ of a digraph D $D$ is the smallest k $k$ for which it admits a k $k$ ‐coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of complete directed minors in digraphs with a given dichromatic number. In this note we exhibit a relation of these problems to Hadwiger's conjecture. Exploiting this relation, we show that every directed graph excluding the complete digraph K↔t ${\overleftrightarrow{K}}_{t}$ of order t $t$ as a strong minor or as a butterfly minor is O(t(log log t)6) $O(t{(\mathrm{log}\unicode{x0200A}\mathrm{log}\unicode{x0200A}t)}^{6})$ ‐colorable. This answers a question by Axenovich, Girão, Snyder, and Weber, who proved an upper bound of t4t $t{4}^{t}$ for the same problem. A further consequence of our results is that every digraph of dichromatic number 22n $22n$ contains a subdivision of every n $n$ ‐vertex subcubic digraph, which makes progress on a set of problems raised by Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé.


| INTRODUCTION
For a given integer ≥ t 1 let m t ( ) χ be the smallest integer for which it is true that every graph with chromatic number at least m t ( ) χ contains a K t -minor. Hadwiger's conjecture [8], which is one of the most important open problems in graph theory, states that m t t ( ) = χ for all ≥ t 1. The conjecture remains unsolved for ≥ t 7. For many years the best general upper bound on m t ( ) χ was due to Kostochka [13,14] and Thomason [27], who independently proved that every graph of average degree at least O t t ( log ) contains a K t -minor, implying that m t O t t ( ) = ( log ) χ . Recently, however, there has been progress. First, Norin, Postle, and Song [22] showed that m t O t t ( ) = ( (log ) ) χ β (for any β > 1 4 ), which was then further improved by Postle [23] to m t O t t ( ) = ( (log log ) ) χ 6 . For more details about Hadwiger's conjecture the interested reader may consult the recent survey by Seymour [26].
This famous conjecture has influenced many researchers and different variations of it have been studied in various frameworks, one of which is directed graphs.
The chromatic number of a digraph was introduced by Neumann-Lara [21] in 1982 as the smallest number of acyclic subsets that cover the vertex set of the digraph. The dichromatic number has received increasing attention since 2000 and has been an extremely active research topic in recent years, we refer to [3,4,9,10] as examples of important results on the topic.
In the case of digraphs there are multiple ways to define a minor. Here we consider three popular variants: strong minors, butterfly minors, and topological minors. The containment of these different minors in dense digraphs as well as their relation to the dichromatic number has already been studied in several previous works, see, for example, [2,12,15] for strong minors, [5,11,16,20] for butterfly minors, and [1,6,7,[17][18][19]25]  They then raised the problem of improving in particular the upper bound and expressed that they think that → sm t ( ) χ should be much closer to the lower than to the upper bound. Here we confirm this belief by improving their upper bound substantially as follows.
Now let us turn to butterfly minors. Given a digraph D and an arc ∈ u v A D ( , ) ( ), this arc is called (butterfly-)contractible if v is the only out-neighbor of u or if u is the only in-neighbor of v in D. Given such a contractible arc e, the digraph ∕ D e is obtained from D by merging u and v into a new vertex and joining their in-and out-neighborhoods, ignoring parallel arcs. A butterfly minor of a digraph D is any digraph that can be obtained by repeatedly deleting arcs, deleting vertices or contracting arcs.
In [20], inspired by Hadwiger's conjecture, Millani, Steiner, and Wiederrecht raised the following question: For a given integer ≥ k 1, what is the largest butterfly minor-closed class  k of k-colorable digraphs? They gave a precise characterization of  2 as the noneven digraphs. The question concerning a characterization of  k for ≥ k 3 is closely related to the question of forcing complete butterfly minors in digraphs. For an integer ≥ t 1, let us define contains ↔ K t as a butterfly minor, and put Let us further denote by  t the class of all digraphs with no ↔ K t as a butterfly minor. Then, on the one hand, every digraph excluding ↔ K b k ( +1) as a butterfly minor is colorable with ≤ → bm b k k ( ( + 1)) − 1 χ colors. On the other hand, every digraph in  k must exclude ↔ K k+1 as a butterfly minor, since its dichromatic number exceeds k. Therefore, for every k we have To see how tight the above inclusions are one needs to obtain good lower bounds on b k ( + 1), or equivalently good upper bounds on In this direction, as an application of Theorem 1 we prove the following corollary. The previously best-known upper bound on → bm t ( ) χ mentioned in [20] was t 4 ( − 1) + 1 t t − 2 and followed from the work of Aboulker et al. [1].
For the sake of completeness we remark that a lower bound of where G is the complete graph on t + 2 vertices with a 5-cycle removed. It is a simple exercise to verify that ↔ χ D t ( ) = and that it contains no butterfly ↔ K t -minor.
Finally, we consider topological minors. Given a digraph H , a subdivision of H is any digraph obtained by replacing every arc ∈ u v A H ( , ) ( ) by a directed path from u to v, such that subdivision paths of different arcs are internally vertex-disjoint. Then H is said to be a topological minor of some digraph D if D contains a subdivision of H as a subgraph.
Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé [1] initiated the study of the existence of various subdivisions in digraphs of large dichromatic number. For a digraph H they introduced the parameter → H mader ( ) χ , the dichromatic Mader number of H , as the smallest integer such that any digraph D with In their main result they proved that if H is a digraph with n vertices and m arcs, then Gishboliner, Steiner, and Szabó [6] conjectured that ↔ ≤ → K Ct mader ( ) χ t 2 for some absolute constant C. However, it seems surprisingly hard to find a polynomial upper bound even for quite simple digraphs H. An indication for this increased difficulty compared with the undirected case could be that for digraphs it is not even possible to force a ↔ K 3 -subdivision by means of large minimum out-and in-degree (compare Mader [17]).
Gishboliner et al. [6] still managed to identify a wide class of graphs, called octus graphs, 1 for which the lower bound above is tight. Their result means that given a digraph it contains the subdivision of every octus graph on at most n vertices. Here, along the same line of thinking, as a corollary of Theorem 1 we prove a similar result for another class of digraphs. By slightly abusing the terminology, we call a digraph D subcubic if D is an orientation of a graph with maximum degree at most three such that the in-and outdegree of any vertex is at most two. 22 then it contains a subdivision of every subcubic digraph on at most n vertices. [ ] is an acyclic digraph. We call D strongly connected if for every ordered pair u v , of vertices in D there is a directed path in D from u to v. An in-/outarborescence is a rooted directed tree where every arc is directed towards/away from the root. For the starting/ending point of an arc we will also use the names tail/head. 1 We note that this class, in particular, includes orientations of cactus graphs (and hence orientations of cycles), as well as bioriented forests.

| Notation
A (proper) coloring of an undirected graph G with colors in a set A is a map is an acyclic set for every ∈ a A. The minimum k for which a k-coloring exists is the chromatic (resp., dichromatic) number of the undirected graph G (resp., digraph D), which we shall denote by χ G ( ) (resp., → χ D ( )).

| Strong minors
The proof of Theorem 1 will be based on the following result.
Theorem 2. For every digraph D there is an undirected graph G such that Proof. To start with, let us first fix a partition X X X , , …, m . Note that the X i 's are well defined since the one vertex-digraph is strongly connected and 2-colorable. Now we define G to be the undirected simple graph with vertex set X X { , …, } .
Assume for contradiction that this is not the case, and there is a directed cycle C in D which is monochromatic. We may, without loss of generality, assume that C is a shortest such cycle, in particular, it is an induced cycle. Let i 0 be the smallest index for which C contains a vertex from X i 0 . Note that, in particular, and, as f D is a proper coloring on D X [ ] i 0 , the cycle C cannot be fully contained in X i 0 . Hence, C contains a subsequence u w w v , , …, , , and ℓ > 0. Let ∈ s {1, …, ℓ} be the smallest index such that w s has an out-neighbor in X i 0 , and denote this out-neighbor by ∈ x X i 0 . Note that s is well defined, since ∈ w v A D ( , ) ( ) ℓ and ∈ v X i 0 . We claim that w s has no in-neighbor in D that is contained in X i 0 . Suppose towards a contradiction that there exists ∈ y X i 0 such that ∈ y w A D ( , ) ( ) s . Let j i > 0 be such that ∈ w X s j . Then, because of the arcs ∈ y w w x A D ( , ), ( , ) ( ) s which contradicts the monochromaticity of C. Hence, we may assume that w s has no in-neighbor contained in X i 0 . In particular, this implies ≥ s 2. Let us now consider the set It is clearly strongly connected, as X i 0 is so and u w w x , , …, , In any case, it is not monochromatic. However, the existence of the set X then contradicts with the maximality of X i 0 , which finishes the proof. □ Now we can easily deduce Theorem 1 from Theorem 2. We would like to remark that the above proof of Theorem 2 actually yields a slightly stronger conclusion: Let the partition X X , …, m 1 of V D ( ) and the graph G be defined as in the proof of Theorem 2. We then claim that for every edge ∈ X X E G ( ) i j with i j < there are at least two arcs in D which go from X i to X j , and at least two arcs which go from X j to X i .
To see this, note that by the definition of G there are edges in both directions spanned between X i and X j , which implies that j is also a strongly connected digraph. Then suppose that contrary to our claim, there would be at most one edge from X i to X j (or at most one edge from X j to X i ) in D. Let ∈ e A D ( ) be such an edge, and note that removing e from ∪ D X X [ ] i j destroys the strong connectivity and creates the two strong components X i and X j of i j intersecting both X i and X j must use the edge e and can therefore not be monochromatic. This however means that ∪ X X i j induces a strongly connected and 2-colorable subgraph of D which properly contains X i and is disjoint from ∪ ⋯∪ X X i 1 −1 . Finally, this contradicts the maximality of X i in our choice of the partition of V D ( ), and proves our above claim. This stronger conclusion can then be used in the proof of Theorem 1 to yield the stronger conclusion that every digraph of dichromatic number at least m t 2 ( ) − 1 χ in fact contains a strong ↔ K t -minor model in which between every pair of branch sets, at least two arcs are spanned in each direction.

| Butterfly minors
Corollary 1 follows directly from Theorem 1 and the following proposition.
an in-arborescence rooted at r i − and T i + is an out-arborescence rooted at r i + . Let us consider the spanning subdigraph D′ of D consisting of the arcs contained in as well as all arcs of D starting in X i + and ending in X j − for ≠ i j. Then every arc of D′ contained in T is either the unique arc in D′ emanating from its tail or the unique arc in D′ entering its head. It follows that all arcs in T are butterfly-contractible. Note that the contraction of an arc does not affect the butterfly-contractibility of other arcs, hence the digraph ∕ D T ′ , obtained from D′ by successively contracting all arcs in T , is a butterfly minor of D.
Proof of Corollary 2. As a first step note that given ∈ n , every undirected graph G with a minimum degree at least ⋅ n n n 10.5 > + 6.291 3 2 contains every n-vertex subcubic graph as a minor. This follows directly from a result of Reed and Wood [24], who proved that every graph with an average degree at least n m + 6.291 contains every graph with n vertices and m edges as a minor.
Let now D be any digraph with → ≥ χ D n ( ) 22 , F a subcubic digraph on ≥ n 2 vertices and H its underlying undirected subcubic graph. By Theorem 2 there exists an undirected graph G such that D is a strong ↔ G -minor model and ≥ χ G n ( ) 11 . In particular, G contains a subgraph of minimum degree at least n n 11 − 1 > 10. 5 This claim holds trivially if d = 0, and if d = 1 then we can simply put b u v ( ) = 1 and let P u 1 be the trivial one-vertex path consisting of v 1 . If d = 2 then, without loss of generality, by the symmetry of reversing all arcs in D and F , we may assume that u is the head of e 1 . We then can put ≔ b u v ( ) 2 , let P u 1 be any directed path in D X [ ] u from v 1 to v 2 , and take P u 2 to be the trivial one-vertex path consisting only of v 2 .
Finally suppose d = 3. Since F is subcubic, u either has in-degree one and out-degree two, or vice versa. As before, without loss of generality, by symmetry we may assume that the first case occurs, and it is e 1 that enters u and e 2 and e 3 that emanate from it. Take now P 12 and P 13 to be directed paths in D X [ ] u starting at v 1 and ending at v 2 and v 3 , respectively. We define now b u ( ) as the first vertex in V P ( ) 12 that we meet when traversing P 13 backwards (starting at v 3 ), P u 1 as the subpath of P 12 directed from v 1 to b u ( ), P u 2 as the subpath of P 12 directed from b u ( ) to v 2 , and P u 3 as the subpath of P 13 directed from b u ( ) to v 3 . It follows by definition that P P P , , 1 2 3 are internally vertex-disjoint, and hence the claim follows.
To finish the proof, let ⊆ S D be a subdigraph with vertex set S is a digraph isomorphic to a subdivision of F in which a vertex ∈ u V F ( ) is represented by the branch-vertex b u ( ). This concludes the proof. □