On locally irregular decompositions and the 1-2 Conjecture in digraphs

The 1-2 Conjecture raised by Przybyło and Wo´zniak in 2010 asserts that every undirected graph admits a 2 -total- weighting such that the sums of weights “incident” to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally irregular decompositions) to digraphs, we here introduce and study several variants of the 1-2 Conjecture for digraphs. For every such variant, we raise conjectures concerning the number of weights necessary to obtain a desired total-weighting in any digraph. We verify some of these conjectures, while we obtain close results towards the ones that are still open.


Introduction
An edge-weighting w : E(G) → N * of an undirected graph G is called sum-colouring if the sums of weights "incident" to the vertices yield a proper vertex-colouring of G.More precisely, for each vertex v of G one can compute σ e (v) := w(vu) and we require σ e to be a proper vertex-colouring.The smallest k ≥ 1 such that G admits a sum-colouring k-edge-weighting (if any) is denoted χ e σ (G). (i) The 1-2-3 Conjecture, posed by Karonski, Łuczak and Thomason (9) in 2004, reads as follows (where an isolated edge refers to a connected component isomorphic to K 2 ).
Conjecture 1.1 (Karonski, Łuczak, Thomason (9)).For every graph G with no isolated edge, we have χ e σ (G) ≤ 3. Since its introduction, the 1-2-3 Conjecture has been attracting growing attention, resulting in many research works considering either the conjecture itself or variants of it.As a best result towards it, it was proved by Kalkowski, Karoński and Pfender that χ e σ (G) ≤ 5 holds for every graph G with no isolated edge (8).For more information, we refer the interested reader to (14) for a survey by Seamone on this wide topic.
In this paper, we mainly focus on two notions related to the 1-2-3 Conjecture.The first one is the total version of the 1-2-3 Conjecture, called the 1-2 Conjecture, which was introduced by Przybyło and Woźniak in (12).Quite similarly as in the context of weighting edges only, we say that a total-weighting w : V (G) ∪ E(G) → N * of G is sum-colouring if the vertex-colouring σ t defined as for every vertex v is a proper vertex-colouring.We then denote by χ t σ (G) the least k ≥ 1 such that G admits a sum-colouring k-total-weighing.It is believed that being granted the possibility to "locally" modify the sums of weights incident to the vertices (by altering the vertex weights) should, compared to the original edge version, reduce the number of needed weights.Conjecture 1.2 (Przybyło,Woźniak (12)).For every graph G, we have χ t σ (G) ≤ 2. The 1-2 Conjecture is known to hold for several families of graphs, such as 3-colourable graphs, complete graphs, and 4-regular graphs (12).As for upper bounds on χ t σ , the best known one is due to Kalkowski (7), who proved that χ t σ (G) ≤ 3 holds for every graph G. Actually, Kalkowski even proved that stronger sumcolouring 3-total-weightings, assigning weights in {1, 2} to the vertices and in {1, 2, 3} to the edges, exist for all graphs.
It is worthwhile mentioning that both the 1-2-3 Conjecture and the 1-2 Conjecture were considered in the more general list context.In that context, instead of weighting all the graph elements (edges and possibly vertices) with values from the same list ({1, 2, 3} for the 1-2-3 Conjecture, {1, 2} for the 1-2 Conjecture), each element now has a private list from which a weight must be chosen.Given a list assignment to the elements we want to weight, the goal is then to design a sum-colouring weighting where the weights are picked from the assigned lists.The list version of the 1-2-3 Conjecture, posed by Bartnicki, Grytczuk and Niwcyk (1), asserts that such sum-colouring edge-weightings can be constructed for every assignment of lists of size 3 to the edges.The list version of the 1-2 Conjecture, posed by Przybyło and Woźniak (13), asserts that such sum-colouring total-weightings can be constructed for every assignment of lists of size 2 to the vertices and edges.Again, we refer the interested reader to (14) for more details on these two list variants.A remarkable result we should mention, however, is one due to Wong and Zhu (15), who proved that sum-colouring total-weightings can be constructed for every assignment of lists of size 3 to the edges and lists of size 2 to the vertices.In contrast, it is still not known whether there is a constant k ≥ 3 such that sum-colouring edge-weightings can be constructed for every assignment of lists of size k to the edges.
The second notion considered in this paper is the one of locally irregular decompositions.We say that a graph G is locally irregular if every two of its adjacent vertices have distinct degrees.A locally irregular edge-colouring of G is then an edge-colouring where each colour induces a locally irregular subgraph.We denote by χ e irr (G) the least number of colours in a locally irregular edge-colouring of G (if any).Intuitively, the parameter χ e irr can be seen as a measure of how "far" from (locally) irregular a graph is.This parameter was introduced and studied by the current authors in (2) mainly because of its link to the 1-2-3 Conjecture and some of its variants.In particular, let us mention that in very particular settings, such as when dealing with regular graphs and only two colours, finding a sum-colouring edge-weighting is equivalent to finding a locally irregular edge-colouring.
Since their introduction, locally irregular edge-colourings gave birth to several investigations.An important first result to have in hand is the exhaustive list of those exceptional graphs for which the parameter χ e irr is not defined at all.In (2), it was proved that these exceptional graphs are exactly those in the family P ∪ C ∪ T , where P is the class of all odd-length paths, C is the class of all odd-length cycles, and T is a peculiar class of graphs obtained by joining, via disjoint paths with particular lengths, disjoint triangles in a tree-like fashion (refer to (2) for the exact definition).For graphs not in P ∪ C ∪ T , hence when χ e irr is defined, the following conjecture was raised.
Conjecture 1.3 (Baudon, Bensmail, Przybyło, Woźniak (2)).For every graph G not in P ∪ C ∪ T , we have χ e irr (G) ≤ 3. Conjecture 1.3 has been mainly verified for several families of graphs, including regular graphs of large degree (2) and graphs of large minimum degree (11).It should be noted that it is NP-complete in general to compute the exact value of χ e irr (G) for a given graph G, as shown in (3) by Baudon, Bensmail and Sopena.In a recent work (5), Bensmail, Merker and Thomassen provided the first constant upper bound on χ e irr , showing that χ e irr (G) ≤ 328 holds for every graph G admitting locally irregular decompositions.Later on (10), Lužar, Przybyło and Soták proved that χ e irr (G) ≤ 220 always holds.This paper is mainly inspired by two papers, namely (4) (authored by Baudon, Bensmail and Sopena) and (6) (authored by Bensmail and Renault), which brought Conjectures 1.1 and 1.3 in the context of digraphs in the particular setting where all notions of "incident weights" and "locally irregular graphs" are with respect to the outdegree parameter.So that we avoid any confusion, we omit the formal definitions and statements here and will rather recall them in the corresponding upcoming sections.Let us just mention that the directed version of Conjecture 1.1 from (4) was completely verified in the same paper, while, towards the directed version of Conjecture 1.3 from (6), only partial results, proved in that same paper, are known to date.To the best of our knowledge, nothing was known for total variants of these problems.
Section 2 is dedicated to sum-colouring edge-weightings and total-weightings in digraphs, while Section 3 is devoted to irregular decompositions in digraphs.The three series of results from these sections are comparable, and should hence be regarded in parallel.We start, in Section 2, by filling in the space showing that the directed version of the 1-2 Conjecture in the setting of ( 4) is false in a strong sense, and introduce a holding variant.In Section 3, we start by improving the main result of ( 6), going one step closer to the main conjecture in that paper.We then investigate two total versions of the same problem inspired by the 1-2 Conjecture.For these two versions, we provide bounds which are close to what we conjecture to be optimal.Some conclusions are gathered in Section 4.
Notation and terminology: Throughout this paper, we focus on simple digraphs, i.e. loopless digraphs with no two arcs directed in the same direction between any pair of distinct vertices.Note that this definition allows our digraphs to have digons, i.e. directed cycles of length 2. Any arc (u, v) of a digraph D will be denoted − → uv to lighten the notation and make the arc's direction apparent.The outdegree (resp.indegree) of outgoing (resp.incoming) incident arcs.In case no ambiguity is possible, the subscript in this notation will be freely omitted.The maximum outdegree (resp.maximum indegree) of D, denoted ∆ + (D) (resp.∆ − (D)) refers to the maximum outdegree (resp.indegree) over the vertices of D.

Sum-colouring arc-and total-weightings in digraphs
In this section, we investigate how the results from ( 4), which concern a directed version of Conjecture 1.1, can be extended to the total context (i.e. when also the vertices are weighted).We start in Section 2.1 by recalling the investigations from ( 4).Then we consider, in Sections 2.2 and 2.3, two directed analogues of the 1-2 Conjecture derived from the problem considered in that paper.The first such variant is shown to be false, even in a strong sense, while the second one is shown to hold.

Outsum-colouring arc-weightings
Let D be a digraph, and w an arc-weighting of D. From w, one can compute, for every vertex v, the sum (outsum) σ e + (v) of "outgoing weights", formally defined as In case σ e + is a proper vertex-colouring, we call w outsum-colouring.The least number k ≥ 1 of weights needed to obtain an outsum-colouring k-arc-weighting of D is denoted χ e σ+ (D).Using a very simple argument, Baudon, Bensmail and Sopena showed in (4) that the tightest upper bound on χ e σ+ is 3, which cannot be improved as deciding whether χ e σ+ (D) ≤ 2 holds for a given digraph D is NP-complete in general.Since this upper bound will be of some use in the upcoming sections, we state it here.

Outsum-colouring total-weightings
We now consider the natural directed variant of the 1-2 Conjecture, where the terminology we use is inspired by that introduced in Section 2.1.Assume w is a total-weighting of a digraph D. To every vertex v, we associate the colour σ t + (v), where We say that w is outsum-colouring if σ t + is a proper vertex-colouring.Again, the least number k ≥ 1 of weights needed to deduce an outsum-colouring k-total-weighting of D is denoted χ t σ+ (D).Due to Theorem 2.1, clearly we have χ t σ+ (D) ≤ 3 for every digraph D (start from an outsum-colouring χ e σ+ (D)-arc-weighting, and put weight 1 on all vertices).As a straight directed analogue of the 1-2 Conjecture, one could naturally wonder about the following question.Unfortunately, easy counterexamples to Question 2.2 can be exhibited, showing that 3 is actually the best general upper bound on χ t σ+ .It can even be proved that Question 2.2 is far from being true, in the sense that there exists no constant k ≥ 3 such that every digraph admits an outsum-colouring (k, 2)-total-weighting, i.e. an outsum-colouring total-weighting using weights among {1, ..., k} on the vertices and among {1, 2} on the arcs.Proposition 2.3.For every k ≥ 1, there exist digraphs admitting no outsum-colouring (k, 2)-total-weighting.
Proof: Given k, choose any odd integer n ≥ 5 such that k < n 2 , and let

− →
T n be the tournament on n vertices defined as follows.Denote 0, 1, ..., n − 1 the vertices of − → T n , and, for every vertex i of From this, we deduce that there has to be at least two vertices of − → T n having the same outsums by w.Since − → T n is a tournament, this implies that w is not outsum-colouring.
Due to Proposition 2.3, digraphs may not admit outsum-colouring (k, 2)-total-weightings with k being any fixed constant.Hence, to design outsum-colouring (k, 2)-total-weightings, in general we should rather consider values of k depending on the given digraph.This is illustrated in the following result, which is actually tight.Proposition 2.4.Every digraph D admits an outsum-colouring (∆ + (D) + 1, 2)-total-weighting.Furthermore, there exist digraphs for which we cannot decrease the number of vertex weights.
Proof: Let w be an outsum-colouring 3-arc-weighting of D. Such exists according to Theorem 2.1.Now for every vertex v ∈ V (D), define the number of arcs outgoing from v weighted 3 by w.Clearly, we have n 3 (v) ≤ ∆ + (D).Now consider the (∆ + (D) + 1, 2)-total-weighting w of D defined as By the way w is defined, the value σ t + (v) induced by w is exactly 1 + σ e + (v), for the value of σ e + (v) induced by w.Since w is outsum-colouring, then w is also outsum-colouring.
To conclude the proof, we just note that the construction from the proof of Proposition 2.3 confirms the last part of the statement, as every considered tournament We end this section by mentioning that Proposition 2.3 remains true even if one requires the adjacent "incident outmultisets" (rather than the "incident outsums") to be different.This will justify our investigations in Section 3.2.Let w be an arc-weighting (resp.total-weighting) of a digraph D. Here, by the outmultiset of any vertex v of D, we mean the multiset { {w(a 1 ), ..., w(a d + (v) )} } (resp.{ {w(a 1 ), ..., w(a d + (v) ), w(v)} } of weights "outgoing" from v, where a 1 , ..., a d + (v) denote the arcs outgoing from v. Note that, for any two adjacent vertices u, v, the outmultisets of u and v differ whenever their outsums differ (but the contrary is not always true).Hence, distinguishing vertices via their outmultisets is easier than distinguishing vertices via their outsums.

Pair-colouring total-weightings
As pointed out in the previous section, the directed analogue of the 1-2 Conjecture in the setting of ( 4) is false in a strong sense (recall Proposition 2.3 and Remark 2.5).In this section, we show that by modifying the aggregate to be distinguished on the adjacent vertices, we get another directed variant of the 1-2 Conjecture, which, here, is verified.
Let w be a total-weighting of a digraph D. From w, we compute at every vertex v the value In case the vertex-colouring ρ t + is proper, we call w pair-colouring.The least number k ≥ 1 of weights needed to obtain a pair-colouring k-total-weighting of D is denoted χ t ρ+ (D).We now prove the analogue of the 1-2 Conjecture for pair-colouring total-weighting.which again exists according to the same argument as above.
We construct a pair-colouring 2-total-weighting w of D by considering the vertices v 1 , ..., v n from "left" to "right", i.e. in increasing order of their indexes.Assume v 1 , ..., v i−1 have already been correctly treated, i.e. ρ t + (v 1 ), ..., ρ t + (v i−1 ) are defined (these vertices and their outgoing arcs have been each assigned a weight) and ρ t + (v j ) = ρ t + (v j ) for every j, j ∈ {1, ..., i − 1} such that v j and v j are adjacent.Let D i := D − {v i+1 , ..., v n }.We now assign a weight to v i and its outgoing arcs by w in such a way that no conflict arises.An important thing to keep in mind is that weighting an arc − − → v i v j does not alter be the number of already treated adjacent vertices which have been assigned weight α.There has to be a value of α ∈ {1, 2} for which Let us assume α = 1 in what follows.Set w(v i ) = 1.Then v i is already distinguished from all its already treated adjacent vertices which received weight 2 by w.Now what remains to do is to weight the arcs outgoing from v i in D so that v i is distinguished by its outsum from all its already treated adjacent vertices which received weight 1 by w (refer to Figure 2 for an illustration).The possible outsums for v i in D by w are those among forming a set with cardinality d + D (v i ) + 1.But the outsum of v i has to be different from the outsums of its n 1 previously treated adjacent vertices which also received weight 1 by w.By the ordering of the vertices of D, we have There is thus at least one value s among S which does not appear as the outsum by w of any vertex v j with j < i neighbouring v i which received weight 1.Then just weight the arcs outgoing from v i so that the outsum of v i is s.Now v i also gets distinguished from its previously considered neighbours weighted 1.
By repeating the above procedure until v 1 is treated, we eventually get the claimed pair-colouring 2-totalweighting w, concluding the proof.

Irregular arc-and total-decompositions in digraphs
We now focus on irregular decompositions of digraphs, where the notion of irregularity is with respect to the one introduced in (6) by Bensmail and Renault.We start by recalling the needed terminology and notation in Section 3.1.In the same section, we then improve the main result from ( 6) by showing that every digraph is decomposable into at most five locally irregular digraphs (while six is proved there).Total counterparts for irregular decompositions of the notions we have introduced in Sections 2.2 and 2.3, are then studied in Sections 3.2 and 3.3.

Locally irregular arc-colourings
A digraph D is called locally irregular if its adjacent vertices have distinct outdegrees.An arc-colouring of D is called locally irregular if its every colour class induces a locally irregular subdigraph.The smallest number of colours in a locally irregular arc-colouring of D is denoted by χ e irr+ (D).The main conjecture stated in (6) is the following.Conjecture 3.1 (Bensmail,Renault (6)).For every digraph D, we have χ e irr+ (D) ≤ 3. It is important to mention, as pointed out in (6), that the upper bound of 3 in Conjecture 3.1 cannot be reduced to 2 in general.For an easy illustration, consider the directed cycle − → C 3 of length 3 (i.e. the digraph with vertex set {v 0 , v 1 , v 2 } and arc set It turns out that determining whether χ e irr+ (D) ≤ 2 holds for a given digraph D is NP-complete in general (6); hence, many more digraphs needing three colours exist.To date, we are not aware of any digraph D verifying χ e irr+ (D) > 3. The originators of Conjecture 3.1 proved its following weakening.Theorem 3.2 (Bensmail,Renault (6)).For every digraph D, we have χ e irr+ (D) ≤ 6.The proof of Theorem 3.2 consists in first arc-decomposing any D into two acyclic digraphs, i.e. two digraphs with no directed cycles.The claimed upper bound then follows by showing that Conjecture 3.1 holds for acyclic digraphs (as first proved in ( 6)).We formally state this result as it will be used in some of our upcoming proofs.Lemma 3.3 (Bensmail,Renault (6)).For every acyclic digraph D, we have χ e irr+ (D) ≤ 3. The NP-completeness result mentioned earlier actually also holds for acyclic digraphs (6).Namely, deciding whether χ e irr+ (D) ≤ 2 is an NP-complete problem, even when restricted to acyclic digraphs D. This implies that there exist infinitely many acyclic digraphs that cannot be arc-coloured with only two colours, hence that the bound in Lemma 3.3 cannot be improved in general.
Our improvement on the bound in Theorem 3.2 from 6 down to 5, consists in showing that every digraph admits an arc-decomposition into one acyclic digraph one degree-decreasing acyclic digraph, which we define as an acyclic digraph admitting an ordering v 1 , ..., v n of its vertices such that 1. all arcs are directed "to the right" (i.e. for every two adjacent vertices v i , v j with i < j, the arc is Since acyclic digraphs D verify χ e irr+ (D) ≤ 3 (according to Lemma 3.3) and degree-decreasing acyclic digraphs D verify χ e irr+ (D) ≤ 2 (which we show in Lemma 3.6 below), our result follows.Theorem 3.4.For every digraph D, we have χ e irr+ (D) ≤ 5. Towards Theorem 3.4, as a first step we start by pointing out that every digraph indeed admits an arcdecomposition into one acyclic digraph and one degree-decreasing acyclic digraph.Proof: Consider the following ordering v 1 , ..., v n over the vertices of D. Start with v 1 being one vertex of D with largest outdegree (if there are several choices as v 1 , pick any of them).Now remove v 1 from D and choose v 2 to be one vertex of D − {v 1 } with the largest outdegree.Then remove v 2 from D − {v 1 } and continue the procedure until all vertices are labelled.Basically, if we just read the vertices from "left" (i.e.v 1 ) to "right" (i.e.v n ) we get that for every two vertices v i and v j with i < j, vertex v i has at least as many outneighbours as v j towards the right.Now let A 2 be the subset of arcs of D containing all arcs of the form − − → v i v j with i < j (i.e. the arcs going to the right).Clearly D[A 2 ] cannot have a directed cycle.Besides, due to the ordering of the vertices, D[A 2 ] is degree-decreasing.Now let A 1 be the subset of the remaining arcs, i.e. those going to the left.For the same reason as previously, D[A 1 ] is acyclic (but clearly it may not be degree-decreasing).Then A 1 and A 2 yield the desired arc-decomposition.
We now prove the second ingredient of our proof of Theorem 3.4, namely that degree-decreasing acyclic digraphs D verify χ e irr+ (D) ≤ 2. The proof is algorithmic and the result is also of interest as, as mentioned earlier, there are acyclic digraphs D verifying χ e irr+ (D) = 3.So our result provides a new class of acyclic digraphs for which two colours are sufficient.Lemma 3.6.For every degree-decreasing acyclic digraph D, we have χ irr (D) ≤ 2.
Proof: We prove the claim by induction on the order n of D. As it can easily be proved by hand for small values of n, we proceed to the induction step.Let v 1 , ..., v n be an ordering of V (D) such that all arcs are directed to the right (i.e. for every two adjacent vertices v i , v j with i < j, the arc joining them is − − → v i v j ), and verifying d + (v 1 ) ≥ ... ≥ d + (v n ).Set v := v 1 , and note that D − {v} is a degree-decreasing acyclic digraph, which hence admits a locally irregular 2-arc-colouring, say with colour red and blue, according to the induction hypothesis.Our goal is to extend this arc-colouring to the arcs outgoing from v, without creating any outsum conflict.
Let u 1 , ..., u k denote the outneighbours of v in D; recall that d + (v) ≥ d + (u i ) for every i.Note that colouring any arc outgoing from v with red or blue does not affect the red and blue outdegrees of the u i 's.Thus, when extending the colouring to the arcs outgoing from v, we just have to make sure that its red Figure 3: A locally irregular 2-total-colouring of a digraph with colours red and blue (left), the resulting red total subdigraph (middle), and the resulting blue total subdigraph (right).In the middle and right pictures, vertices filled in white are hollow while the other ones are solid.outdegree does not meet that of any of the u i 's, and similarly for its blue outdegree.In the following, we say that one of the u i 's is a (d 1 , d 2 )-vertex if it has red outdegree d 1 and blue outdegree d 2 .Now consider the following procedure for colouring the arcs outgoing from v. As Step 1, we start by colouring all arcs outgoing from v red.If this extension of the arc-colouring is not locally irregular (otherwise we are done), then it means that at least one outneighbour of v, say u 1 , has red outdegree k.Actually, by the ordering of the vertices of D, we even get that u 1 is a (k, 0)-vertex.As Step 2, we then colour the arc −→ vu 1 blue, and all other arcs outgoing from v red.If this does not result in a locally irregular 2-arc-colouring, then, since u 1 has blue outdegree 0, this means that v has at least one outneighbour, say u 2 , which is a (k − 1, ≤ 1)-vertex different from u 1 .As Step 3, we then colour the arcs −→ vu 1 and −→ vu 2 blue, and all other arcs red.This time, if an outsum conflict arises, then this means that at least one third outneighbour of v, say, u 3 , different from u 1 and u 2 , is a (k − 2, ≤ 2)-vertex.
We carry on this procedure step by step: at Step i, we colour the arcs −→ vu 1 , ..., − −− → vu i−1 blue, and the remaining arcs outgoing from v red.This cannot cause any blue outsum conflict because the vertices u 1 , ..., u i−1 where revealed, in the previous steps, to have blue outdegree strictly less than i − 1.So if a conflict arises, this is because at least one new outneighbour of v, denoted u i , different from u 1 , ..., u i−1 , is revealed to have red outsum k − i + 1.More precisely, u i is revealed to be a (k − i + 1, ≤ i − 1)-vertex.For these reasons, once Step k is achieved, if no locally irregular 2-arc-colouring has been obtained at any point, then it means that the u i s are precisely (k, ≤ 0)-, (k − 1, ≤ 1)-, (k − 2, ≤ 2)-, ..., (1, ≤ k − 1)-vertices.Then colouring all arcs outgoing from v blue results in a correct colouring.

Locally irregular total-colourings
Let us now discuss a total counterpart of locally irregular decompositions, which is inspired from (2), where the authors introduced similar concepts for undirected graphs, as an attempt for binding the 1-2 Conjecture and locally irregular decompositions (the latter concept being originally motivated by applications to the 1-2-3 Conjecture).
By a total digraph, we mean a triplet D t := (V 0 , V 1 ; A) with vertex set V 0 ∪ V 1 (V 0 ∩ V 1 = ∅) and arc set A. The vertices in V 0 are called hollow, while those in V 1 are said solid.The main difference between total digraphs and usual digraphs lies in the definition of vertex degrees.Namely, for a total digraph D t , the total outdegree (resp.indegree) of a vertex v, denoted d + t (v) (resp.d − t (v)), is understood as only the number of arcs outgoing from (resp.incoming to) v if v is a hollow vertex, or this quantity plus 1 if v is solid.In other words, solid vertices have their indegrees and outdegrees being naturally altered by 1.
We call a total digraph D t := (V 0 , V 1 ; A) locally irregular if d + t (u) = d + t (v) for every arc − → uv ∈ A. A locally irregular k-total-colouring of a usual digraph D is then a total-colouring c of D with k colours such that each colour class induces a locally irregular total subdigraph (where for any given colour i, the vertices of D coloured with i define V 1 for the corresponding total digraph, and the other vertices of D yield V 0 ).The least number of colours needed to colour D in this way is denoted by χ t irr+ (D).These concepts are illustrated in Figure 3

Figure 2 :
Figure 2: Illustration of the proof of Theorem 2.6.

Lemma 3 . 5 .
Every digraph D admits an arc-decomposition into one acyclic digraph and one degreedecreasing acyclic digraph.