On DP-Coloring of Digraphs

DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle and was introduced as an extension of list-colorings of (undirected) graphs. It transforms the problem of finding a list-coloring of a given graph $G$ with a list-assignment $L$ to finding an independent transversal in an auxiliary graph with vertex set $\{(v,c) ~|~ v \in V(G), c \in L(v)\}$. In this paper, we extend the definition of DP-colorings to digraphs using the approach from Neumann-Lara where a coloring of a digraph is a coloring of the vertices such that the digraph does not contain any monochromatic directed cycle. Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number, which extends various results on the (list-)chromatic number of digraphs.

chromatic number of a graph is always at most its maximum degree plus 1 and that the only connected graphs for which equality hold are the complete graphs and the odd cycles was proven by Mohar [14]. As usual, a digraph D is k-critical if χ D k ( ) = but ≤ χ D k ( ′) − 1 for every proper subdigraph D′ of D. Mohar [14] proved the following: Theorem 1 (Mohar [14]). Suppose that D is a k-critical digraph in which each vertex v . Then, one of the following cases occurs: (a) k = 2 and D is a directed cycle of length ≥2.
(b) k = 3 and D is a bidirected cycle of odd length ≥3.
(c) D is a bidirected complete graph.
Moreover, some results regarding the list-chromatic number can also be transferred to digraphs. Given a digraph D, some color set C, and a function → L V D : ( ) 2 C (a so-called listassignment), an L-coloring of D is a function contains no directed cycle for each ∈ c C (if such a coloring exists, we say that D is L-colorable). Harutyunyan and Mohar [11] proved the following, thereby extending a theorem of Erdős et al [8] for undirected graphs. Recall that a block of a digraph is a maximal connected subdigraph that does not contain a separating vertex.
Theorem 2. Let D be a connected digraph, and let L be a list-assignment such that Suppose that D is not L-colorable. Then, D is Eulerian and for every block B of D one of the following cases occurs: for all ∈ v V D ( ) is already sufficient for implying the above statement. That this is not the case was shown by Harutyunyan and Mohar [11]. More precisely, they proved that it is even NPcomplete to decide whether a planar digraph satisfying this condition is L-colorable, or not.
Recently, Dvořák and Postle [6] introduced a new coloring concept, the so-called DP-colorings (they call it correspondence colorings). DP-colorings are an extension of list-colorings, which is based on the fact that the problem of finding an L-coloring of a graph G can be transformed to that of finding an appropriate independent set in an auxiliary graph with vertex set In Section 3, we extend the concept of DP-coloring from graphs to digraphs. In particular, we introduce the DP-chromatic number of a digraph and show that the DP-chromatic number of a bidirected graph is equal to the DP-chromatic number of its underlying graph (see Corollary 4). As the main result of our paper, we provide a characterization of DP-degree colorable digraphs (see Theorems 7 and 15) that generalizes Theorem 2.
vertices of D is called the order of G and is denoted by D | |. Digraphs in this paper may have neither loops nor parallel arcs; however, it is allowed that there are two arcs going in opposite directions between two vertices (in this case we say that the arcs are opposite). We denote by uv the arc whose initial vertex is u and whose terminal vertex is v; u and v are also said to be the end-vertices of the arc uv. Let ⊆ X Y V D , ( ), then A X Y ( , ) D denotes the set of arcs that have their initial vertex in X and their terminal vertex in Y . Two vertices u v , are adjacent if at least one of uv and vu belongs to A D ( ). If u and v are adjacent, we also say that u is a neighbor of v and vice versa. If ∈ uv A D ( ), then we say that v is an out-neighbor of u and u is an in-neighbor of v.
the set of in-neighbors of v. Given a digraph D and a vertex set X , by D X [ ] we denote the subdigraph of D that is induced by the vertex set X , that is, Similarly, the number of arcs whose terminal vertex is v is called the in-degree of v and is denoted by if and only if at least one of uv and vu belongs to A D ( ).
we denote the set of all blocks of D.
where the v i are all distinct. A directed cycle of length 2 is called a digon. If D is a digraph and if C is a cycle in the underlying graph G D ( ), we denote by D C the maximal subdigraph of D satisfying G D C ( ) = C . A bidirected graph is a digraph that can be obtained from an undirected (simple) graph G by replacing each edge by two opposite arcs, we denote it by D G ( ). A bidirected complete graph is also called a complete digraph.

| The DP-chromatic number
Let D be a digraph. A cover of D is a pair X H ( , ) satisfying the following conditions: and each X v is an independent set of H . For each arc ∈ a uv A D = ( ) , the arcs from A X X ( , ) is the smallest integer ≥ k 0 such that D is DP-k-colorable. DP-coloring was originally introduced for undirected graphs by Dvořák and Postle [6] and, independently, by Fraigniaud et al [9]; they call it conflict coloring. Let G be an undirected (simple) graph. A cover of G is a pair X H ( , ) satisfying (C1) and (C2) where the matching M e associated to an edge ∈ e uv E G = ( )is an undirected matching between X u and X v (and H is therefore an undirected graph). An X H ( , )-coloring of G is an independent transversal T of X H ( , ), that is, T is a transversal of X H ( , ) such that H T [ ] is edgeless. The definitions of DP-fcolorable, DP-k-colorable and the DP-chromatic number are analogous.
We now investigate the relation between undirected and directed DP-colorings.
Proof. We prove the two implications separately. First, assume that D is DP-f -colorable.
D is an f -cover of D. By assumption, there is an acyclic transversal T of X H ( , ) D . As H D is bidirected, T is an independent transversal of X H ( , ) G and so G is DP-f -colorable. The converse is less obvious since even if D is bidirected, its covers do not have to be bidirected. Let X H ( , ) D be a cover of a bidirected graph D. We say that the cover is symmetric if and only if for every pair of opposite arcs uv and vu in D, the matchings M uv and M vu are opposite, that is, each arc in M vu is opposite to some arc in M uv . We say that the cover is locally symmetric around a given vertex ∈ v V D ( ) if M uv and M vu are opposite for every vertex u adjacent to v.
Let f be such that D is not DP-f -colorable. We claim that G G D = ( ) is not DP-fcolorable. To prove this, we choose an f -cover X H BANG-JENSEN ET AL.

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Consequently, X H ( , ) D is symmetric and, as a consequence, for An important property of the chromatic number of a digraph is that the chromatic number of a bidirected graph coincides with the chromatic number of its underlying graph. Theorem 3 implies that this property also holds for DP-coloring.
Corollary 4. The DP-chromatic number of a bidirected graph is equal to the DPchromatic number of its underlying graph.
DP-colorings are of special interest because they constitute a generalization of list-colorings: let D be a digraph, let C be a color set, and let . Hence, the list-chromatic number ℓ χ of D, which is the smallest integer k such that D admits an L-coloring for every list- Moreover, by using a sequential coloring algorithm it is easy to verify that . Hence, we obtain the following sequence of inequalities: In the following, we will give a characterization of the non-DP-degree-colorable digraphs as well as a characterization of the edge-minimal corresponding "bad" covers (see Theorem 7). Clearly, it suffices to do this only for connected digraphs. For undirected graphs, those characterizations were given by Kim and Ozeki [13]; for hypergraphs it was done by Schweser [18].
A feasible configuration is a triple D X H ( , , ) consisting of a connected digraph D and a cover X H  (a) For every vertex ∈ v V D ( ) and every vertex ∈ The above proposition leads to the following concept. We say that a feasible configuration As usual, H a − denotes the digraph obtained from H by deleting the arc a. Clearly, it follows from the above Proposition that if D X H ( , , ) is an uncolorable feasible configuration, then there is a spanning subdigraph H′ of H such that D X H ( , , ′) is a minimal uncolorable feasible configuration.
To characterize the class of minimal uncolorable degree-feasible configurations, we first need to introduce three basic types of degree-feasible configurations.
We say that D X H ( , , ) is a K-configuration if D is a complete digraph of order n for some ≥ n 1, and X H ( , ) is a cover of D such that the following conditions hold: An example of a K-configuration with n = 4 is given in Figure 1. It is an easy exercise to check that each K-configuration is a minimal uncolorable degree-feasible configuration. Note that for . Note that in this case, H is a copy of D. Clearly, each C-configuration is a minimal uncolorable degree-feasible configuration.
We say that D X H ( , , ) is an odd BC-configuration if D is a bidirected cycle of odd length ≥5 and X H ( , ) is a cover of D such that the following conditions are fulfilled:

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Note that It is easy to verify that every odd BC-configuration is a minimal uncolorable degree-feasible configuration.
We call D X H ( , , ) an even BC-configuration if D is a bidirected cycle of even length ≥4, X H ( , ) is a cover of D, and there is an arc ∈ uu A D ′ ( ) such that: Again, it is easy to check that every even BC-configuration is a minimal uncolorable degreefeasible configuration. By a BC-configuration we either mean an even or an odd BCconfiguration.
Our aim is, to show that we can construct every minimal uncolorable degree-feasible configuration from the three basic configurations by using the following operation. Let D X H ( , , ) 1 1 2 be two feasible configurations, which are disjoint, that is, . Furthermore, let D be the digraph obtained from D 1 and D 2 by identifying two vertices ∈ v V D ( ) Now we define the class of constructible configurations as the smallest class of feasible configurations that contain each K-configuration, each C-configuration, and each BCconfiguration and that is closed under the merging operation. We say that a digraph is a DPbrick if it is either a complete digraph, a directed cycle, or a bidirected cycle. Thus, if D X H ( , , ) is a constructible configuration, then each block of D is a DP-brick. The next proposition is straightforward and left to the reader.
Our aim is to prove that the class of constructible configurations and the class of minimal uncolorable degree-feasible configurations coincide. This leads to the following theorem. For DP-colorings of undirected graphs, an analogous result was proven by Bernshteyn et al in [3]. However, they only characterized the graphs that are not DP-degree colorable, but not the corresponding bad covers. This was done later by Kim and Ozeki [13]. The third author of this paper extended the characterization of the non-DP-degree colorable graphs to hypergraphs [18] and characterized also the minimal uncolorable degree-feasible configurations; since he used the same terminology as we do and since we need to refer to the undirected version in our proof, we only state the part of his theorem examining simple undirected graphs.
Regarding undirected graphs, a degree-feasible configuration is a triple G X H ( , , ), where G is an undirected (simple) graph and X H ( , ) is a cover of G such that Furthermore, for undirected graphs, the definition of a K-configuration and a BC-configuration can be deduced from the above definition for digraphs by considering the underlying undirected graphs (see Figure 2). Finally, for undirected graphs we define the class of constructible configurations as the smallest class of configurations that contains each K-configuration and each BC-configuration and that is closed under the merging operation. The proof of the following theorem can be found in [18]. In the following, given a feasible configuration D X H ( , , ), we will often fix a vertex ∈ v V D ( ) and regard the feasible configuration D X H For the sake of readability, we will write ∕ X H X H v ( ′, ′) = ( , ) . First, we state some important facts about minimal uncolorable degree-feasible configurations. Recall that the digraph D of a degree-feasible configuration D X H ( , , ) is connected by definition. Proof.
(a) The proof is by induction on the order of D. The statement is clear if D | | = 1 as in this case . By applying the induction hypothesis to D X H By symmetry, we may assume ∈ wv A D ( ). But then, which is impossible. This proves (a). x X v such that no vertex of T is an out-neighbor of x in H . Then, similarly to the proof of (a), we conclude that ∪ T x { } is an acyclic transversal of X H ( , ), a contradiction. Hence, each vertex ∈ x X v has in H at least one out-neighbor belonging to T. Moreover, for each vertex ∈ u N v ( ) The next proposition shows the usefulness of the merging operation.
2 , this implies that at least one of v 1 and v 2 (by symmetry, we can assume it , the set ∪ T T T = 1 2 is an acyclic transversal of X H a ( , − ) and so D X H a ( , , − ) is colorable. Thus, (b) holds.
To prove that (b) implies (a), we first show that D X H ( , 1 is minimal uncolorable.
Assume that D X H ( , 1 is colorable, that is, X H ( , ) 1 1 has an acyclic transversal T 1 . Since D X H ( , , ) is a minimal uncolorable degree-feasible configuration and as H X − v 2 is minimal uncolorable, too.
It remains to show that D X H ( , , ) j j j is degree-feasible for ∈ j {1, 2}. As D X H ( , , ) is an uncolorable degree-feasible configuration, Proposition 9(a) implies that Consequently, each vertex from D v − , and so D j is Eulerian for ∈ j {1, 2}. Moreover, it follows from (1)  To prove Theorem 7, we need some more tools. The first one, which will be frequently used in the following, is the so-called shifting operation. Let D X H ( , , ) be a minimal uncolorable degree-feasible configuration, let D D v ′ = − for some ∈ v V D ( ), and let T be an acyclic transversal of X H X H v ( ′, ′) = ( , )/ (which exists by Proposition 9(b)). Then it follows from Proposition 9(c) that for each vertex ∈ x X v there is exactly one vertex ∈ the vertex x has no out-neighbor (respectively no in-neighbor) and, hence, x cannot be contained in a directed cycle. We say that T′ (respectively T″) evolves from T by shifting the color x′ (respectively x″) to x. Of course, the shifting operation may be applied repeatedly. The next proposition can be easily deduced from Proposition 9 by applying the shifting operation. The statements of the proposition are illustrated in Figure 3.
Proof. Statement (a) is a direct consequence of Proposition 9(c). To prove (b) let ∈ u N v ( ) Furthermore, T must contain both x u and x v as otherwise T would be an acyclic transversal of X H ( , ), a contradiction. Then, Let T* be the transversal that evolves from T′ by shifting x u to x v . Then, x u has an in-neighbor x* from T* in H (by Proposition 11(a)) and ). Moreover, x* is contained in the transversal T that evolves from T′ by shifting x u to x and so , a contradiction. □ In particular, the above proposition implies the following concerning the shifting operation. Let D X H ( , , ) be a minimal uncolorable degree-feasible configuration, let ∈ v V D ( ) and let T be an acyclic transversal of X H X H v ( ′, ′) = ( , )/ (which exists by Proposition 9(b)). Then it follows from the above proposition together with Propositions 11(b) and (c) that for each vertex u that is in D adjacent to v and for the unique vertex Hence, x v is the unique vertex from X v to which we can shift the color x u . Thus, in the following we may regard the shifting operation as an operation in the digraph D rather than in H and write → u v to express that we shift the color from the corresponding vertex x u to x v .
As another consequence of Proposition 12 we easily obtain the following corollary.
Corollary 13. Let D X H ( , , ) be a degree-feasible minimal uncolorable configuration such that D is a bidirected graph. Then H is a bidirected graph, too.
Having all those tools available, we are finally ready to prove our main theorem.

| Proof of Theorem 7
This subsection is devoted to the proof of Theorem 7, which we recall for convenience. and so D X H ( , , ) is a K-configuration. Thus, we may assume that ≥ D | | 2. By Proposition 9(a), We distinguish between two cases. and so D j is Eulerian for As D X H ( , , ) is uncolorable and degree-feasible, both 1  and 2  are non-empty (by Proposition 9(b)). Moreover, for ∈ j {1, 2}, let X j be the set of all vertices of X v * that do not occur in any set from j  . We claim that 1 has an acyclic transversal T. Then T is in j j j is uncolorable and D j is connected, it follows from Proposition 9(a) that . So, D X H ( , , ′) is a degree-feasible configuration that is obtained from two isomorphic copies of D X H ( , , ) In the remaining part of the proof we will show that under the assumption (3), the and The remaining part of the proof is straightforward: As D is a block, G D ( ) contains an induced cycle C. Then, D C is a directed cycle by Claim 6. We claim that D D = C . Otherwise, there would be a vertex ∈ v V D V C ( )\ ( ). Moreover, since D and therefore G D ( ) is a block, there are two internally disjoint paths P and P′ in G D ( ) from v to vertices ≠ w w′ such that ∩ V P V C w ( ) ( ) = { } and ∩ V P V C w ( ′) ( ) = { ′}. Since all cycles of G D ( ) are induced (by Claim 7), w and w′ are not consecutive in C. Let P C and P′ C denote the two internally disjoint paths between w and w′ contained in C. Then, P P , ′ together with P C , respectively P P , ′ together with P′ C form induced cycles C 1 and C 2 of G D ( ). Since D C is a directed cycle, either D C 1 or D C 2 is not a directed cycle, contradicting Claim 6. Hence, D D = C , that is, D is a directed cycle. As D X H ( , , ) is a minimal uncolorable degreefeasible configuration, we easily conclude that D X H ( , , ) is a C-configuration. This completes the proof. □

| CONCLUDING REMARKS
The next two statements are direct consequences of Theorem 7 and Proposition 6. In particular, Theorem 15 is a generalization of Theorem 2. Finally, we deduce a Brooks-type theorem for DP-colorings of digraphs. For undirected graphs, the theorem was proven by Bernshteyn et al [3].