Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of P 4 ∗

An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph D is H -free if D does not contain H as an induced sub(di)graph. The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest F , there is some function f such that every F -free graph G with clique number ω ( G ) has chromatic number at most f ( ω ( G )) . Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph D is the minimum number of colors required to color the vertex set of D so that no directed cycle in D is monochromatic. Aboulker, Charbit, and Naserasr’s −→ χ -boundedness conjecture states that for every oriented forest F , there is some function f such that every F -free oriented graph D has dichromatic number at most f ( ω ( D )) , where ω ( D ) is the size of a maximum clique in the graph underlying D . In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr’s −→ χ -boundedness conjecture by showing that it holds when F is any orientation of a path on four vertices.


Introduction
In a simple graph, the size of a maximum clique gives a lower bound on its chromatic number.But if a graph contains no large cliques, does it necessarily have small chromatic number?This question has been answered in the negative.In the mid-twentieth century, Mycielski [15] and Zykov [21] gave constructions for triangle-free graphs with arbitrarily large chromatic number.Hence we may ask the following question instead: Given some xed graph H, do graphs with a bounded clique number that do not contain H as an induced subgraph have bounded chromatic number?In 1959, Erdős showed that there exist graphs with arbitrarily high girth and arbitrarily high chromatic number [6].Hence, the answer to the previous question is "no" whenever H contains a cycle, and thus we need only consider the question when H is a forest.Around the 1980s, Gyárfás and Sumner independently conjectured [8,20] that for any forest H, all graphs with bounded clique number and no induced copy of H have bounded chromatic number.The conjecture has been proven for some speci c classes of forests but remains largely open; see [18] for a survey of related results.This paper concerns an extension of the Gyárfás-Sumner conjecture to directed graphs proposed by Aboulker, Charbit, and Naserasr [2].We will state the Gyárfás-Sumner conjecture and its extension to directed graphs more formally after introducing some necessary terminology.
A directed graph, or digraph, is a pair D = (V, E) where V is the vertex set and E is a set of ordered pairs of vertices in V called the arc set.We call a digraph oriented if it has no digon (directed cycle of length two).This paper will focus on nite, simple, oriented graphs.
For a digraph D = (V, E) we de ne the underlying graph of D to be the graph D * = (V, E * ) where E * is the set obtained from E by replacing each arc e ∈ E by an undirected edge between the same two vertices.We say two vertices in D are adjacent or neighbors if they are adjacent in D * .If (v, w) is an arc of D we say say that v is an in-neighbor of w and that w is an out-neighbor of v.We denote the set of neighbors of a vertex v ∈ V (D) by N (v) and we denote N (v) ∪ {v} by N [v].For a set of vertices S ⊆ V (D) we let N (S) and N [S] denote the sets ∪ v∈S N (v) \ S and ∪ v∈S N [v].We call N (S) the neighborhood of S and N [S] the closed neighborhood of S. For a subdigraph H ⊆ D we let N (H) denote the set N (V (H)).
We let P t denote the path on t vertices.We say an oriented path is a directed path if its vertices are p 1 , p 2 , . . ., p t , in order, and its orientation is p 1 → p 2 → • • • → p t .We let − → P t denote the directed path on t vertices.We say a digraph D is strongly connected if for every v, w ∈ V (D) there is a directed path starting at v and ending at w.An induced subdigraph H of a digraph D is a strongly connected component of D if it is strongly connected and every induced subgraph H of D such that H ⊆ H is not strongly connected.We call a strongly connected H component a source (sink) component of D if every arc between V (H) and V (D \ H) begins (ends) in V (H).
We call a digraph whose underlying graph is a clique a tournament.As we consider only oriented digraphs, this de nition corresponds to the standard de nition of a tournament in the literature.Given a (di)graph G and S ⊆ V , we denote the sub(di)graph of G induced by S as G [S].We say that a (di)graph G contains a (di)graph H if G contains H as an induced sub(di)graph.If G does not contain a (di)graph H we say that G is H-free.If G does not contain any of the (di)graphs H 1 , H 2 , . . ., H k we say G is (H 1 , H 2 , . . ., H k )-free.The clique number and the chromatic number of a digraph are the chromatic number and clique number of its underlying graph, respectively.We denote the clique number and the chromatic number of a (di)graph G by ω(G) and χ(G), respectively.We say that a graph H is χ-bounding if there exists a function f with the property that every H-free graph G satis es χ(G) ≤ f (ω(G)).In this language, [6] implies all χ-bounding graphs are forests.We are now ready to state the Gyárfás-Sumner conjecture more formally.
Today, the conjecture is only known to hold for restricted classes of forests.For example, Gyárfás showed that it holds for paths [9] via a short and elegant proof.Subsequently, the conjecture was proven for other classes of forests.For example, the following classes of trees have been proven to be χ-bounding: • Trees of radius two by Kierstead and Penrice in 1994 [12], • Trees that can be obtained from a tree of radius two by subdividing every edge incident to the root exactly once by Kierstead and Zhu in 2004 [14], and • Trees that can be obtained from a tree of radius two by subdividing some of the edges incident to the root exactly once by Scott and Seymour in 2020 [17].
Note that the class of trees described by the third bullet contains the classes described in both the rst and second bullet.See the survey of Scott and Seymour [18] for an overview of the state of the conjecture from 2020.How can the Gyárfás-Sumner conjecture be adapted to the directed setting?A rst idea is to call an oriented graph H χ-bounding if there exists a function f with the property that every H-free oriented graph D satis es χ(D) ≤ f (ω(D)).Then, once again, by [6], all χbounding oriented graphs are oriented forests.Note that if an oriented graph H is χ-bounding, its underlying graph H * is also χ-bounding.However, the converse does not hold, as, for instance, P 4 is χ-bounding, but there exist orientations of P 4 that are not χ-bounding.There are four di erent orientations of P 4 , up to reversing the order of the vertices on the whole path: →→→, →←→, →←←, ←←→ Only the last two oriented graphs in the list are χ-bounding: • Recall, we denote the oriented P 4 with orientation →→→ by − → P 4 .In 1991, Kierstead and Trotter [13], showed that − → P 4 is not χ-bounding.Their construction was inspired by Zykov's construction of triangle-free graphs with a high chromatic number [21], and builds − → P 4 -free oriented graphs with arbitrarily large chromatic number and no clique of size three.
• Around 1990, Gyárfás pointed out that ←→← is not χ-bounding, as witnessed by an orientation of the shift graphs on pairs [10].We will denote the P 4 with orientation ←→← by − → A 4 .
• Chudnovsky, Scott and Seymour [5] showed that →←← and ←←→ are both χ-bounding in 2019.In the same article, the authors show that orientations of stars are also χ-bounding (stars are the class of complete bipartite graphs K 1,t for any t ≥ 1).We will denote →←← and ←←→ by − → Q 4 and − → Q 4 , respectively.
Our rst attempt at adapting the Gyárfás-Sumner conjecture to oriented graphs failed for oriented paths such as − → P 4 and − → A 4 .Hence, we focus on a di erent approach proposed by Aboulker, Charbit, and Naserasr [2] which uses a concept called "dichromatic number".Directed coloring, or dicoloring, is a weakening of coloring de ned on digraphs and was proposed by Neumann-Lara and subsequently developed by Erdős and Neumann-Lara [7,16].A dicoloring of a digraph D is a partition of V (D) into classes, or colors, such that each class induces an acyclic digraph (that is, there is no monochromatic directed cycle).The dichromatic number of D, denoted as − → χ (D), is the minimum number of colors needed for a dicoloring of D. Notice that every coloring of a directed graph D is also a dicoloring, thus − → χ (D) ≤ χ(D).
We say a class of digraphs D is − → χ -bounded if there exists a function f such that every D ∈ D and we call such an f a − → χ -binding function for D. We say that a digraph H is − → χ -bounding if the class of H-free oriented graphs is − → χ -bounded.
We can now state Aboulker, Charbit, and Naserasr's dichromatic analogue to the Gyárfás-Sumner conjecture for digraphs.For brevity, we will call this conjecture the "ACN − → χboundedness" conjecture in the remainder of this paper.Note, the ACN − → χ -boundedness conjecture was originally published as Conjecture 4.4 in [2].
The converse of the ACN − → χ -boundedness conjecture holds; all − → χ -bounding digraphs must be oriented forests.Indeed, Harutyunyan and Mohar proved that there exist oriented graphs of arbitrarily large undirected girth and dichromatic number [11].Oriented graphs of suciently large undirected girth (and no digon) forbid any xed digraph that is not an oriented forest.Hence, no digraph containing a digon or a cycle in its underlying graph is − → χ -bounding.
Moreover, for any nite list of digraphs . ., D k must be a forest.One might ask whether the situation changes when we forbid an in nite list of oriented graphs.We list some results related to this: • In [3], Carbonero, Hompe, Moore, and Spirkl provided a construction for oriented graphs with clique number at most three, arbitrarily high dichromatic number, and no induced directed cycles of odd length at least 5.They use this construction to disprove a wellknown stronger version of the Gyárfás-Sumner conjecture sometimes referred to as "Esperet's conjecture" (see also [18]).
• In [1], Aboulker, Bousquet, and de Verclos showed that the class of chordal oriented graphs, that is, oriented graphs forbidding induced directed cycles of length greater than three, is not − → χ -bounded, answering a question posed in [3].
• In [4], Carbonero, Hompe, Moore, and Spirkl extended the result of [3] to t-chordal graphs.A digraph is t-chordal if it does not contain an induced directed cycle of length other than t.In [4] the authors showed that t-chordal graphs are not − → χ -bounded, but t-chordal − → P t -free graphs are − → χ -bounded.
Note that Conjecture 1.2 only considers oriented graphs.This is the only sensible case.By the result of Harutyunyan and Mohar [11] if F contains a digon, then the class of F -free oriented graphs is not − → χ -bounded.If F contains no digons and at least one edge, then the class of F -free digraphs is not − → χ -bounded; Any digraph obtained from a graph by replacing every edge with a digon does not contain any oriented graph with at least one edge as an induced subgraph.Hence by [15,21], for any choice of an oriented graph with at least one edge F , there exist F -free digraphs (with digons) that have arbitrarily high dichromatic number and do not contain a triangle in their underlying graph.The ACN − → χ -boundedness conjecture is still widely open.It is not known whether the conjecture holds for any orientation of any tree T on at least ve vertices that is not a star.
In particular, it is not known whether the conjecture holds for oriented paths.In contrast, Gyárfás showed that every path is χ-bounding in the 1980's [8,9].We will introduce some terminology before discussing the status of the ACN − → χ -boundedness conjecture for oriented paths in more detail.For t ≤ 3, P t is − → χ -bounding.(This can be proven by, for example, noting that for t ≥ 3 the graph P t is a star and applying Chudnovsky, Scott, and Seymour's result [5] that every orientation of a star is χ-bounding and therefore also − → χ -bounding.)However, for t ≥ 4, the picture gets more complicated: • Let T be any xed orientation of K 3 .In [2], Aboulker, Charbit and Naserasr showed that class of (T , − → P 4 )-free oriented graphs have bounded dichromatic number.The authors also show that − → P 4 -free oriented graphs with clique number at most three have bounded dichromatic number.

• Let
− → K t denote the transitive tournament on t vertices.In [19], Steiner showed that the class of ( − → K 3 , − → A 4 )-free oriented graphs has bounded dichromatic number.In the same paper Steiner asked whether the class of (H, − → K t )-free oriented graphs has bounded dichromatic number for t ≥ 4 and H ∈ { − → P 4 , − → A 4 }.We explain in the next subsection that our main result answers this question in the a rmative.In particular, for any H-free oriented graph D, 5) .

Our contributions
Our result also answers the question of [19] in the a rmative, that is, for H ∈ { − → P 4 , − → A 4 } and any k ≥ 4 the class of H-free oriented graphs not containing a transitive tournament of order k has bounded dichromatic number.Indeed, any tournament of order 2 k−1 must contain a transitive tournament of order k.Thus, forbidding a given transitive tournament forbids any large enough tournament.Note that there is no analogous result for non-transitive tournaments since all sub-tournaments of a transitive tournament are transitive.The conjectures raised in [2] are aimed at characterizing heroic sets, that is, sets F such that digraphs (allowing digons) forbidding all elements of F have bounded dichromatic number.If we ignore the degenerate cases where heroic sets include the empty graph or the graph consisting of a single vertex, there are no heroic sets of size one and the only heroic set of size two consists of an arc and a digon.Therefore, heroic sets of size three are the smallest interesting case.Hence, our work in this paper can be seen as a continuation of the investigation of [2] into heroic sets of order three containing a digon.In the language of heroic sets, if the class of all H-free oriented graphs is − → χ -bounded, then the set consisting of H, a digon, and a transitive tournament is a heroic set.Then, Theorem 1.3 can be restated by saying that the heroic sets of the form { ← → K 2 , H, K}, where ← → K 2 denotes a digon, H is an orientation of P 4 , are exactly those where K is a transitive tournament.
Structure of the paper and proof overview.Let H be any orientation of P 4 .We prove Theorem 1.3 by induction on the clique number.We x an integer ω(D) ≥ 2. We de ne a function f and assume that H-free oriented graphs with clique number ω where 1 ≤ ω < ω have dichromatic number at most f (ω ).We then consider an oriented graph with clique number ω and show that D can be dicolored using at most f (ω) colors.
Our strategy to bound − → χ (D) crucially relies on a tool we call dipolar sets which were introduced by the name "nice sets" in [2].Dipolar sets have the following useful property [2]: In order to bound the dichromatic number of a class of oriented graphs closed under taking induced subgraphs, it su ces to exhibit a dipolar set of bounded dichromatic number for each of the members in the class.We give a few preliminary observations as well as an introduction to dipolar sets in Section 2.
In Section 3, we show how to construct a dipolar set for any H-free oriented graph D of clique number ω.Our goal is to bound the dichromatic number of this set.The backbone of our construction is an object we call a closed tournament.
De nition 1.4 (path-minimizing closed tournament).We say K and P form a closed tournament C = K ∪ V (P ) if K is a tournament of maximum order and P is a directed path from a source component to a sink component of the directed graph induced by K.
We say K and P form path-minimizing closed tournament if |P | is minimized amongst all choices of K, P that form a closed tournament.
It follows from the de nition of closed tournament that the graph induced by a closed tournament is strongly connected and that every strongly connected oriented graph has a path-minimizing closed tournament.We will de ne a set S consisting of the closed neighborhood of a path-minimizing closed tournament C and a subset of the second neighbors of C. We will show that if D is H-free, then S is a dipolar set.This proof will rely heavily on the fact that C is strongly connected.
The strong connectivity of C is a powerful property in showing that S is a dipolar set.However, ensuring C is strongly connected by adding P to K makes it harder to bound the dichromatic number of N (C).We explain in Section 2 that we can easily bound the dichromatic number of the rst neighborhood of any bounded cardinality set.Unfortunately, we have no control over the cardinality of P in a path-minimum closed tournament.In fact, P , and thus C, might be arbitrarily large with respect to ω.This signi cantly increases the diculty of the task of bounding the dichromatic number of N (C).Fortunately, since D is H-free and we may choose C to be a path-minimizing closed tournament, there are a lot of restrictions on what arcs may exist between vertices of N (C).Ultimately able to exploit these restrictions to bound the dichromatic number of N (C).
Interestingly, we can de ne S and prove that it is a dipolar set in the same way for each possible choice of an oriented P 4 .We describe our construction of a dipolar set S in Section 3.However, we used di erent (but similar) proofs to show that S has bounded dichromatic number for In Section 4, we bound the dichromatic number of C, the vertices of S in the second neighborhood of C, N (K) for H-free graphs where H is an arbitrary choice of an orientated P 4 .In Section 5, we bound the dichromatic number of the vertices in S not handled in Section 4. These remaining vertices are the set N (P ) \ N [K].Here we use separate (but similar) proofs for In Section 6, we put the pieces together to obtain our main result that any orientation of P 4 is − → χ -bounding.We discuss some related open questions in Section 7.

Preliminaries
In this section, we lay the groundwork for our proof by making a few observations useful in later sections and introducing dipolar sets.In the rest of the paper, we will only consider strongly connected oriented graphs since the dichromatic number of an oriented graph is equal to the maximum dichromatic number of one of its strongly connected components.In particular, we will work with the following assumptions: Scenario 2.1 (Inductive Hypothesis).Let H be an oriented P 4 and let ω > 1 be an integer.We let γ be the maximum of − → χ (D ) over every H-free oriented graph D satisfying ω(D ) < ω.We assume γ is nite.We let D be an H-free oriented graph with clique number ω and assume D is strongly connected.
We will aim to bound the − → χ (D) in terms of γ and ω.We begin with some easy observations about the dichromatic number of the neighborhood of any sets of vertices in D. For any vertex v ∈ V (D), by de nition ω(D[N (v)]) ≤ ω − 1 as otherwise D would contain a tournament of size greater than ω.Hence, for any v ∈ V (D), − → χ (N (v)) ≤ γ.This can be directly extended to bounding the dichromatic number of the neighborhood of a set of a given size as follows: Observation 2.2.Let D be an oriented graph and let γ be the maximum value of − → χ (N (v)) for any v ∈ V (D).Then every X ⊆ V (D) satis es: We now formally de ne dipolar sets, one of the main tools used in this paper.Note, dipolar sets were rst introduced in [1] as "nice sets".

De nition 2.3 (dipolar set).
A dipolar set of an oriented graph D is a nonempty subset S ⊆ V (D) that can be partitioned into S + , S − such that no vertex in S + has an out-neighbor in V (D \ S) and no vertex in S − has an in-neighbor in V (D \ S).
We will use the following lemma from [1] which reduces the problem of bounding the dichromatic number of D to bounding the dichromatic number of a dipolar set in every induced oriented subgraph of D. Lemma 2.4 (Lemma 17 in [1]).Let D be a family of oriented graphs closed under taking induced subgraphs.Suppose there exists a constant c such that every D ∈ D has a dipolar set S with

Building a dipolar set
In this section we give a construction for a dipolar set in an H-free oriented graph D where H is a oriented P 4 .We will then show that the dipolar set we construct has bounded dichromatic number if D satis es the properties given in Scenario 2.1.

Closed Tournaments
The simplest case for our construction is when D contains a strongly connected tournament J of order ω(D).Then, we can build a dipolar set consisting of the union of J and a subset of vertices at distance at most two from K.
Let K be a tournament of order ω(D) contained in D. By de nition every vertex v ∈ N (K) has a non-neighbor in K. Hence, the graph underlying D[K ∪ {v}] contains an induced P 3 .Now, suppose K is strongly connected.Then we get an even more powerful property: Since K is strongly connected there is both an arc from K \ N (v) to N (v) ∩ K and to K \ N (v) from N (v) ∩ K.This means that D[K ∪ {v}] contains an induced P 3 starting at v whose last edge is oriented as → and an induced P 3 starting at v whose last edge is oriented as ←.This property will give us more power to build speci c induced orientations of P 3 in N [K].In particular, this restricts the way vertices at distance at most two interact with the rest of the graph and allows us to choose a dipolar set.
To overcome the fact that D may not contain a strongly connected tournament of order ω(D), we use closed tournaments.By de nition of closed tournament every strongly connected oriented graph has a path-minimizing closed tournament.We will base our construction of a dipolar set on some path-minimizing tournament in order to gain some additional structure that we can use to bound the dichromatic number of our dipolar set.In the next subsection we formally give the de nition of our dipolar set.

Extending a closed tournament into a dipolar set
In order to build a dipolar set from a closed tournament, we need to make some distinctions between di erent types of neighbors of a set of vertices.For a set of vertices A and v ∈ N (A) we say v is a strong neighbor of A if v has both an in-neighbor and an out-neighbor in A. Then, the strong neighborhood of A is the set of strong neighbors of A.
Given a closed tournament C, we let X denote the set of strong neighbors of C. The following lemma proves that N [C ∪ X] is a dipolar set.
Suppose v ∈ Y .Then, by de nition, the following statements all hold: • There is some x ∈ X such that x and v are adjacent.
• There are vertices c 1 , c 2 ∈ C where c 1 is an in-neighbor of x and c 2 is an out-neighbor of x.
• b 1 , b 2 are not adjacent to any of x, c 1 , c 2 .
Thus for some choice of i, j ∈ {1, 2} the set {c i , x, v, b j } induces a copy of H. (See Figure 1.)This proves (1).Since D is an H-free oriented graph, it follows from (1) that v ∈ Z. Then by de nition of Z, the neighbors of v in C are either all in-neighbors of v or all out-neighbor of v.
There exist arcs (q 1 , p 1 ), (p 2 , q 2 ) ∈ E(C) such that v is adjacent to q 1 , q 2 and nonadjacent to p 1 , p 2 . ( It follows from the fact that ω(C) = ω(D) that v has some non-neighbor in C. Since C is strongly connected, N (v) ∩ C must have both an incoming arc and an outgoing arc from C \ N (v).Let p 1 , p 2 be vertices of C\N (v) witnessing this fact and let q 1 , q 2 their respective neighbors in N (v) ∩ C.This proves (2).
It follows that for some i, j ∈ {1, 2} the graph induced by {p i , q i , v, b j } is a copy of H, a contradiction.(See Figure 1).

First steps towards bounding the dichromatic number of N [C ∪ X]
As usual, we suppose D satis es the assumptions given in Scenario 2.1 all hold.We choose a tournament of order ω(D) and a directed path P that form a path-minimizing tournament C in D. Let X be the strong neighborhood of C and Y = N (X) \ N [C] as before.In the previous section we showed that N [C ∪ X] is a dipolar set.Thus, by Lemma 2. By de nition, We We will require separate proofs for

Bounding the dichromatic number of V (P ) and N [K]
We bound the dichromatic number of V (P ) and N [K] by an easy observation about "forwardinduced" paths.We say a directed path p 1 → p 2 → • • • → p t is forward-induced if no arc of the form (p i , p j ) exists where j > i + 1 and i, j ∈ {1, 2, . . ., t}.
Observation 4.1.Let D be an oriented graph and let P ⊆ D be a forward-induced directed path.Then − → χ (P ) ≤ 2.
Proof.Let the vertices of P be p 1 → p 2 → . . .→ p , in order.We assign colors to the vertices of P by alternating the colors along P .Suppose there is some monochromatic directed cycle Q in the oriented graph induced by V (P ).Then Q contains no arc of P .Hence Q must contain some arc (p i , p j ) with i, j ∈ {1, 2, . . ., } and j > i + 1, contradicting the de ntion of forwards-induced.Now, we turn to bounding the dichromatic number of our dipolar set, N [C ∪ X] by bounding − → χ (N [K] ∪ V (P )).
Observation 4.2.Let D be an oriented graph satisfying − → χ (N (v)) ≤ γ for all v ∈ V .Let K be a maximum tournament and P be a directed path in D such that K and P form a path-minimizing closed tournament   Lemma 4.3.Let H be an oriented P 4 and let D be an H-free oriented graph.Suppose there is a partition of V (D) into sets Q, R, S such that there is no arc between Q and S, every r ∈ R has both an in-neighbor and an out-neighbor in Q and every s ∈ S has a neighbor in R. Let γ be an integer such that for every r ∈ R, we have − → χ (N (r)) ≤ γ.Then − → χ (S) ≤ 2γ.
Proof.We proceed by induction on |S|.Suppose S = ∅.Then there is some r ∈ R that has a neighbor in S.
We may partition N (r) ∩ S into two sets S 1 , S 2 such that no vertex in S 1 has an in-neighbor in S \ N (r) and no vertex in S 2 has an out-neighbor in S \ N (r).
By de nition r has an in-neighbor q 1 and an out-neighbor q 2 in Q. Suppose some s ∈ N (r)∩S has both an in-neighbor s i and an out-neighbor s j in S \ N (r).Then there is an copy of H induced by {q i , r, s, s j } for some choice of i, j ∈ {1, 2}, a contradiction.(See Figure 2.) This proves (3).
Let D 1 , D 2 be two disjoint sets of γ colors each.By induction we can dicolor S \ N (r) with D 1 ∪ D 2 .By (3), we can extend this coloring to a dicoloring of S by dicoloring S i with colors from D i for i ∈ {1, 2}.
Corollary 4.4.Let H be an oriented P 4 and let D be a strongly connected H-free oriented graph.
Let K be a maximum tournament and P be a directed path in D such that K, P form a pathminimizing closed tournament.Let γ denote the maximum value of − → χ (N (v)) for any v ∈ D.
Proof.By de nition we may applying Lemma 4.3 to the induced subgraph D[C ∪ X ∪ Y ] with Q := C, R := X and S := Y (see Figure 1).Hence, Then by Observation 4.2 and since we obtain: Thus, if we can bound − → χ (N (P ) \ N [K]) we can bound the dichromatic number of our dipolar set N [C ∪ X] by Corollary 4.4.We will handle this in the next section.
5 Completing the bound on the dichromatic number of our dipolar set In this section, we will prove a bound on the dichromatic number of N (P ) \ N [K] where K is a maximum tournament and P is a directed path that forms a path-minimizing closed tournament C in an oriented graph that forbids some orientation of P 4 .By Corollary 4.4, this will imply that every oriented graph has which forbids some orientation of P 4 has a dipolar set of bounded dichromatic number.Thus, by Lemma 2.4, this will give us our main result.By de nition of path-minimizing closed tournament, P is a forward-induced directed path.In Subsection 5.1, we start by giving some structural properties on properties of the neighborhood of forwards-induced paths.Then, in Subsection 5.2 we show how to use these properties to bound the dichromatic number of the rst neighborhood of C for − → Q 4 -free graphs.(Recall, the bound for − → Q 4 -free oriented graphs implies the bound for − → Q 4 -free oriented graphs.)When H is one of the other two orientations, − → P 4 and − → A 4 , we required a ner analysis of N (P ) in order to bound − → χ (N (P )).We handle this case in Subsections 5.3.1-5.3.3.

Forbidden arcs among neighbors of a forward-induced directed path
We de ne two partitions of the rst neighborhood of a directed path and show how to forbid some of the arcs between classes of each partition in an H-free oriented graph.
De nition 5.1.Let P = p 1 → p 2 → • • • → p be a forward-induced directed path in an oriented graph.For brevity, for any v ∈ N (P ) and i, j ∈ {1, 2, . . ., } we say p i is the rst neighbor of v on P if v is adjacent to p i and non-adjacent to p i for each 1 ≤ i < i ≤ .Similarly, p j is the last neighbor of v on P if v is adjacent to p j and non-adjacent to each p j for each 1 ≤ j < j ≤ .We will de ne two partitions of N (P ) according to their rst and last neighbors in V (P ), respectively.
• For each i ∈ {1, 2, . . ., } we say v ∈ N (P ) is in F i if p i is the rst neighbor of v on P .This yields partition (F 1 , F 2 , . . ., F ), which we call the partition of N (P ) by rst attachment (on P ).
• Symmetrically, for each j ∈ {1, 2, . . ., } we say v ∈ L j if p j is the last neighbor of v on P .This yields partition (L 1 , L 2 , . . ., L ), which we call the partition of N (P ) by last attachment (on P ).
For each i ∈ {1, 2, . . ., } we re ne each partition by dividing each F i , L i into the in-neighbors and out-neighbors of v.We de ne F + i and L + i to be the sets consisting of all the in-neighbors of p i in F i , L i , respectively.Similarly, we de ne F − i and L − i to be the sets consisting of all the out-neighbors of p i in F i , L i , respectively.Observation 5.2.Let P = p 1 → p 2 → • • • → p be a forward-induced directed path in an oriented graph D. Let (F 1 , F 2 , . . ., F ), (L 1 , L 2 , . . ., L ) be the partitions of N (P ) by rst attachment and last attachment on P , respectively.Let 2 ≤ i < j ≤ − 1.Then the following statements all hold: free, there are no arcs from F j to F i .
• If D is − → P 4 -free, there are no arcs from F − i to F j , and no arcs from L i to L + j .
• If D is − → A 4 -free, there are no arcs from F + i to F j , and no arcs from L i to L − j .
Suppose for some v ∈ F j and w ∈ F i that (v, w) ∈ E(D).Then the vertices p i−1 , p i , w, v, induce a P 4 in D with orientation In either case we obtain an induced If D is − → P 4 -free, there are no arcs from F − i to F j , and no arcs from L i to L + j . ( Suppose for some v ∈ F − i and w ∈ F j that (v, w) ∈ E(D).Then p i−1 → p i → v → w is an induced − → P 4 (see the dark blue arcs in Figure 3).Hence D is not − → P 4 -free.This proves the rst part of the statement (5).The argument that there are no arcs from L i to L + j in an − → P 4 -free graph is symmetric.This proves (5).

If D is
− → A 4 -free, there are no arcs from F + i to F j , and no arcs from By symmetry it is enough to show that if D is − → A 4 -free then there is no arc from F + i to F j .Suppose for some v ∈ F + i and w ∈ F j that (v, w) ∈ E(D).Then p i−1 → p i ← v → w is an induced − → A 4 in D (see the dark green arcs in Figure 3).This proves (6).In the Subsection 5.2 we use Observation 5.2 to bound the dichromatic number of N (P ) \ N [K] in the − → Q 4 -free case.In the − → P 4 -free case and the − → A 4 -free case we need to perform a more careful analysis of N (P ) \ N [K] in order to bound its dichromatic number because the conditions guaranteed by Observation 5.2 are weaker in these two cases.In Subsection 5.3.1, we use Observation 5.2 to bound the dichromatic number of the following subsets of when D is − → P 4 -free and when D is − → A 4 -free.The vertices in N (P )\(N [K]∪W p ) and N (P )\(N [K]∪W a ) have restrictions on how they may have neighbors in V (P ).We will use this to bound their dichromatic number in Subsections 5.3.2 and 5.3.3,respectively.

The
− → Q 4 -free case In section, we bound the dichromatic number of a path-minimizing closed tournamentin D when D is a − → Q 4 -free oriented graph satisfying the conditions of Scenario 2.1.
Lemma 5.3.Let P be a forward-induced directed path in D and γ be an integer satisfying Proof.Assume D is − → Q 4 -free oriented graph.Let (F 1 , F 2 , . . ., F ) be the partition of N (P ) by rst attachment on P .By de nition Hence we may use the same set of γ colors for each of F 2 , F 3 , . . ., F −1 .Thus, − → χ (N (P ) \ N ({p 1 , p }) ≤ γ.
Lemma 5.3 allows us to demonstrate a bound on our dipolar set N [C ∪ X] as follows: Lemma 5.4.Let D be a strongly connected − → Q 4 -free oriented graph.Let γ be an integer satisfying − → χ (N (v)) ≤ γ for each v ∈ V (P ).Then D has a dipolar set with dichromatic number at most Proof.Let K be a maximum tournament and P be a directed path in D such that K and P form a path-minimizing closed tournament C in D. Let X denote the strong neighborhood of C. By Lemma 3.1, N [C ∪ X] is a dipolar set.By Corollary 4.4 we obtain: Let p 1 , p denote the ends of P .Then by de nition p 1 , p ∈ K. Hence the result follows by Lemma 5.3.

5.3
The − → P 4 -free case and the − → A 4 -free case In this subsection, we bound the dichromatic number our dipolar set in the case where D is − → P 4 -free or − → A 4 -free.

Bounding
In this subsection, we bound the dichromatic number of of W p and W a using Observation 5.2 in − → P 4 -free and − → A 4 -free oriented graphs, respectively.
Lemma 5.5.Let P be a forward-induced directed path in D and γ be an integer satisfying − → χ (N (v)) ≤ γ for each v ∈ V (P ).Let W p and W a be de ned with respect to P .Then, the following statements both hold: Proof.Let the vertices of P be p 1 → p 2 → . . .→ p , in order.Let (F 1 , F 2 , . . ., F ) and (L 1 , L 2 , . . ., L ) be the partitions of N (P ) by rst attachment and last attachment on P , respectively., respectively.We begin by proving the rst bullet.Suppose D is − → P 4 -free.Then by Observation 5.2, every directed cycle in By assumption, − → χ (N (p i )) ≤ γ for every p i ∈ P .Hence, we may use the same Therefore, since W p is the union of these two sets, we obtain The case is symmetric when D is − → A 4 -free.The third item of Observation 5.2 allows us to use the same set of colors for each of F + 2 , F + 3 , . . ., F + −1 , and the same set of colors for each of

5.3.2
Completing the bound on the dichromatic number of our dipolar set in the − → P 4 -free case In this section, we will consider the dichromatic number of the following set of vertices.
De nition 5.6.Let P be a forward-induced directed path in an oriented graph.Let the vertices of P be p 1 → p 2 → • • • → p , in order.We let where W p is de ned with respect to P .
We will assume that D satis es the conditions of Scenario 2.1 with H = − → P 4 for the remainder of Subsubsection 5.3.2.In other words, D is a strongly connected − → P 4 -free oriented graph with clique number ω, and there is some nite γ such that every − → P 4 -free oriented graph with clique number less than ω has dichromatic number at most γ.Let K be a maximum tournament and P be a directed path in D such that K and P form a path-minimizing closed tournament C in D.Then, in terms of ω and γ in order to demonstrate that D as a dipolar set of bounded dichromatic number.By de nition, W p is the set of vertices in N (P ) \ N ({p 1 , p }) whose rst neighbor on P is an in-neighbor or whose last neighbor in V (P ) is an out-neighbor.Hence, R p consists exactly of the vertices in N (P ) \ N ({p 1 , p 2 , p }) whose rst neighbor in V (P ) is an out-neighbor and whose last neighbor in V (P ) is an in-neighbor..We will show that since C = K ∪ V (P ) is a path-minimizing closed tournament there is no tournament of order ω in R p and thus − → χ (R p ) ≤ γ.In particular, we will show that for a contradiction, if R p has a tournament J of order ω, then we can nd a directed path P that is shorter than P such that J and P form a closed tournament.In order to prove this, we will need the following lemma, which will allow us to exhibit a relatively short path between two adjacent vertices in R p .
Lemma 5.7.Let D be a P 4 -free oriented graph.Let P = p 1 → p 2 → . . .→ p be a forwardinduced directed path in D. Let R p be de ned with respect to P .Let v, w ∈ R p , if (w, v) ∈ E(D), there is a directed path from v to w on at most max{6, − 1} vertices.
Proof.Let p v 1 denote the rst neighbor of v in V (P ) and let p w 2 denote the last neighbor of w in V (P ).Then, since v, w ∈ ∪ i=1 F − i ∪ L + i , the corresponding arcs are (v, p v 1 ), (p w 2 , w) ∈ E(D).By de nition of R p , we obtain that 3 ≤ v 1 , w 2 ≤ − 1.Hence, we may assume that w w is a directed path from v to w with at most − 1 vertices, as desired.
Since w 2 < v 1 the vertices v, w have no common neighbors in V (P ).Now, consider the directed path w is a directed path from v to w of length three, as desired.Hence we may assume that (p w 2 , p v 1 ) ∈ E(D).
Since P is a forward-induced directed path and w 2 < v 1 , it follows that v 1 = w 2 + 1.Consider the directed path -free it cannot be induced.Therefore, the vertices p w 2 −1 and p v 1 , the vertices p w 2 and p v 1 +1 , or the vertices p w 2 −1 and p v 1 +1 are adjacent.Furthermore, since P is a shortest path, this means that at least one of (p v 1 +1 , p w 2 −1 ), (p v 1 +1 , p w 2 ), (p v 1 , p w 2 −1 ) is an arc of D, see Figure 4. We consider each case separately: In every case, the oriented graph induced by {v, p v 1 , p v 1 +1 , p w 2 −1 , p w 2 , w} contains a directed path from v to w on at most six vertices.Since one of the cases must hold, this completes the proof.
With the last lemma in hand, we are ready to bound the dichromatic number of R p .
Proof.Let K be a maximum tournament and P be a directed path in D such that K and P form a path-minimizing closed tournament C in D. Then by Lemma 3.1, N [C ∪ X] is a dipolar set.We will use Lemma 5.7 to bound − → χ (R p ). Then we will combine this bound with the results from the previous sections to bound − → χ (N [C ∪ X]).
Suppose ≥ 7. If ω(D[R p ]) < ω(D) then by assumption − → χ (R p ) ≤ γ, so we may assume that there is an ω(D)-tournament J ⊆ R. Since C is a minimum closed tournament and P is non-empty, J is not strongly connected.Hence there must be exactly one strongly connected component of J that is a sink and exactly one strongly connected component of J that is a source (and they are not equal).Let v be a vertex in the sink component of J and w be a vertex in the source component of J. Therefore, (w, v) ∈ E(D).Thus by Lemma 5.7 there is a path Q from v to w of length less than that that of P .Hence, J, P form a closed tournament.By de nition since K, P were chosen to form a path-minimizing closed tournament P cannot be shorter than P , a contradiction.This proves (9).
By combining (10) with Corollary 4.4, we have Let W a and R a be de ned with respect to P .By de nition p 1 , p ∈ K and so N (P ) \ N [K] ⊆ W a ∪ R a .Thus by combining Lemmas 5.5 and 5.12, we obtain Then the lemma follows by combining (11) and (12).

Computing the − → χ -binding function
We will show that an element from the following family of functions is a − → χ -binding function for any class of oriented graphs forbidding a particular orientation of P 4 .
De nition 6.2.For any integer c ≥ 3 we let for any non-negative integer x.
We will need that f c satis es the following recursive properties in order to show that for some c the function f c is χ-bounding for any class of oriented graph forbidding a particular orientation of P 4 .Observation 6.3.Let c ≥ 3. Then: • f c (x) = 2(x + c)f c (x − 1) + 4 for any integer x ≥ 2, and • f c (x) ≤ (x + c) x+c+1.5 for any integer x ≥ 1.
Proof.By de nition since c ≥ 3 we obtain that f c (1) > 2c! > 1.Hence, the second bullet holds and we will now prove the rst bullet.Let x ≥ 2 be an integer.Then, f c (x) = 2(x + c) 2 x−1 (x + c − 1)! + 1 2(x + c) Thus, by combining the previous two equations we obtain that f c (x) = 2(x + c)f c (x − 1) + 4.This proves the rst bullet.We will complete the proof by showing the third bullet holds.Let x ≥ 1.By de nition, . ( Since c ≥ 3 we obtain the following by combining ( 13) and ( 14).This proves the third bullet.

− → χ -boundedness
We are now ready to prove the following more precise version of our main result, Theorem 1.3.
Theorem 6.4.Let H be an orientation of P 4 , then H-free graphs are − → χ -bounded.Speci cally, We have f c (1) > 1 by the rst bullet of Observation 6.3, so the statement holds for oriented graphs with no arcs.We complete the proof by induction on the clique number.Let ω > 1 be an integer.Suppose every H-free oriented graph D with clique number less than ω satis es − → χ (D ) ≤ f (ω(D )).Let D be an H-free oriented graph with clique number equal to ω.We Since ω ≥ 2 this implies − → χ (D) ≤ f c (ω) by the second bullet of Observation 6.3.This completes the proof.

Conclusion
Our result is an initial step towards resolving the ACN − → χ -boundedness conjecture for orientation of paths in general.However, we think we are still far from this result.Our construction of a dipolar set with bounded chromatic number relies heavily on the length of P 4 and we do not expect that our techniques can be directly extended to show that any oriented P t for t ≥ 5 is − → χ -bounding.It would already be interesting to hear the answer to the easier question: Does there exists an integer t ≥ 5 and an orientation H of P t such that the class of oriented graphs forbidding H and all tournaments of size 3 has unbounded dichromatic number?Recall that the classes of − → Q 4 -free oriented graphs and − → Q 4 -free oriented graphs were already shown to be χ-bounded in [5].The χ-binding function f for these two classes from [5] is de ned using recurrence f (x) := 2(3f (x − 1)) 5   which leads to a double-exponential bound on χ, and cannot guarantee a better bound on − → χ .
In this paper, Theorem 1.3 provides an improved − → χ -binding function when any orientation of P 4 is forbidden.It would interest us to know of any improvements to the − → χ function.In particular, we would like to know whether any orientation of P 4 is polynomially − → χ -bounding.
In other words, is there some oriented P 4 so that the class of oriented graphs forbidding it has a polynomial − → χ -binding function?

Lemma 3 . 1 .
Let H be an orientation of P 4 and D be an H-free oriented graph.Let C be a closed tournament in D and let X denote the strong neighborhood of C. Then N [C ∪ X] is a dipolar set.Proof.Let Z denote the neighbors of C that are not strong, and let Y = N (X)\N [C].These sets satisfy N [C ∪ X] = C ∪ X ∪ Z ∪ Y and the graph on N [C ∪ X] is illustrated in Figure 1.Then by de nition, N [C ∪X] = N [C]∪Y and the only vertices of N [C ∪X] with neighbors in V (D) \ N [C ∪ X] are in Y ∪ Z. Suppose for a contradiction that some v ∈ Z ∪ Y has both an in-neighbor b 1 and an out-neighbor b 2 in V \ (N [C] ∪ Y ).Let us rst deal with the case where v ∈ Y .

Figure 1 :
Figure 1: An illustration of the extension of a closed tournament C into the dipolar set N [C ∪ X].Highlighted in blue, Z consists of neighbors of C that are not strong, i.e., do not have both an in-neighbor and an out-neighbor in C. The set X consists of the strong neighborhood of C, while set Y contains all neighbors of X not in N [C].Note that arcs between Z and X or Y are not represented here.In Lemma 3.1, we prove that if there is some vertex in N [C ∪X] with both an in-neighbor and an out-neighbor in the rest of the oriented graph (drawn in dashed red), then N [C ∪ X] contains all orientations of P 4 as an induced oriented subgraph.
4 we can prove that all orientations of P 4 are − → χ -bounding by proving that − → χ (N [C ∪ X]) is bounded in terms of ω(D) and γ, the maximum value of − → χ (D ) for any H-free D with clique number less than ω(D).
will bound − → χ (N [C ∪ X]) by bounding each of the terms on the right-hand side of the equation.We bound the dichromatic number of N [K] and P in Subsection 4.1 and we bound the dichromatic number of Y in Subsection 4.2.We are able to use the same techniques for each choice of H when proving these bounds.As already hinted, bounding − → χ (N [C ∪ X]) is non-trivial because we have no control over the cardinality of P .Hence, we cannot obtain a useful bound on − → χ (N (P ) \ N [K]) by simply applying Observation 2.2.In the next section, we will show how to bound − → χ (N (P ) \ N [K]).

Proof.
Since |K| = ω(D) > 1, we have N [K] = x∈K N (x), and hence − → χ (N [K]) ≤ ω • γ by Observation 2.2.This proves the rst statement.By de nition, N [C] = (N (P ) \ N [K]) ∪ N [K] ∪ P .Since C is path-minimizing, P is forward-induced.Thus, we obtain the second statement by Observation 4.1.Thus, it only remains to bound the dichromatic number of Y and N (P ) \ N [K] in order to bound the dichromatic number of our dipolar set N [C ∪ X].

4. 2
Bounding the dichromatic number of YIn this subsection, we bound the dichromatic number of Y = N (X)\N [C].We rst state a more general lemma, which gives the bound on − → χ (Y ) as a direct corollary.

Figure 2 :
Figure 2: Vertex sets Q, R, S, such that no arc lies between Q and S, the vertices in R are all strong neighbors of Q and S is a subset of neighbors of R. We illustrate the case of graphs forbidding − → P 4 .Other orientations behave symmetrically.In dark green, a vertex r ∈ R is depicted with an out-neighbor s ∈ S. Then if s has an out-neighbor s 1 ∈ S \ N (r) there would be an induced − → P 4 , a contradiction.Symmetrically in dark blue, an in-neighbor s ∈ S of r cannot admit an in-neighbor s 2 ∈ S\N (r).

Figure 3 :
Figure 3: A shortest path directed path P = p 1 → ... → p along with the partition (F 1 , ..., F ) of N (P ) by rst attachment on P .Note that the setting is symmetric for the partition of N (P ) by the last attachment.Each class of the partition F i is represented as a circle and further split into F + i in green and F − i in blue, all possible arcs towards P are drawn in gray.An arc from F − i to F j with j > i would induce a − → P 4 using (p i−1 , p i ), as highlighted in dark blue.An arc from F + i to F j would induce a − → A 4 , represented in dark green.

Figure 4 :
Figure 4: On the bottom, a shortest path P in D, and an arc (w, v) between neighbors of P in R p .Illustrated here is the case where the last neighbor p w 2 of w on P appears just before the rst neighbor p v 1 of v, all possible arcs are shown in gray.Then, path p w 2 −1 , p w 2 , p v 1 , p v 1 +1 cannot induce a − → P 4 .Any arc possibly preventing this, shown in dash-dotted green, yields a path from v to w of length at most ve.

5. 3 . 3
Completing the bound on − → χ of our dipolar set in the − → A 4 -free caseIn this section, we prove a bound on the remaining vertices of N (P )\N [K] and use the results of the previous sections to show that D contains a dipolar set of bounded dichromatic number in the − → A 4 -free case.

Figure 5 :
Figure 5: A shortest directed path P , along with v ∈ F i \ W a and w ∈ F j \ W a of R a .Since vertex w /∈ W a , it must also belong to some L + x with x > j, meaning its last neighbor on P is an out-neighbor.Then, since x > j > i + 2, an arc (v, w) would yield a shorter path from p 1 to p .
Theorem 1.3.Let H be an oriented P 4 .Then, the class of H-free oriented graphs is − → χ -bounded.
In this section, we consider an oriented graph D satisfying Scenario 2.1.The previous sections show that D has a dipolar set of bounded dichromatic number.We will use this result and Lemma 2.4 to show that oriented graphs not containing some orientation of P 4 are − → χ -bounded., then D contains a dipolar set with dichromatic number at most (ω + 6) • γ + 2. , then D contains a dipolar set with dichromatic number at most (ω + 7) • γ + 2.