# The Oriented Coloring Page

This page is maintained by Éric Sopena and is intended to collect results on oriented colorings that have been published in refereed journals or conferences. *Some arXiv papers, yet unpublished, might also be cited, in which case the corresponding paragraphs appear in italics.*

If you have any suggestion, or new result not mentioned here, please do not hesitate to contact me at *eric.sopena@labri.fr*.

## Contents

- Definitions and notation (oriented graphs, oriented colorings, homomorphisms, oriented chromatic number, oriented cliques, oriented clique numbers)
- General results (bounds, forests, cycles, graphs with bounded acyclic chromatic number, graphs with bounded maximum average degree, critical graphs, complexity)
- Planar graphs (grids, outerplanar graphs, 2-outerplanar graphs, planar graphs, planar graphs with large girth)
- Graphs with bounded degree (general bounds, graphs with maximum degree three, graphs with larger maximum degree)
- Other graph classes (Cartesian products, Halin graphs, hypercubes, graphs with given excess, fully subdivided graphs)
- Miscellaneous (oriented arc-coloring, strong oriented colorings, pushable chromatic number, injective oriented colorings, oriented list coloring, oriented coloring game)
- References

## Definitions and notation

### Oriented graphs

An *oriented graph* is a digraph with no loops and no opposite arcs. An oriented graph is thus an *orientation* of a simple undirected graph *U*, obtained from *U* by giving to each edge of *U* one of its two possible orientations.

We denote by *V*(*G*) and *A*(*G*) the set of vertices and the set of arcs, respectively, of an oriented graph *G*.

### Oriented colorings

An *oriented k-coloring* of an oriented graph *G* is a mapping *c: V*(*G*) *→ C*, where *C* is a set of *k* colors (usually *C =* {1,...,k}), such that:

*c*(*u*) ≠*c*(*v*) for every arc*uv*in*A*(*G*),*c*(*u*) ≠*c*(*y*) for every two arcs*uv*and*xy*with*c*(*v*) =*c*(*x*).

Note in particular that if *uvw* is a directed path in *G* (that is *uv*, *vw ∈ A*(*G*)), then *c*(*u*) ≠ *c*(*v*) ≠ *c*(*w*) ≠ *c*(*u*).

An oriented *k*-coloring of an oriented graph *G* thus corresponds to a partition of the vertex set *V*(*G*) of *G* into *k* independent subsets such that all the arcs linking any two of these subsets have the same direction.

** Example.** Figure 1 shows an oriented 4-coloring of an oriented graph. It can be observed that this graph admits no oriented 3-coloring.

Colorings of oriented graphs first appear in the work of Courcelle (Co94) on the monadic second order logic of graphs, as a way to encode graph orientations by means of vertex labels. Since then, these colorings have been considered by many authors, following the work of Raspaud and Sopena (RS94) on oriented colorings of planar graphs (see (So01) for a survey on oriented colorings).

### Homomorphisms

Let *G* and *H* be two oriented graphs. A *homomorphism* from *G* to *H* is a mapping *h: V*(*G*) *→ V*(*H*) which preserves the arcs: *uv ∈ A*(*G*)* ⇒ h*(*u*)*h*(*v*)* ∈ A*(*H*). We write *G → H* whenever there exists a homomorphism from *G* to *H*.

A homomorphism from *G* to *H* can therefore be viewed as an oriented coloring of *G* that uses the vertices of *H* as colors. An oriented *k*-coloring of *G* can thus be equivalently defined as a homomorphism of *G* to some oriented graph *H* on *k* vertices.

** Example.** Figure 2 shows how the oriented 4-coloring of Figure 1 can be interpreted as a homomorphism to an oriented graph of order 4.

### Oriented chromatic number

The *oriented chromatic number* χ_{o}(*G*) of an oriented graph *G* is defined as the smallest *k* for which *G* admits an oriented *k*-coloring or, equivalently, as the smallest order of an oriented graph *H* such that *G* admits a homomorphism to *H*.

The oriented chromatic number χ_{o}(*U*) of an *undirected* graph *U* is defined as the maximum oriented chromatic number of its orientations: χ_{o}(*U*) = max { χ_{o}(*G*), *G* is an orientation of *U* }.

*Examples.*

- As observed above, the oriented chromatic number of the graph depicted in Figure 1 is 4.
- Let
*C*be the_{5}*directed*cycle on 5 vertices. Since any two vertices of*C*are linked by a directed path of length at most 2, we get χ_{5}_{o}(*C*) = 5._{5}

### Oriented cliques (or o-cliques)

An oriented graph *G* is an *oriented clique* (or, simply, an *o-clique*) if χ_{o}(*G*) = |*V*(*G*)|. Therefore, an oriented graph is an o-clique if and only if every two of its vertices are linked (in either direction) by a directed path of length 1 or 2 (in other words, its *weak diameter* is at most 2).

Every tournament is obviously an o-clique but being a tournament is not a necessary condition for being an o-clique. Figure 3 depicts some sample o-cliques of order up to 6.

Let *O _{n}* be an o-clique of order

*n*. Take now two copies of

*O*, a new vertex

_{n}*v*, and add all possible arcs from vertices of the first copy towards the vertex

*v*and all possible arcs from the vertex

*v*to vertices of the second copy. It is easy to check that the so-obtained oriented graph is an o-clique of order 2

*n*+ 1. (The o-clique

*O*in Figure 3 is obtained in this way from the o-clique

_{3}*O*.)

_{1}In (KMG04a), Klostermeyer and MacGillivray proved that the maximum order of an o-clique whose underlying graph is a partial 2-tree graph is 7 (which implies, in particular, that the maximum order of an outerplanar o-clique is 7; Figure 4 below shows an outerplanar o-clique of order 7). They also proved that every planar o-clique has order at most 36 and conjectured that the maximum order of a planar o-clique is 15. Observe that applying the above-described construction to the outerplanar o-clique of Figure 4 produces a planar o-clique of order 15. This conjecture has been proved by Sen in (Se12) (another proof was given later by Nandy, Sen and Sopena in (NSS16)).

In (FHPZ98) and (KLSS99), Füredi, Horak, Pareek and Zhu, and Kostochka, Łuczak, Simonyi and Sopena, respectively, independently proved that the minimum number of edges in an o-clique of order

*n*is (1 +

*o*(1))

*n*log

_{2}

*n*.

### Oriented clique numbers

Two types of clique numbers are defined for oriented graphs. The *absolute oriented clique number* ω_{ao}(*G*) of an oriented graph *G* is the maximum order of an o-clique subgraph of *G*. The *relative oriented clique number* ω_{ro}(*G*) of an oriented graph *G* is the maximum cardinality of a set of vertices in *G* with the property that every two vertices in this set are connected by a directed path of length 1 or 2 in *G* (in other words, in every oriented coloring of *G*, all the vertices in such a set must be assigned distinct colors; such a set is called a *relative oriented clique*, or *relative o-clique* of *G*). It follows from these definitions that we have ω_{ao}(*G*) ≤ ω_{ro}(*G*) ≤ χ_{o}(*G*) for every oriented graph *G*.

In (NSS16), Nandy, Sen and Sopena studied the absolute oriented clique number of planar graphs with given girth. They prove that ω_{ao}(*G*) ≤ 15, 6, 5 or 3, if *G* is a planar graph with girth at least 3, 4, 5 or 6, respectively, and that all these bounds are sharp. (The fact that ω_{ao}(*G*) ≤ 15 for every planar graph *G* was already proved in (Se12).)

Relative oriented clique numbers of oriented graphs have been studied by Das, Prabhu and Sen in (DPS19). They proved the following results:

- ω
_{ro}(*G*) ≤ 7 if*G*has maximum degree at most 3, and this bound is sharp, - ω
_{ro}(*G*) ≤ 7, 5 or 3 if*G*is an outerplanar graph with girth at least 3, 4 or 5, or 6, respectively, and all these bounds are sharp, - ω
_{ro}(*G*) ≤ 32, 14, 6, 4, 3 if*G*is a planar graph with girth at least 3, 4, 5, 6 or 7, respectively. Moreover, there exist planar graphs with relative oriented clique number 15, and triangle-free planar graphs with relative oriented clique number 10, while the bounds for girth 5, 6 and 7 are sharp.

They also proved that determining the relative oriented clique number of an oriented bipartite graph is a NP-hard problem.

## General results

### Bounds

In (OM17), Ochem and Movarraei proved that for every undirected graph *G*, χ_{o}(*G*) ≤ χ(*G*)χ_{2}(*G*) and χ_{2}(*G*) ≤ χ(*G*)χ_{o}(*G*), where χ(*G*) is the ordinary chromatic number of *G*, and χ_{2}(*G*) the *2-edge-colored chromatic number* of *G* (see (OM17) for details).

### Forests

It is not difficult to observe that every oriented forest *F* admits a homomorphism to *C _{3}*, the directed cycle of order 3, so that χ

_{o}(

*T*) ≤ 3.

### Cycles

As observed above, the directed cycle on 5 vertices is an o-clique and, therefore, has oriented chromatic number 5. It is not difficult to prove that every other oriented cycle has oriented chromatic number at most 4.

More precisely, if *C* is an oriented cycle, we define the *level* *λ*(*C*) of *C* as the number of forward arcs minus the number of backward arcs in *C*. Then we have (1) if the orientation of *C* is alternating, then χ_{o}(*C*) = 2, (2) if *λ*(*C*) = 0 (mod 3) and the orientation of *C* is not alternating, then *C* admits a homomorphism to *C _{3}* and χ

_{o}(

*C*) = 3, (3) if

*C*does not contain three consecutive arcs with the same direction and the orientation of

*C*is not alternating, then

*C*admits a homomorphism to

*T*, the transitive tournament or order 3, and χ

_{3}_{o}(

*C*) = 3, (4) otherwise, χ

_{o}(

*C*) = 4 if and only if

*C*is not the o-clique

*O*(depicted in Figure 3).

_{5}### Graphs with bounded acyclic chromatic number

A proper vertex coloring of an undirected graph *U* is *acyclic* if for every two colors *a* and *b*, the subgraph of *U* induced by the vertices with color *a* or *b* is a forest. In others words, every cycle of *U* uses at least three colors. The *acyclic chromatic number* of *U*, denoted *χ _{a}*(

*U*), is then the smallest number of colors needed by an acyclic coloring of

*U*.

In (RS94), Raspaud and Sopena proved that every graph with acyclic chromatic number at most

*k*has oriented chromatic number at most

*k*.2

^{k-1}. (Ochem (Oc05) proved that this bound is tight for every

*k ≥ 3*.) Conversely, Kostochka, Sopena and Zhu proved in (KSZ97) that every class of graphs with bounded oriented chromatic number has bounded acyclic chromatic number.

### Graphs with bounded maximum average degree

The *maximum average degree* mad(*U*) of an undirected graph *U* is defined as the maximum of the average degrees of its subgraphs (the *average degree* of *H* is given by 2.|*E*(*H*)| / |*V*(*H*)|).

Recall that the *girth* of *U* is the length of a shortest cycle in *U*.
If *U* is a planar graph with girth at least *g*, then, using Euler's formula, it can be proved that *U* has maximum degree less than 2*g*/(*g*-2). Therefore,
several authors have considered the problem of determining the oriented chromatic number of graphs with bounded maximum average degree (and, sometimes, with high girth), in order to get upper bounds on the oriented chromatic number of planar graphs with high girth:
Borodin, Kostochka, Nešetřil, Raspaud and Sopena (BKNRS99),
Borodin, Ivanova and Kostochka (BIK07),
Pinlou (Pi09).

The best known upper bounds are summarized in the table below (some of them giving the best known upper bounds for planar graphs with high girth).

Mad | Girth | Χ_{o} | Author(s) |
---|---|---|---|

< 10/3 | ≤ 19 | Borodin, Kostochka, Nešetřil, Raspaud and Sopena (BKNRS99) | |

< 10/3 | ≥ 5 | ≤ 16 | Pinlou (Pi09) |

< 3 | ≤ 11 | Borodin, Kostochka, Nešetřil, Raspaud and Sopena (BKNRS99) | |

< 11/4 | ≥ 5 | ≤ 7 | Borodin, Kostochka, Nešetřil, Raspaud and Sopena (BKNRS99) |

< 12/5 | ≥ 5 | ≤ 5 | Borodin, Ivanova and Kostochka (BIK07) |

### Critical graphs

An oriented graph *G* is *vertex-critical* (resp. *arc-critical*) if, for every vertex *u* in *V*(*G*) (resp. for every arc *uv* in *A*(*G*)), χ_{o}(*G - u*) < χ_{o}(*G*) (resp. χ_{o}(*G - uv*) < χ_{o}(*G*)).

In (BFKRS01), Borodin, Fon-der-Flaass, Kostochka, Raspaud and Sopena proved that for every *k* > 0, there exists an oriented graph *G _{k}* such that χ

_{o}(

*G*) - χ

_{k}_{o}(

*G*) ≥

_{k}- u*k*for every vertex

*u*in

*V*(

*G*) and χ

_{k}_{o}(

*G*) - χ

_{k}_{o}(

*G*) = 2 for every arc

_{k}- uv*uv*in

*A*(

*G*) (it is not difficult to observe that χ

_{k}_{o}(

*G*) - χ

_{o}(

*G - uv*) ≤ 2 for every oriented graph

*G*and every arc

*uv*in

*A*(

*G*)).

### Complexity

In (KMG04b), Klostermeyer and MacGillivray proved the following: for a given oriented graph *G*, whether χ_{o}(*G*) ≤ *k* can be decided in polynomial time if *k* ≤ 3 and is NP-complete if *k* ≥ 4, even when *G* is connected. Culus and Demange (CD06) proved that deciding whether χ_{o}(*G*) ≤ 4 is NP-complete even if *G* is a bounded degree bipartite, or a bounded degree acyclic oriented graph. Ganian and Hliněný (GH10) proved that this problem is also NP-complete when restricted to graphs with K-width 1 and DAG-depth 3 (see (GH10) for details), and Coelho, Faria, Gravier and Klein (CFGK16) proved that this problem is stll NP-complete when restricted to acyclic oriented graphs whose underlying graph is connected, planar, bipartite and has maximum degree 3.

Ganian (Ga09) proved that there exists an FPT-algorithm for determining the oriented chromatic number of an oriented graph, parameterized by the treewidth of its underlying undirected graph.

## Planar graphs

### Grids

The *rectangular grid* *G _{m,n}* is the Cartesian product of

*P*by

_{m}*P*, the two paths with

_{n}*m*and

*n*vertices, respectively. In (FRR03), Fertin, Raspaud and Roychowdhury proved that the oriented chromatic number of every rectangular grid is at most 11, and that there exists such grids with oriented chromatic number 7 (they provided an orientation of the grid

*G*with oriented chromatic number 7). They also conjectured that the oriented chromatic number of every rectangular grid is at most 7, and that every such grid admits a homomorphism to the tournament

_{4,5}*T*(see below). They proved that this conjecture is true when

_{7}*m = 2,3*.

The tournament

*T*is built from the non-zero quadratic residues of 7; it has vertex set

_{7}*V*(

*T*) = Z/7Z and

_{7}*ij*is an arc if and only if

*j*-

*i*∈ {1,2,4}.

Szepietowski and Targan (ST04) proved that this conjecture is also true when

*m = 4*but that the second part of the conjecture fails to be true whenever

*m ≥ 5*by providing an orientation of

*G*which admits no homomorphism to the tournament

_{5,33}*T*. The first part of the conjecture was finally disproved by Dybizbański and Nenca (DN12) who exhibited an orientation of the rectangular grid

_{7}*G*with oriented chromatic number at least 8.

_{7,212}In (Bi06), Bielak studied hexagonal grids

*H*, made of

_{m,n}*m*rows of

*n*hexagons. She proved that χ

_{o}(

*H*) = 5 for every

_{1,n}*n*≥ 3 and that χ

_{o}(

*H*) ≤ 6 for every

_{m,n}*m, n*.

### Outerplanar graphs

In (So97), it was proved that every partial 2-tree has oriented chromatic number at most 7. Since every outerplanar graph is a partial 2-tree, we get that every outerplanar graph has oriented chromatic number at most 7. This bound is tight, as shown by the outerplanar o-clique *O _{7}* depicted in Figure 4.

In (PS06), Pinlou and Sopena determined the oriented chromatic number of outerplanar graphs of given girth. They proved that every outerplanar graph (in fact, more generally, that every partial 2-tree) with girth

*g*has oriented chromatic number at most 6 is

*g*≥ 4, and at most 5 if

*g*≥ 5. Moreover, all these bounds are tight.

### 2-Outerplanar graphs

A graph is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the external face is outerplanar. In (EO07), Esperet and Ochem proved that every 2-outerplanar graph has oriented chromatic number at most 67. This bound was later decreased to 40 by Ochem and Pinlou (OP14).

### Planar graphs

In (Bo79), Borodin proved that every undirected planar graph is acyclically 5-colorable. Therefore, using the result of Raspaud and Sopena (RS94), we get that every planar graph has oriented chromatic number at most 5.2^{4} = 80. Despite many efforts, this upper bound has not been improved yet.

As observed above, there exists a planar o-clique of order 15. This lower bound on the oriented chromatic number of the class of planar graphs has been improved to 16 by Sopena (So02), to 17 by Marshall (Ma07), and to 18, again by Marshall (Ma12).

### Planar graphs with large girth

Several authors have considered the problem of determining the oriented chromatic number of the families of planar graphs with given girth, successively improving various lower or upper bounds:
Raspaud and Sopena (RS94),
Nešetřil, Raspaud and Sopena (NRS97),
Borodin, Kostochka, Nešetřil, Raspaud and Sopena (BKNRS98, BKNRS99),
Ochem (Oc04),
Borodin, Kim, Kostochka and West (BKKW04),
Borodin and Ivanova (BI05a, BI05b),
Esperet and Ochem (EO07),
Borodin, Ivanova and Kostochka (BIK07),
Montassier, Ochem and Pinlou (MOP08),
Ochem and Pinlou (OP08, OP14),
Pinlou (Pi09),
Marshall (Ma12, Ma13).

The best known lower and upper bounds are summarized in the table below (first reference is for the lower bound, second reference for the upper bound).

Girth | Χ_{o} | Author(s) |
---|---|---|

≥ 3 | 18 ≤ Χ_{o} ≤ 80 | Marshall (Ma12) - Raspaud and Sopena (RS94) |

≥ 4 | 11 ≤ Χ_{o} ≤ 40 | Ochem (Oc04) - Ochem and Pinlou (OP14) |

≥ 5 | 7 ≤ Χ_{o} ≤ 16 | Marshall (Ma13) - Pinlou (Pi09) |

≥ 6 | 7 ≤ Χ_{o} ≤ 11 | Marshall (Ma13) - Borodin, Kostochka, Nešetřil, Raspaud and Sopena (BKNRS99) |

≥ 7 | 6 ≤ Χ_{o} ≤ 7 | Nešetřil, Raspaud and Sopena (NRS97) - Borodin and Ivanova (BI05a) |

≥ 8 | 5 ≤ Χ_{o} ≤ 7 | Nešetřil, Raspaud and Sopena (NRS97) - Borodin and Ivanova (BI05a) |

≥ 11 | 5 ≤ Χ_{o} ≤ 6 | Nešetřil, Raspaud and Sopena (NRS97) - Ochem and Pinlou (OP08) |

≥ 12 | Χ_{o} = 5 | Nešetřil, Raspaud and Sopena (NRS97) - Borodin, Ivanova and Kostochka (BIK07) |

In (BKKW04), Borodin, Kim, Kostochka and West proved that every oriented graph with girth at least 13 embeddable on the torus or the Klein bottle has oriented chromatic number at most 5.

## Graphs with bounded degree

### General bounds

It was proved in (So97) that every oriented graph with maximum degree *Δ* has oriented chromatic number at most (2*Δ* - 1).2^{2Δ-2}.
This bound was decreased to 2.*Δ*^{2}.2^{Δ} by Kostochka, Sopena and Zhu (KSZ97).
(These authors also proved that for every *Δ* ≥ 2, there exists an oriented graph with maximum degree *Δ* and oriented chromatic number at least 2^{Δ/2}.)

This upper bound has been specialised by Aravind and Subramanian (AS09) as follows. Recall that the *degeneracy* of a graph *G* is the smallest integer *d* such that every subgraph of *G* contains a vertex of degree at most *d*. They proved that every graph with maximum degree *Δ* and degeneracy *d* has oriented chromatic number at most 16.*Δ.d*.2^{d}.

In (Wo07), Wood proved that every oriented graph of order *n* with maximum degree *Δ* has oriented chromatic number at most 2*Δ*.sqrt(*n* - 1) and constructed such graphs with oriented chromatic number at least *Δ*.sqrt(*n*/8).

### Graphs with maximum degree three

It was proved in (So97) that every oriented graph with maximum degree 3 has oriented chromatic number at most 16. This upper bound has been decreased to 11 by Sopena and Vignal (SV96). This bound was then decreased to 9 for connected graphs by Duffy, MacGillivray and Sopena in (DMGS19).

*The connectivity condition has been proved to be unnecessary by Ochem and Pinlou in a yet unpublished paper, available on arXiv: Oriented coloring of graphs with low maximum degree. Later, this bound has been decreased to 8 for connected graphs by Duffy in a paper available on arXiv: A note on colourings of connected oriented cubic graphs*.

The following conjecture was proposed in (So97):

**Conjecture.** Every *connected* oriented graph with maximum degree 3 has oriented chromatic number at most 7.

If the conjecture is true, then the bound of 7 cannot be improved, as shown by the oriented o-clique of order 7 depicted in Figure 5.

### Graphs with larger maximum degree

Duffy, MacGillivray and Sopena proved in (DMGS19) that every connected oriented graph with maximum degree 4 has oriented chromatic number at most 69. *In the arXiv paper Oriented coloring of graphs with low maximum degree, Ochem and Pinlou decreased this bound to 26, showing again that the connectivity condition is unnecessary. They also prove that every graph with maximum degree 5, 6 or 7 has oriented chromatic number at most 90, 306 or 1322, respectively.*

## Other graph classes

### Cartesian products

The *Cartesian product* of two undirected graphs *U* and *U′*, denoted *U* □ *U′*, is the graph defined by *V*(*U* □ *U′*) = *V*(*U*) × *V*(*U′*) and (*u*,*u′*)(*v*,*v′*) ∈ *E*(*U* □ *U′*) if and only if either *u* = *u′* and *vv′* ∈ *E*(*U′*) or *v* = *v′* and *uu′* ∈ *E*(*U*).

In (ANS11), Aravind, Narayanan and Subramanian proved that for every undirected graph *U*:

- χ
_{o}(*U*□*P*) ≤ (2_{k}*k*- 1).χ_{o}(*U*), for every*k*≥ 2, - χ
_{o}(*U*□*C*) ≤ 2_{k}*k*.χ_{o}(*U*), for every*k*≥ 3,

where *P _{k}* and

*C*stands for the path and the cycle on

_{k}*k*vertices, respectively.

These results were immproved in (So12), where it is proved that χ

_{o}(

*U*□

*T*) ≤ 6χ

_{o}(

*U*) for every undirected tree

*T*, χ

_{o}(

*U*□

*C*) ≤ 12χ

_{k}_{o}(

*U*) for every

*k*≥ 3,

*k*≠ 5 and χ

_{o}(

*U*□

*C*) ≤ 15χ

_{5}_{o}(

*U*).

### Halin graphs

A *Halin graph* is a planar graph which has a planar embedding such that deleting the edges of its external face gives a tree with at least three vertices. Hosseini Dolama and Sopena proved in (HDS06b) that every oriented Halin graph has oriented chromatic number at most 9 and that there exist oriented Halin graphs with oriented chromatic number at least 8. In (DS14), Dybizbański and Szepietowski finally proved that every oriented Halin graph has oriented chromatic number at most 8, and thus this bound is tight.

### Hypercubes

In (Wo07), Wood proved that, for the *d*-dimensional hypercube *Q _{d}*, 0.8007...sqrt(2

^{d}) ≤ χ

_{o}(

*Q*) ≤ 2

_{d}*d*.sqrt(2

^{d}- 1).

### Graphs with given excess

The *excess* of a graph *G* is the minimum number of edges that must be deleted from *G* in order to get a forest. In (HDS06a), Hosseini Dolama and Sopena proved that every oriented graph with excess *k* has oriented chromatic number at most *k* + 3, except for graphs with excess 1 containing a directed cycle of length 5 which have oriented chromatic number 5. This bound is tight for every *k* ≤ 4.

### Fully subdivided graphs

Let *U* be an undirected graph. We denote by *U ^{s}* the (undirected) graph obtained from

*U*by subdividing exactly once every edge of

*U*. Wood (Wo05) proved that for every undirected graph

*U*, the oriented chromatic number of

*U*is at most 7 is

^{s}*Χ*(

*U*) ≤ 7, at most 9 if

*Χ*(

*U*) = 8 and equals

*Χ*(

*U*) if

*Χ*(

*U*) ≥ 9, where

*Χ*(

*U*) stands for the ordinary chromatic number of

*U*.

## Miscellaneous

### Oriented arc-coloring

Let *G* be an oriented graph. The *line-digraph* of *G*, denoted *LD*(*G*), is the oriented graph whose set of vertices is the set of arcs of *G*, and whose set of arcs if the set of pairs (*uv,vw*) such that *uv* and *vw* are both arcs in *G*.
An *oriented k-arc-coloring* of *G* is then defined as an oriented (vertex) *k*-coloring of *LD*(*G*). In other words,
an oriented *k*-arc-coloring of *G* is a mapping *c* from *A*(*G*) to a set *C* of *k* colors such that:

*c*(*uv*) ≠*c*(*vw*) for every arcs*uv*and*vw*in*A*(*G*),- if
*uv*,*vw*,*xy*and*yz*are four arcs in*A*(*G*) with*c*(*uv*) =*c*(*yz*), then*c*(*vw*) ≠*c*(*xy*).

The *oriented chromatic index* of *G*, denoted *Χ′ _{o}*(

*G*), is then the smallest

*k*such that

*G*admits an oriented

*k*-arc-coloring.

Let

*G*be an oriented graph with oriented chromatic number

*k*and

*γ*be an oriented (vertex)

*k*-coloring of

*G*. It is not difficult to check that the mapping

*c*defined by

*c*(

*uv*) =

*γ*(

*u*) for every arc

*uv*in

*G*is an oriented

*k*-arc-coloring of

*G*. Therefore,

*Χ′*(

_{o}*G*) ≤ χ

_{o}(

*G*) for every oriented graph

*G*.

The main results concerning the oriented chromatic index of oriented graphs are the following:

- If the underlying undirected graph of
*G*has acyclic chromatic number at most*k*, then*Χ′*(_{o}*G*) ≤ 2*k*(*k*- 1) - ⌊*k*/2⌋ - Ochem, Pinlou and Sopena, (OPS08). - As a corollary of the above result, we get that every oriented planar graph has chromatic index at most 38; moreover, there exist oriented planar graphs with oriented chromatic index at least 10 - Ochem, Pinlou and Sopena, (OPS08).
- If
*G*is a planar oriented graph with girth*g*≥ 46, then*Χ′*(_{o}*G*) ≤ 4 - Ochem, Pinlou and Sopena, (OPS08). - For every partial 2-tree
*G*,*Χ′*(_{o}*G*) ≤ 7 and this bound is tight - Pinlou and Sopena, (PS06). - Every outerplanar graph with girth
*g*has oriented chromatic index at most 7 if*g*≥ 3, at most 6 is*g*≥ 4, at most 5 if 5 ≤*g*≤ 9 and at most 4 if*g*≥ 10; moreover, all these bounds are tight - Pinlou and Sopena, (PS06). - Every partial 2-tree with girth
*g*has oriented chromatic index at most 7 if*g*≥ 3, at most 6 is*g*≥ 5, at most 5 if 7 ≤*g*≤ 17 (tight) and at most 4 if*g*≥ 18 (tight) - Ochem and Pinlou, (OP07). - If
*G*is an oriented graph with maximum degree*Δ*, then*Χ′*(_{o}*G*) ≤ 2.(⌊*Δ*^{2}/2⌋ +*Δ*) - Ochem, Pinlou and Sopena, (OPS08). - If
*G*is an oriented graph with maximum degree 3, then*Χ′*(_{o}*G*) ≤ 7, and there exists such oriented graphs with oriented chromatic index at least 6 - Pinlou (Pi06). - Given an oriented graph
*G*, deciding whether*Χ′*(_{o}*G*) ≤*k*is polynomial time if*k*≤ 3 and NP-complete if*k*≥ 4 - Ochem, Pinlou and Sopena, (OPS08).

### Strong oriented coloring

Strong colorings of oriented graphs have been introduced by Nešetřil and Raspaud in (NR99).
A *strong coloring* of an oriented graph *G* is a homomorphism of *G* to an oriented Cayley graph *X* = *X*(*M*,*A*), where *M* is a group and *A* a subset of *M* such that *A* ∩ *A ^{-1}* = ∅.
The smallest order of a group

*M*for which there exists a subset

*A*of

*M*satisfying the previous conditions is the

*strong oriented chromatic number of G*, denoted

*Χ*(

_{s}*G*).

Nešetřil and Raspaud proved in (NR99) that if

*G*is an oriented graph with acyclic chromatic number

*k*, then

*Χ*(

_{s}*G*) ≤ 6

^{k}. Thanks to Borodin's Theorem (Bo79), this yields that every oriented planar graph has strong oriented chromatic number at most 7776. This upper bound has then be decreased to 672 by Šamál (Sa03) and to 271 by Marshall (Ma06). In (Sa04), Šamál proved that every oriented graph with oriented chromatic number at most

*q*has strong oriented chromatic number at most (1 +

*o*(1))).

*q*and that this bound is tight. (He also proved that every oriented graph with acyclic chromatic number at most

^{2}*k*has strong oriented chromatic number at most 64/49.

*k*.2

^{2}^{k}which, for planar graphs, gives an upper bound of 1045 for the strong oriented chromatic number.)

Nešetřil and Raspaud (NR99) also considered planar graphs with large girth. They proved that if

*G*is an oriented planar graph with girth at least

*g*then

*Χ*(

_{s}*G*) is at most 5 if

*g*≥ 14, at most 7 if

*g*≥ 8, at most 11 if

*g*≥ 6 and at most 19 if

*g*≥ 5.

Several results about upper bounds on the oriented chromatic number of certain graph classes were obtained by proving that every member of the class admits a homomorphism to some oriented graph which is indeed a Cayley graph, so that these results also hold for the strong oriented chromatic number. Hence we get for instance:

- if
*G*is an oriented outerplanar graph (or, more generally, an oriented partial 2-tree), then*Χ*(_{s}*G*) ≤ 7, Sopena (So97), - if
*G*is an oriented 2-outerplanar graph, then*Χ*(_{s}*G*) ≤ 67, Esperet and Ochem (EO07), - if
*G*is an oriented graph with maximum average degree less than 10/3 (resp. 3, 11/4), then*Χ*(_{s}*G*) ≤ 19 (resp. 11, 7), Borodin, Kostochka, Nešetřil, Raspaud and Sopena (BKNRS99), - if
*G*is an oriented rectangular grid, then*Χ*(_{s}*G*) ≤ 11, Fertin, Raspaud and Roychowdhury (FRR03), - if
*G*is a graph with maximum degree 3, then*Χ*(_{s}*G*) ≤ 11, Sopena and Vignal (SV96).

### Pushable chromatic number

Klostermeyer and MacGillivray introduced in (KMG04b) (see also (KMG04c)) the notion of the *pushable chromatic number* of an oriented graph *G*. If *Y* is a subset of *V*(*G*), we denote by *G ^{Y}* the oriented graph obtained from

*G*by reversing the orientation of all the arcs that have exactly one end in

*Y*. The pushable chromatic number of

*G*, denoted

*Χ*(

_{p}*G*), is then defined as the minimum oriented chromatic number of the oriented graphs

*G*, taken over all possible subsets

^{Y}*Y*of

*V*(

*G*).

It is not difficult to observe that

*Χ*(

_{p}*G*) ≤ χ

_{o}(

*G*) ≤ 2.

*Χ*(

_{p}*G*) for every oriented graph

*G*. Klostermeyer and MacGillivray proved that every oriented partial 2-tree has pushable chromatic number at most 4. Moreover, they proved that the problem of deciding if an oriented graph has pushable chromatic number at most

*k*is polynomial if

*k*≤ 2 and NP-complete if

*k*≥ 3.

A

*push-clique*is an oriented graph whose pushable chromatic number equals its order. In (BNS17), Bensmail, Nandi and Sen proved that deciding whether an undirected graph can be oriented in such a way that the resulting oriented graph is a push-clique is an NP-complete problem. Moreover, they proved that the maximum order of a planar push-clique is 8 (and this bound is sharp), and characterized all planar push-cliques in terms of spanning subraphs (see (BNS17) for details).

### Injective oriented colorings

Oriented colorings that are injective on in-neighborhoods first appeared in the work of Courcelle (Co94), under the name of *good and semi-strong* colorings. An oriented *k*-coloring is said to be *injective* if, for every vertex *u* in *V*(*G*), all the in-neighbors of *u* are assigned distinct colors. It follows from Courcelle's Theorem on the monadic second order logic of graphs that deciding, for a fixed number *k* of colors, if an oriented graph *G* admits an injective oriented *k*-coloring is polynomially solvable if *G* has bounded treewidth.
In (RS94), Raspaud and Sopena proved that every oriented planar graph admits an injective oriented 320-coloring.

The interest in injective oriented colorings has been reactivated by MacGillivray, Raspaud and Swarts with their WG'09 paper (MGRS10).

Injective oriented colorings may obviously be interpreted in terms of *(locally) injective homomorphisms* of oriented graphs, that is arc-preserving vertex mappings such that in-neighbors are mapped to distinct vertices. In (MGS10), MacGillivray and Swarts proved that if *T* is a tournament, the injective *T*-coloring problem is NP-complete, except when *T* has at most three vertices or consists of a directed 3-cycle dominated by a source vertex. Moreover, the problem of deciding if an oriented connected graph admits an injective oriented *k*-coloring is polynomial if *k* ≤ 3 and NP-complete if *k* ≥ 4 (MGRS10). If *T* is an oriented tree, there exists a linear time algorithm to determine if *T* admits an injective oriented *k*-coloring, for every given fixed *k* (MGRS10).

In (MGRS11), MacGillivray, Raspaud and Swarts exhibited sets of forbidden subgraphs for an oriented graph to admit an injective oriented 2- or 3-coloring and described efficient algorithms that either produce such colorings or a forbidden
structure.

An *io-clique* is an oriented graph *G* that admits no injective oriented *k*-coloring if *k* < |*V*(*G*)|. (Observe that every oriented graph admits an injective oriented *k*-coloring with *k* = |*V*(*G*)|.)
MacGillivray, Raspaud and Swarts proved that *G* is an io-clique if and only if every two vertices in *G* have a common out-neighbor or are joined by a directed path of length at most 2 (MGRS10).

Let *G* be an oriented graph and let *Δ ^{-}* (resp.

*Δ*) denote the maximum in-degree (resp. out-degree) of

^{+}*G*. Then

*G*admits an injective oriented

*k*-colouring with

*k*≤ 2

^{p}- 1,

*p*=

*Δ*+(

*Δ*-1)

^{-}*Δ*+(

^{+}*Δ*)

^{+}^{2}(with

*k*≤ 2.

*Δ*+ 1 if

^{-}*G*is an oriented tree and, in that case, this bound is tight) (MGRS10).

### Oriented list coloring

Sali and Simonyi introduced in (SS99) the notion of oriented list coloring.
An oriented graph *G* is *k-choosable* if, for every possible assignment of lists of size at least *k* to its vertices, there is an oriented coloring of *G* such that every vertex is assigned one of the colors of its list. The minimum *k* for which *G* is *k*-choosable is the *oriented choice number of G*, denoted ch_{o}(*G*).

If *U* is an *undirected* graph, the oriented choice number ch_{o}(*U*) of *U* is defined as the *minimum* oriented choice number of its orientations. (Note that this is in contrast to the oriented chromatic number of *U*, defined has the *maximum* oriented chromatic number of its orientations.)

Sali and Simonyi proved the following:

- For every
*k*> 0, there is an integer*g*(*k*) such that, if an undirected graph*U*contains at least*g*(*k*) independent edges, then ch_{o}(*U*) >*k*. (As a corollary, we get that for every*k*> 0, there exists an undirected graph with (ordinary) choice number 2 and oriented choice number greater than*k*.) - For every undirected graph
*U*, ch_{o}(*U*) ≤*τ*(*U*) + 1, where*τ*(*U*) denotes the minimum number of vertices covering all edges of*U*.

In (TV01), Tuza and Voigt characterized matchings, connected graphs, and graphs containing at least one cycle, with oriented choice number 2. In particular, they proved that the smallest number of edges in a matching with oriented choice number 3 is 12.

### Oriented coloring game

Nešetřil and Sopena introduced in (NS01) the oriented version of the graph coloring game. Let *H* be any fixed oriented graph. The *oriented H-coloring game* is played on an oriented graph *G* by two players, Alice and Bob, who alternate in turn, Alice having the first move.
Each move consists in choosing an uncolored vertex *u* in *G* and assigning it a color *α* from *V*(*H*) such that:

- if
*uv*(resp.*vu*) is an arc in*A*(*G*) and*v*has color*β*, then*αβ*(resp.*βα*) is an arc in*A*(*H*), - if there exists a directed 2-path from
*u*to some vertex*w*with color*γ*, then*α*≠*γ*.

Alice's goal is to eventually produce a *H*-coloring of *G* (that is, a homomorphism from *G* to *H*) while Bob tries to prevent her from doing so. The *oriented game chromatic number* of *G*, denoted ogcn(*G*), is defined as the smallest *k* for which there exists an oriented graph *H* of order *k* such that Alice has a winning strategy when playing the oriented *H*-coloring game on *G*.

The authors proved the following:
if *P* is an oriented path, then ogcn(*P*) ≤ 7 (and this bound is tight),
if *T* is an oriented tree, then ogcn(*T*) ≤ 19,
the oriented game chromatic number of the class of oriented outerplanar graphs is finite.

In (KT01), Kierstead and Trotter proved that for every *k*, there exists an integer *t* and a tournament *T* of order *t* such that, if Γ is a topologically closed class of graphs (that is, every subgraph or subdivision of a member of Γ belongs to Γ) and Γ does not contain the complete graph *K _{k}*, then Alice wins the oriented

*T*-coloring game on every orientation of any member of Γ. In particular, this proves that the oriented game chromatic number of the class of oriented planar graphs is finite (this question was posed in (NS01)).

In (KT03), Kierstead and Tuza proved that for every

*k*, the oriented game chromatic number of the class of oriented partial

*k*-trees is finite.

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