The diachromatic number of digraphs

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichromatic and diachromatic numbers are, respectively, the smallest and the largest number of colors in a complete acyclic coloring. We give some general results for the diachromatic number and study it for tournaments. We also show that the interpolation property for complete acyclic colorings does hold and establish Nordhaus-Gaddum relations.


Introduction
A complete coloring of an undirected graph G is a vertex coloring of G such that for every pair of colors there is at least one edge in G whose endpoints are colored with this pair of colors. The chromatic and achromatic numbers of G are the smallest and the largest number of colors in a complete proper coloring of G, respectively. The concept of achromatic number have been intensely studied in graphs since it was introduced by Harary, Hedetniemi and Prins [14] in 1967, for more references of results related to this parameter see for instance [2,3,1,5,8,15,26]. The achromatic number has been extended to digraphs with two different colorings one by Edwards [11] and another by Sopena [24]. Edwards considered colorings such that the underlying graph is proper colored and the completeness such that for each ordered pair (c, c ′ ) of distinct colors, there is at least one arc xy such that x has color c and y has color c ′ , and he proved that a directed graph does not necessarily have a complete coloring and that determining whether one exists is an NP-complete problem. Sopena proposed another extension of the achromatic number, the oriented achromatic number, using the oriented chromatic number defining a completeness that corresponds to complete homomorphisms of oriented graphs. He proved that for every integers a and b with 2 ≤ a ≤ b, there exists an oriented graph G a,b with oriented chromatic number a and oriented achromatic number b. He also studied the behavior of the oriented achromatic number adding or deleting a vertex or an arc. In both extensions of the achromatic number for digraphs, it is proven that the interpolation property does not hold. In this paper, we propose yet another extension of the achromatic number to digraphs en terms of acyclic colorings. We stress that with our extension, the interpolation property does hold.
A vertex coloring of a digraph D is called acyclic if each chromatic class induces a subdigraph with no directed cycles. The dichromatic number dc(D) of a digraph D is the smallest k such that D admits an acyclic coloring, it was introduced by Neumann-Lara in [20] as a generalization of the chromatic number, for more references of results related to this parameter see for instance [4,12,16,17,18,19,21,22]. A coloring of a digraph D is called complete if for every ordered pair (i, j) of different colors there is at least one arc (u, v) such that u has color i and v has color j [11]. It is not hard to see that any acyclic coloring of D with dc(D) colors is a complete coloring. We define the diachromatic number dac(D) of a digraph D as the largest number of colors for which there exists a complete and acyclic coloring of D. Hence, the dichromatic and diachromatic numbers of a digraph D are, respectively, the smallest and the largest number of colors in a complete acyclic coloring of D. The pseudoachromatic number ψ(D) of a digraph D is the largest number k for which there exists a complete coloring of D using k colors (see [11,13]).
Since the dichromatic number of a symmetric digraph is equal to the chromatic number of the underlying graph, the diachromatic number of a symmetric digraph is equal to the achromatic number of the underlying graph; and the pseudoachromatic number of a symmetric digraph is equal to the pseudoachromatic number of the underlying graph, the dichromatic number, the diachromatic number and the pseudoachromatic number of a digraph generalizes the chromatic number, the achromatic number and the pseudoachromatic number of a graph; recall that the underlying graph G D of a digraph D is obtained from D changing all symmetric and asymmetric arcs by edges. Note that the chromatic number of the underlying graph is an upper bound for the dichromatic number of a digraph and the pseudoachromatic number of a digraph is a lower bound for the pseudoachromatic number of the underlying graph, but this kind of relation can not be established between the diachromatic and the achromatic numbers.
This paper is organized in six sections: the second section contains general results for the diachromatic number of a digraph, in the third section we study the diachromatic number of tournaments; in the four and five sections we generalize some results on graphs, given in Chapter 12 of [10], for digraphs; also in four section, we define the concept of dihomorphisms and show that the interpolation property does hold for complete acyclic colorings; and five section establishes the Nordhaus-Gaddum relations. Finally, the last section has some conclusions and future work.

Definitions and basic results
In this paper, we consider finite digraphs. The arc uv ∈ A(D) is symmetric if vu ∈ A(D) and asymmetric if vu / ∈ A(D). A digraph is symmetric (resp. asymmetric) if every arc of D is symmetric (resp. asymmetric); for general concepts see [9]. Let k be a natural number. A vertex coloring ς of a digraph D with k colors is a surjective function that assigns to each vertex of D a color of [k] := {1, . . . , k}. For each i ∈ [k], the set ς −1 (i) ⊆ V (D) will be called chromatic class. The coloring ς of D with k colors is called acyclic if no chromatic class induces a subdigraph with a directed cycle. An acyclic coloring with k colors is denoted (for short) as k-coloring. The dichromatic number dc(D) of a digraph D is the smallest k such that D admits an acyclic coloring [20]. An acyclic coloring of a digraph D with k colors is called complete if for every ordered pair (i, j) of different colors there is at least one arc (u, v) such that u has color i and v has color j [11]. It is not hard to see that any dc(D)-coloring of D is a complete coloring. We define the diachromatic number dac(D) of a digraph D as the largest number k for which there exists a complete and acyclic k-coloring of D.
From the definition, we obtain the following for any digraph D of order n: The converse digraph D op of D is obtained replacing each arc (u, v) of D by the arc (v, u). The complement D c of a digraph D is that digraph whose vertex set is V (D) and where (u, v) is an arc of D if and only if (u, v) is not an arc of D. It is not difficult to prove the following remark. Remark 1. If G is a graph, and − → G is an orientation of G we have that The following theorem gives a rather simple (and sharp) bound for the diachromatic number of a digraph in terms of its size.
and the result follows. Let − → M be an oriented matching of size m. We exhibit a k-complete and acyclic coloring of − → M for m = k(k − 1) and . . , k} that assign different colors to different elements of Y j is bijective because Y j has exactly k − 1 elements.
. . , k}. The k-coloring is complete because for any pair of different colors a and b all the vertices x i for (a − 1)(k − 1) + 1 ≤ i ≤ a(k − 1) satisfies that φ(x i ) = a and, by construction, always exits exactly one vertex in Y a of color b. Furthermore, if m ≥ k(k − 1) and m ′ = k(k − 1), we can repeat some colors used previously on the vertices {x i , y i } for i ∈ {m ′ + 1, . . . , m} preserving the property of φ. The following theorems give some results related to the diachromatic number of digraphs in terms of their dichromatic number and structural properties. Since the dichromatic number of a non acyclic digraph is at least 2, we obtain the following corollary, which is a generalization of a theorem given by Shaoji Xu [25]. It establishes an upper bound for dac(D) − dc(D) in terms of the order of D. Let D be a digraph of order n whose n vertices are listed in some specified order. In a greedy coloring of D, the vertices are successively colored with positive integers according to an algorithm that assigns to the vertex under consideration the smallest available color. Hence, if the vertices of D are listed in the order v 1 , v 2 , . . . , v n , then the resulting greedy coloring ς assigns the color 1 to v 1 , that is, ς(v 1 ) = 1. If v 1 and v 2 are not a 2-cycle, then assign ς(v 2 ) = 1, else ς(v 2 ) = 2. In general suppose that the first j vertices v 1 , v 2 , . . . , v j , where 1 ≤ j < n, in the sequence have been colored with the colors 1, . .
be the set of chromatic classes. Consider the vertex v j+1 , if there exists a chromatic class C i for which, When the algorithm ends, the vertices of D have been assigned colors from the set [k] for some positive integer k. Thus, and so k is an upper bound for the dichromatic number of D and a lower bound for the diachromatic number of D.
It is useful to know how the diachromatic number of a digraph can be affected by the removal of a single vertex.
Proof. Let dac(D) = l, and let ς be a complete l-coloring of D where the set of vertices colored l is U and suppose that u ∈ U . Therefore, the partial l-coloring of ς of D restricted to D − U is a complete (l − 1)coloring. We can obtain a complete coloring using a greedy coloration for the remaining vertices x ∈ U − u.
Let dac(D − u) = k, and consider a complete k-coloring of D − u. We can obtain a complete coloring of D using a greedy coloration for u. Therefore, dac(D) ≥ k = dac(D − u).
The following result is an immediate consequence of Theorem 5.
Proof. Let X = {x 1 , . . . , x r } and Y = {y 1 , . . . , y s }. Suppose that r ≤ s and color the vertices x i and y i with By Theorem 5, the removal of a single vertex from a digraph D can result in a digraph whose diachromatic number is either one less than or is the same as the diachromatic number of D; whereas there exist three possibilities when a single edge is removed.
Proof. Let dac(D) = k, then there exists a complete k-coloring of D, where the colors assigned to u and v are the same or distinct. If u and v have the same color assigned, the complete k-coloring of D is also a complete k- Assume that u and v have different colors, say u is colored k − 1 and v is colored k. If the resulting k-coloring of D − f is not a complete k-coloring, then no directed cycle in D − f is bicolored with the colors k and k − 1. Hence, every vertex colored k may be recolored k − 1, resulting in a complete (k − 1)-coloring of D − f . In Let dac(D − f ) = l, then there exists a complete l-coloring of D − f . The vertices u and v are either assigned distinct colors or the same color. If u and v are assigned distinct colors, then the complete l-coloring of D − f is also a complete l-coloring of D. Hence, we may assume that u and v are assigned the same color, say l. The complete l-coloring of D − f is also a complete coloring of D with l colors. If the set of vertices colored l is acyclic in D, then the complete l-coloring of D −f is also a complete l-coloring of D. Assume that the set of vertices colored l has monochromatic directed cycles in D, where f is an arc of any monochromatic cycle. If there is a chromatic class C j , for some j ∈ [l − 1], such that u ∪ C j (resp. v ∪ C j ) induces an acyclic digraph, then by recoloring the vertex u (v resp.) with the color j, we obtain an acyclic complete coloring using a greedy coloration for the remaining vertices x colored l. Assume that for each i ∈ [l − 1] the vertex set C i ∪ u y C i ∪ v contains a directed cycle, then recoloring v with the color l + 1. The resulting coloring is an acyclic complete (l + 1)-coloring of D. In any case, dac(D) Since k-minimal digraphs have diachromatic number k, the size of every such graph is at least k(k − 1) and every chromatic class induces an arc-less digraph, else if f is an arc of an induced subdigraph of a chromatic class, the complete k-coloring of D is also a complete k-coloring of D − f . Hence, dac(D − f ) ≥ dac(D) > dac(D) − 1, contradicting that D is a k-minimal digraph. The following theorem is a generalization of a result given by Bhave [7] characterized graphs that are k-minimal in terms of their size.
Theorem 9. Let D be a digraph with diachromatic number k. Then D is k-minimal if and only if its size is k(k − 1).
Proof. Assume first that the size of D is k(k − 1). Then for every arc f of D, the size of D − f is k(k − 1) − 1.
Since the size of D − f is less than k(k − 1), it follows that dac(D − f ) < k = dac(D) and that D is k-minimal. We now verify the converse. Assume, to the contrary, that there is a k-minimal graph H whose size m is not k(k − 1). Since dac(H) = k, it follows that m ≥ k(k − 1). Since m = k(k − 1), it follows that m ≥ k(k − 1)+ 1. Let φ be a complete k-coloring of H. For every two distinct colors i, j ∈ [k], there exist an arc such that its vertices are colored i and j. Since there are only k 2 pairs of two distinct colors from the set [k] and two arc for each pair, there are two distinct arc f = (u, v) and then φ is also a complete k-coloring of H − f , contradicting the assumption that H is k-minimal.

Tournaments
A tournament T of order n is an orientation of the complete graph K n . An acyclic tournament T is transitive, the vertex set V (T ) = {v 1 , . . . , v n } of a transitive tournament has a unique (acyclic) order (v 1 , . . . , v n ), where Lemma 10. Let T be a tournament of order n ≥ 3. Then Proof. For every xy ∈ A let C 3 (xy) be the number of directed triangles of T with xy as an arc, and T T * (xy) be the number of transitive triangles C of T such that x is the source and y is the sink of C. Let x 0 y 0 ∈ A such that C 3 (x 0 y 0 ) + T T * (x 0 y 0 ) is maximum. Claim 1 C 3 (x 0 y 0 ) + T T * (x 0 y 0 ) ≥ n−2 3 . Let C 3 (T ) and T T 3 (T ) be the number of directed triangles and transitive triangles in T , respectively. Observe that xy∈A (C 3 (xy) + T T * (xy)) = 3C 3 (T ) + T T 3 (T ) = n 3 + 2C 3 (T ), by an average argument there exists zw ∈ A such that Let j be the minimum integer such that B + j+1 = ∅. Observe that j ≤ log 2 (|B + | + 1) − 1 and that V ′ = {z j , z j−1 , . . . , z 0 , x 0 , y 0 } induces a transitive subtournament of T such that for every w ∈ V \ (B − ∪ V ′ ) there is a pair {x, y} ⊆ V ′ such that {xw, wy} ⊆ A.
Theorem 11. Let T be a tournament of order n ≥ 3. Hence, Proof. Given any complete vertex coloring Γ of T , it follows that there is no pair of colors i, j such that |Γ −1 (i)| = |Γ −1 (j)| = 1, otherwise either there is no Γ(i)Γ(j)-arc or there is no Γ(j)Γ(i)-arc. Thus, except for at most one color, every color appear at least twice in V . Therefore, dac(T ) ≤ ⌈ n 2 ⌉. For the lower bound, V j ], it follows that in T there are V i V j -arcs and V j V i -arcs, and therefore Γ is a complete coloring of T . On the other hand, by Lemma 10, there is a partition 3 ) and the result follows.
Let Z 2m+1 be the cyclic group of integers modulo 2m + 1 (m ≥ 1) and J a nonempty subset of Z 2m+1 \ {0} such that |{−j, j} ∩ J| = 1 for every j ∈ J. The circulant tournament  Proof. Let ς be a dc(T )-coloring and x the order of the largest chromatic class i, therefore, n ≤ xdc(T ). By Corollaries 6 and 12, we obtain x 2 ≤ dac(T ) because the subdigraph induced by ς −1 (i) is a transitive tournament.
Recall that a digraph is strongly connected if for every pair of vertices u and v there exist a directed u − v walk. The strongly components of a digraph D form a partition ∼ into subdigraphs that are themselves strongly connected. Such partition induces the digraphD := D/ ∼. Theorem 14. [9] If T is a tournament with (exactly) k strong components, thenT is the transitive tournament of order k. . . , r + k} and color the remaining vertices using a greedy coloration. The coloring is complete since for the pair of distinct colors (i, j), with i, j ≤ r, there are at least an arc (x i , x j ) or (y i , y j ) such that x i or y i has the color i and x j or y j has the color j; and for each integer j, with r < j ≤ r + k and i < j there are at least the arcs (y i ′ , y k+j ) and (y k+r+1−j , y i ′ ) such that y i ′ has the color i and the vertices y k+j and y k+r+1−j have the color j. Let F be a digraph. The digraph D is F -free if D has no subdigraph isomorphic to F . A tournament F is a hero if and only if there exists c > 0 such that every F -free tournament D has a transitive subset of cardinality at least c|V (D)|, see [6]. Let us say a tournament F is a hero, if there exists c ′ (depending on F ) such that every F -free tournament has dichromatic number at most c ′ . Corollary 18. If T is a hero then dac(T ) ∈ θ(n).
In the following sections, we generalize some results on graphs, given in Chapter 12 of [10], for digraphs; in Section 4, we define the concept of dihomorphisms and show that the interpolation property does hold for complete acyclic colorings; and in Section 5, we establish the Nordhaus-Gaddum relations.

The interpolation theorem
is an elementary dihomomorphism from D to D ′ . The dihomomorphic image ǫ(D) of a digraph D obtained from an elementary dihomomorphism ǫ is also referred to as an elementary dihomomorphic image. Not only is D ′ a dihomomorphic image of D, a digraph F is a dihomomorphic image of a digraph D if and only if F can be obtained by a sequence of elementary dihomomorphisms beginning with D.
The fact that each dihomomorphic image of a digraph D can be obtained from D by a sequence of elementary dihomomorphisms tells us that we can obtain each dihomomorphic image of D by an appropriate partition In particular, if the coloring ς is a complete k-coloring, then F is the complete symmetric digraph of order k. Therefore, if a digraph F is a dihomomorphic image of a digraph D, then there is a dihomomorphism φ from D to F and for each vertex v in F , the set φ −1 (v) of those vertices of D having v as their image is acyclic in D. Consequently, each coloring of F gives rise to a coloring of D by assigning to each vertex of D in φ −1 (v) the color that is assigned to v in F . For this reason, the digraph D is said to be F -colorable. This provide us the following remark: Proof. Suppose that ǫ identifies the nonadjacent vertices u and v of D. We have already noted the inequality dc(D) ≤ dc(ǫ(D)). Let dc(D) = k and consider a k-coloring ς of D. Define a coloring ς ′ of ǫ(D) by Since ς ′ is a (k + 1)-coloring of ǫ(D), dc(ǫ(D)) ≤ k + 1 = dc(D) + 1.
Clearly, if there exists a dc(D)-coloring of D in which u and v are assigned the same color, we have that: Conversely, if dc(ǫ(D)) = dc(D) = k. Let ς ′ be a k-coloring of ǫ(D), then the coloring of D that assign to ǫ −1 (x) the color of x is a coloring with the required property.
Beginning with a noncomplete digraph D, we can always perform a sequence of elementary dihomomorphisms until arriving at some complete graph. As we saw, a complete graph K k obtained in this manner corresponds to a complete k-coloring of D. Consequently, we have the following.
Corollary 21. The largest order of a complete graph that is a dihomomorphic image of a digraph D is the diachromatic number of D.
The following theorem is a generalization of The Homomorphism Interpolation Theorem due to Harary, Hedetniemi, and Prins [14] and is an immediate consequence of Theorem 20 (see also [10]). For every noncomplete digraph D, there is an elementary dihomomorphism ǫ of D such that dac(ǫ(D)) = dac(D).
Suppose that dac(D) = k. Hence, there exists a sequence Therefore, dac(D i ) = k for all i ∈ [t] and the result follows.

Nordhaus-Gaddum relations
The Nordhaus-Gaddum Theorems [13,23] state for every graph G of order n. For digraphs, we have the following results: We have seen that for every digraph D of order n (see Equation 1), With the aid of Theorem 20, we show that dac(D) can never be closer to n than to dc(D).
The following result is the analogue of Theorem 20 for complementary graphs. Proof. We first show that dc(D c )−1 ≤ dc(ǫ(D) c ). Let dc((ǫ(D)) c ) = k for some elementary dihomomorphism ǫ of D that identifies two nonadjacent vertices u and v in D, where the vertex in ǫ(D) obtained by identifying u and v is denoted by v. Let there be given an k-coloring of ǫ(D) c . We may assume that the vertex v in ǫ(D) is assigned the color k. Remember that, if v and a vertex w in ǫ(D) c lie on a (directed) common cycle, w has a different color to k. Assign to each vertex in D c distinct from u the same color assigned to that vertex in ǫ(D) c and assign u the color k + 1. Since no vertex in D c on a common cycle with v is assigned the color k, this produces an (k + 1)-coloring of D and so dc(D c ) ≤ k + 1 = dc(ǫ(D) c ) + 1.

Conclusions and Future work
As we state in the introduction, the diachromatic number generalizes the achromatic number and as it turned out, several classic results for the achromatic number are extended to results for the diachromatic number. To future work we propose extend other known results to the diachromatic number as well as to study proper results for digraphs, and study the diachromatic number in fixed families of digraphs or oriented graphs. For instance, inspired in the definition of the dichromatic number of a graph, the diachromatic number of a graph can be defined as the minimum diachromatic number over all possible orientations of the graph. On the other hand, we also propose study the diGrundy number, defined as the the maximum positive integer k for which a digraph D has a greedy k-coloring, extending the results of undirected graphs to directed and oriented graphs and find this parameter for some families of digraphs. Also, we wish to improve the Nordhaus-Gaddum bounds or to find an infinite family of graphs to establish the sharpness of such bound.