On Panchromatic Digraphs and the Panchromatic Number

Graphs and CombinatoricsDOI 10.1007/s00373-013-1367-z

ORIGINAL PAPER

On Panchromatic Digraphs and the PanchromaticNumber

Hortensia Galeana · Ricardo Strausz

Received: 8 November 2011 / Revised: 20 August 2013© Springer Japan 2013

Abstract Let D = (V, A) be a simple finite digraph, and let π(D), the panchromaticnumber of D, be the maximum number of colours k such that for each (effective, oronto) colouring of its arcs ς : A → [k] a monochromatic path kernel N ⊂ V , asintroduced in Galeana-Sánchez (Discrete Math 184: 87–99, 1998), exists. It is nothard to see that D has a kernel—in the sense of Von Neumann and Morgenstern(Theory of games and economic behaviour. Princeton University Press, Princeton,1944)—if and only if π(D) = |A|. In this note this invariant is introduced and someof its structural bounds are studied. For example, the celebrated theorem of Sandset al. (J Comb Theory Ser 33: 271–275, 1982), in terms of this invariant, settles thatπ(D) ≥ 2. It will be proved that

π(D) < |A| ⇐⇒ π(D) < min{

2√

χ(D), χ(L(D)), θ(D) + max dc(Ki ) + 1}

,

where χ(·) denotes the chromatic number, L(·) denotes the line digraph, θ(·) denotesthe minimum partition into complete graphs of the underlying graph and dc(·) denotesthe dichromatic number. We also introduce the notion of a panchromatic digraphwhich is a digraph D such that for every k ≤ |A| and every k-colouring of its arcs, ithas a monochromatic path kernel. Some classes of panchromatic digraphs are furthercharacterised.

H. Galeana · R. Strausz (B)Instituto de Matemáticas, Universidad Nacional Autónoma de México, CiudadUniversitaria, 04510 Mexico, D.F., Mexicoe-mail: [email protected]

H. Galeanae-mail: [email protected]

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Graphs and Combinatorics

Keywords Digraphs · Kernels · Edge colourings · Chromatic number ·Panchromatic number · Dichromatic number · Line Digraph

1 Introduction

Let D = (V, A) be a finite simple digraph; that is, a finite set V endowed with a nonempty irreflexive relation A ⊂ V × V —no loops, nor parallel arcs are considered;and all walks, paths and cycles are to be considered directed. A stable set N ⊆ V ofits vertices is said to be a kernel of D if for any other vertex v ∈ V \ N there is an arcwhich absorbs it into N ; that is, there exists a vertex u ∈ N and the arc vu ∈ A (cf.Von Neumann and Morgenstern’s [11]).

Now, consider a k-colouring of the arcs ς : A → [k] (here, and in the sequel,all colouring are surjective functions). A subset of the vertices N ⊆ V is said tobe ς -stable—or ς -independent—if there is no monochromatic path between anypair of vertices of N in D. It is ς -absorbent if, for each vertex v ∈ V \ N thereis a monochromatic vu-path, for some u ∈ N . If N is both ς -independent andς -absorbent, it is said to be a monochromatic path kernel—or simply an mp-kernel(cf. Galeana-Sánchez’s [6]).

Kernels in digraphs have found many applications in different topics; interestingsurveys can be found in [3,4]. Clearly, the concepts of ς -absorbency, ς -independenceand mp-kernel in edge-coloured digraphs are natural generalisations of previouslyknown concepts; their study starts with the nice theorem of Sands et al. [10] whichguarantees the existence of an mp-kernel in every 2-coloured digraph (see below).These concepts have been deeply and widely studied by several authors, for examplesee [5–10].

We say that the digraph D is panchromatic if for every 1 ≤ k ≤ |A| and everyeffective (onto) k-colouring ς : A → [k] of its arcs, D has an mp-kernel. For example,as the reader may easily verify (see Proposition 1), if D is { C3, P3}-free, then D ispanchromatic (here and in the sequel, C3 and P3 denote the cyclic orientation of thetriangle and the directed path of length 3, respectively). Clearly, if D is panchromatic,then it has a kernel.

After the preliminaries of Sect. 2, in Sect. 3 we study some classes of panchromaticdigraphs. In particular we show that

• Acyclic digraphs are panchromatic, and• A quasi-transitive digraph is panchromatic if (and only if) it has a kernel.

Let us now introduce the following new parameter. The panchromatic number π(D)

is the maximum number k such that in each k-colouring of the arcs A, we can find anmp-kernel N ⊆ V . With this definition at hand, the celebrated theorem of Sands et al.[10], which motivated such a definition, settles that

• π(D) ≥ 2 for any digraph D (with at least two arcs).

At the other extreme, it is easy to see that

• π(D) = |A| if and only if D has a kernel, and• π(D) = |A| − 1 if and only if D = C3 is the directed triangle.

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Fig. 1 A digraph with γ̌ = 3,γ̂ = 6 and π = 9

In a sense, the panchromatic number allow us to determine when a digraph is notpanchromatic, however the behaviour of the panchromatic number can be quite erratic.To illustrate this, let us introduce two other parameters: the gap numbers γ̌ (D) andγ̂ (D) are respectively the minimum and the maximum (number of colours) k, if any,such that there exists a k-colouring of the arcs A without an mp-kernel. For example,in Fig. 1 we show a digraph G with γ̌ (G) = 3, γ̂ (G) = 6 and π(G) = 9; that is, forall colourings of the arcs with 1, 2, 7, 8 or 9 colours there are mp-kernels, but thereare colourings with 3, 4, 5 and 6 colours with out them.

We found very interesting a question proposed by one of our anonymous refereeswhich ask: does there always exist an interval for which mp-kernels always exist,followed by an interval where they don’t (with endpoints the two gap functions intro-duced), then an interval for which they do (possibly empty in the last two cases?

Section 4 is devoted to studying more closely the class of quasi-transitive digraphs(which includes semi-complete digraphs and tournaments). In particular we show that

• A quasi-transitive digraph D has no kernel (if and) only if π(D) = 2, and therefore• A quasi-transitive digraph D is panchromatic if (and only if) π(D) > 2.

In Sect. 5 some structural bounds for the panchromatic number are exhibited;namely, we show that

• π(D) < |A| ⇐⇒ π(D) < min{2√

χ(D), χ(L(D)), θ(D) + max dc(Ki ) + 1},

and• If D = ∑

G Gv then π(D) < |V (G)| + min {π(Gv) : v ∈ V (G)},where χ(D) denotes the chromatic number, L(D) denotes the (directed) line graph,θ(D) denotes the minimum number w of a w-partition into complete graphs of theunderlying graph of D, the maximum runs over all complete graphs Ki of such adecomposition and dc(Ki ) denotes their dichromatic number; finally,

∑G Gv denotes

the Zykov (or lexicographic) sum of the digraphs Gv , one for each vertex v ∈ V (G)

of the base digraph G.We close with some remarks and an intriguing open problem on Sect. 6.

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2 Preliminaries

In the sequel, D = (V, A) will be a digraph with finite vertex set V and edge setA ⊂ V × V ; we do not consider loops, nor parallel arcs. The order and the size of Dwill be denoted by the letters n = |V | and m = |A|, respectively. We denote by Cn

and Pm the cyclic orientation of the cycle of order n and the oriented path of size m,respectively.

A digraph D is said to be transitive, respectively quasi-transitive, if for every(directed) path P2 = (u, v, w) in it, there exist the arc uw ∈ A, respectively one ofthe two arcs uw ∈ A or wu ∈ A. It is said to be semi-complete if every pair of verticesu, v ∈ V are adjacent; that is, either uv ∈ A or vu ∈ A.

Given D and a family of mutually disjoint digraphs {Hv : v ∈ V (D)} the Zykovsum

∑D Hv is the union of those digraphs on the family where the arcs xy are added

whenever x ∈ V (Hu), y ∈ V (Hv) and uv ∈ A(D).We denote by χ(D), as usual, the chromatic number of D; i.e., χ(D) is the min-

imum number of colours k for which there exists a k-colouring of its vertices witheach chromatic class being a stable (or independent) set. Also, L(D) will denote the(directed) line graph of D; that is, its vertices are the arcs of D and its arcs are itsdirected walks W2 of size equal to two (two consecutive arcs).

We denote by τ(D) the minimum number of parts in a partition of D into transi-tive induced subdigraphs. Analogously, we denote by θ(D) the minimum number oftournaments in which the vertices of the digraph can be partitioned.

Let [k] := {1, 2, . . . , k} be the canonical k-set (of colours). The digraph D is saidto panchromatic if for every k ≤ m and every k-colouring ς : A → [k] of its arcs,we can find an mp-kernel; that is, a stable set N ⊂ V with no monochromatic pathbetween any two of its vertices, and such that for each other vertex v ∈ V \ N thereexists a monochromatic vu-path with u ∈ N . Recall that our colouring are meant tobe surjective.

The panchromatic number π(D) is the maximum k such that in each k-colouringwe can find an mp-kernel. Clearly 2 ≤ π(D) ≤ m and π(D) = m if and only if Dhas a kernel in the classical sense [11] (the lower bound is Lemma 1 below [10]).

Given a subset of vertices U ⊂ V , we denote by N+(U ) := {uv ∈ A : u ∈U and v ∈ V \ U } the outer-neighbourhood of it; analogously, N−(U ) denotes theinner-neighbourhood. Observe that we are deviating a bit of the usual terminologybecause we focus our attention on the arcs and not in the vertices.

Given a coloured digraph D = (V, A), we denote by C(D) its monochromatic pathclosure; that is, the digraph which results from D by adding an edge e = uv ∈ V × Vwhenever there exists a monochromatic path from u to v, and colouring e with thecolour of such a path. Observe that C(D) may have parallel arcs—this is the only casewhere we consider parallel arcs.

Finally, in the sequel the following facts will be used.

Lemma 1 (Sands et al. [10]) Every 2-coloured digraph has an mp-kernel.

Lemma 2 (Galeana et al. [5]) Let D be a coloured digraph. D has an mp-kernel ifand only if C(D) has a kernel.

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Fig. 2 The obstruction of BangJensen

Lemma 3 (Berge-Duchet [2]) Let D be a digraph. If each cycle of D has a symmetricarc, then D has a kernel.

Lemma 4 (Bang Jensen [1]) Let D be a quasi-transitive digraph. If there is a pathfrom u to v, then either u and v are adjacent, or the graph in Fig. 2 is an inducedsubgraph of D.

3 Some Panchromatic Digraphs

Let us warm up by proving that

Proposition 1 A { C3, P3}-free digraph D = (V, A) is panchromatic.

Proof Let us start with a 3-colouring ς : A → {r, g, b} of D; we refer to the coloursas red, green, and blue, respectively. Now, let us identify two chromatic classes, sayred and blue become purple, to form a 2-colouring ς ′ : A → {p, g} of D. Due toLemma 1, we know there is an mp-kernel K ⊂ V in this 2-colouring. If K is an mp-kernel for ς we are done, so let us suppose it is not. Observe that the independence ofK is inherited from ς ′ to ς—i.e., U ⊂ V is ς -stable whenever it is ς ′-stable—so, wejust have to check the obstructions of K to being ς -absorbent.

Let u ∈ V be a vertex which is not absorbed by K in ς . Since K is ς ′-absorbent,there must be a purple path P from u to K . The length of P has to be 2; it clearlycannot be 1, and it cannot be 3 (or greater) because D is P3-free. So, let us supposeP = uvw is that path. We may suppose that K ∪{u} is not ς -stable. Then, either thereis a w′ ∈ K and a w′u-path, or wu is an edge. The last case is forbidden since D isC3-free, and the former induces a P3 which is also forbidden. Thus we conclude thatK —or, K ∪ {u1, . . . , u�}, for some �—is ς -absorbent and therefore an mp-kernel.Furthermore, since green arcs did not play a role, we can argue in the same fashionwithin any k-colouring (2 < k ≤ |A|) and therefore D is panchromatic. ��

As a direct consequence of Lemmas 2 and 3 we have that

Proposition 2 Acyclic digraphs are panchromatic.

Proof Let D be an edge-coloured acyclic digraph (without directed cycles). Clearly,C(D) has no directed cycles either. Then, due to Lemma 3, C(D) has a kernel and,due to Lemma 2, D has an mp-kernel. ��

The following is a nice characterisation that uses Bang Jensen’s obstruction:

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Proposition 3 A quasi-transitive digraph has a kernel if and only if it is panchromatic.

Proof For the non-trivial side, let N ⊂ V be a kernel of D. Clearly N is ς -absorbent forall colourings ς , so we have to check only the obstructions to N being ς -independent.For, let us suppose there is a monochromatic uv-path, with both u and v in N , andsuppose this is of minimum length. Due to Lemma 4, we may suppose that the digraphof Fig. 2 is a part of D, with x, y �∈ N .

Since N is a kernel, there must be an arc from x to N , say xw. If w �∈ {u, v} then uand w must be adjacent which contradicts the fact that N is independent. Otherwise,u and v must be adjacent, leading also to a contradiction. ��Since tournaments are quasi-transitive, it follows that

Corollary 4 A tournament has a kernel if and only if it is panchromatic.

Let us close this section by exhibiting a colouring that we will need, and generalise,in Sect. 4, and a nice criterion to determine if a tournament (or semi-complete digraph)has a kernel.

Proposition 5 A tournament T (with more than two vertices) is panchromatic if andonly if π(T ) > 2.

Proof For the non-trivial side, suppose that T is not panchromatic, and therefore ithas no kernel; we will prove that π(T ) = 2. Let B denote the vertices of a maximaltransitive tournament inside T (B is not V (T ) because T has no kernel). Observe that,if b ∈ V (B) is the sink of such a transitive tournament, since B is maximal, there mustbe a vertex u �∈ V (B) and the arc bu ∈ A(T ) and another vertex v ∈ V (B) with thearc uv ∈ A(T )—since otherwise it absorbs the whole set.

Now, assign colour red to the arcs in N+(B), green to the arcs in N−(B), andblue to all the rest A(T ) \ (

N+(B) ∪ N−(B)). From here, it is not hard to see that

T , endowed with such a colouring, does not have an mp-kernel (just recall that, sincewe are dealing with a tournament, an mp-kernel must consist of a single vertex, andno single vertex absorbs simultaneously B and v); this proves that π(T ) �= 3. Wecan now simply subdivide the chromatic classes to show that π(T ) �= k for eachk ∈ {4, 5, . . . , |A(T )|}, concluding the proof. ��It follows immediately that

Corollary 6 A tournament has a kernel if and only if for every 3-colouring of its arcs,it has an mp-kernel.

A completely analogous argument shows also that

Proposition 7 A semi-complete digraph K is panchromatic if and only if π(K ) > 2.

4 On Quasi-Transitive Digraphs

Proposition 8 An asymmetric quasi-transitive digraph has a kernel if and only ifπ(D) > 2.

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Proof Let us do this by induction. The smallest asymmetric quasi-transitive digraphwithout a kernel is C3 and clearly π( C3) = 2. Now, let us suppose D is an asymmet-ric quasi-transitive digraph without a kernel and suppose that every smaller digraphsatisfies the theorem. We now proceed with a case-by-case analysis.

Case 1: D is strongly connected. Due to a result of Bang-Jensen and Gutin (c.f. [1,Theorem 4.8.5]) we have that

D =∑

K

Di ,

where K is a semi-complete strongly connected digraph and each Di is quasi-transitivenot strongly connected. We have now two possibilities:

Case 1.1: K has a kernel. Since K is semi-complete, a kernel of it must consist of asingle vertex; furthermore, since D is asymmetric, K is also asymmetric and its kernelis unique, say ν ∈ V (K ) (the unique vertex with empty outer-neighbourhood). Let Dν

be the term of D = ∑K Di corresponding to such a vertex. Clearly, Dν has no kernel

(recall that D has no kernel). Since Dν is not strongly connected, |V (Dν)| < |V (D)|(i.e., the sum has at least two terms). It follows from the inductive hypothesis that foreach k such that 3 ≤ k ≤ |A(Dν)|, there is a k-colouring of Dν without an mp-kernel;consider such a colouring and extend it arbitrarily to D.

Claim: With such a colouring, D has no mp-kernel. Seeking a contradiction, supposethat D has an mp-kernel, say N ⊂ V (D). Observe that, since ν is the kernel ofK , which is semi-complete and asymmetric, it follows that N+(V (Dν)) = ∅ andtherefore, N ∩ V (Dν) �= ∅. Furthermore, N ∩ V (Dν) must be an mp-kernel of Dν ,yielding a contradiction and proving the claim.

Case 1.2: K does not have a kernel. Due to Proposition 7, a semi-complete graphK without a kernel has panchromatic number π(K ) = 2 and therefore for every k,with 3 ≤ k ≤ m, there is a k-colouring without an mp-kernel.

Fix a colouring of K without a mp-kernel. We define a colouring of D as follows:for each coloured arc uv ∈ A(K ) in K , we paint all arcs from Du to Dv with thesame colour; the internal arcs in Du and Dv are coloured arbitrarily—we have tobe careful here since it may be the case that K did not used all the k colours (itmay have less than k arcs); we may use those unused colours inside some of theaddends.

Claim: With such a colouring, D has no mp-kernel. If D admits an mp-kernelN ⊂ V (D), since we are adding over the semi-complete digraph K , there exists aterm Dν such that N ⊂ V (Dν). But then, the vertex ν ∈ V (K ) would be an mp-kernelof K contradicting our assumption.

Case 2: D is not strongly connected. Due to a result of Bang-Jensen and Gutin (c.f.[1, Theorem 4.8.5]) we have that

D =∑

T

Di , (*)

where T is transitive and acyclic, and each Di is quasi-transitive and strongly conn-ected—indeed, T is the condensation digraph of D.

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Since D has no kernel, there exists a terminal component of D such that it hasno kernel, say Dν (a strongly connected component with no arc going out of it; cf.[1, p.16]). Such a component is clearly a term of the sum (∗). Since Dν is terminal,N+(Dν) = ∅. Since Dν is asymmetric, quasi-transitive, strongly connected withouta kernel, by induction we have that π(Dν) = 2 and there is a k′-colouring, for eachk′ such that 3 ≤ k′ ≤ |A(Dν)|, which we extend arbitrarily to D—just being carefulto use all k colours.

Claim: With such a colouring, D has no mp-kernel. Suppose N ⊂ V (D) is anmp-kernel. Since Dν is a terminal component, N ∩ Dν is an mp-kernel of Dν whichcontradicts the way we coloured Dν.

Now, observe that for each k such that |A(Dν)| ≤ k ≤ |A(D)| we can colourD in such a way that Dν receives k′ colours while the rest of the digraph uses thek − k′ remaing colours arbitrarily. Such a colouring will not admit an mp-kernel.

��

5 Structural Upper Bounds

Proposition 9 Let D = (V, A) be a digraph of size m = |A|. Then

π(D) < m ⇐⇒ π(D)

< min{

2√

χ(D), χ(L(D)), 2√

τ(D) + 1, θ(D) + max dc(Ki ) + 1}

.

Theorem 9 is an immediate consequence of the following lemmas.

Lemma 5 A digraph D has no kernel if and only if π(D) < 2√

χ(D).

Proof Let k = χ(D). Clearly, if π(D) < k then π(D) < m and therefore D has nokernel. For the converse, suppose that D has no kernel and let ς : V → [k] be a propercolouring of its vertices. Let us denote by Vi = ς−1(i), for i ∈ [k], its chromaticclasses.

We may suppose that each chromatic class has an arc going out of it—the othercase is analogous. Now, we colour the arcs of D with k colours letting each arc receivethe colour of its starting vertex; that is, if uv ∈ A is an arc in the outer-neighbourhoodN+(Vi ), then it receives the colour i . Since the colouring ς is proper, each Vi is stable,and all arcs receive a colour. Denote by Ai , for i ∈ [k], the edge-chromatic classes ofsuch a colouring. Clearly, this colouring does not add structure to the monochromaticpaths of D—there are no two arcs with the same colour following one another—therefore, since D has no kernel, D has not mp-kernel with such a colouring. Thisproves that π(D) �= k.

We can now subdivide an edge-chromatic class, say Ai = Ai1 ∪ Ai2 , in two non-empty classes finding a k + 1 colouring without an mp-kernel. This would prove thatπ(D) �= k + 1. We can proceed up with this and find a colouring for each numberbetween k and m without mp-kernels, proving that π(D) �= �, for � = k, . . . , m, andtherefore π(D) < χ(D).

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We can now improve this colouring as follows. Let � < k be a natural number anddefine a partition of the set {Vi : i ∈ [k]} with each part of order “nearly” k/�, say

U1 = {V1, V2, . . . , V�k/��},U2 = {V�k/��+1, V�k/��+2, . . . , V2�k/��},

...

U� = {V(�−1)�k/��+1, V(�−1)�k/��+2, . . . , Vk}.

Denote by Ai = {uv ∈ A : u, v ∈ ⋃Ui } the set of arcs completely contained in part

Ui , for i ∈ [�]. Each part in such a partition can now be coloured with �k/�� coloursand we need � extra colours to those arcs “joining” the parts. That is, we use a box of�k/�� colours, say inks, to assign to the arcs going within each class (e.g., the arcs inN+(V1)∩ A1 can be coloured, as those of N+(V�k/��+1)∩ A2, with ink colour 1) andwe use a second box of � colours, say of oils, to assign to those arcs which go from onepart to another (e.g., the arcs in N+(U1) can be coloured with oil colour 1). It is nothard to see that such a colouring does not change the monochromatic-paths structureof D providing a colouring without mp-kernels and proving that π(D) �= �k/�� + �.Since this can be done for each � = 1, . . . , k, we can minimise the previous functionof � concluding that π(D) < 2

√χ(D). ��

Lemma 6 A digraph D has no kernel if and only if π(D) < χ(L(D)).

Proof One direction of the statement is obvious. For the other, let k = χ(L(D)) andsuppose that D has no kernel and let ς : A → [k] be a proper colouring of its arcs.Let us denote by Ai = ς−1(i), for i ∈ [k], its chromatic classes.

As in the previous proof, this colouring does not add structure to the monochromaticpaths of D—since the colouring ς is proper, each Ai is stable and there are no two arcswith the same colour following one another—therefore, since D has no kernel, D doesnot have an mp-kernel with such a colouring. This proves that π(D) �= k. Subdividingthe chromatic classes up to reach the number of arcs we can find �-colourings, for� = k, . . . , m, without mp-kernels and proving the lemma. ��This argument can be generalised in the following result; however the bound is a bitweaker:

Lemma 7 A digraph D has no kernel if and only if π(D) < 2√

τ(D) + 1.

Proof For the non-trival side, let D be a digraph without a kernel. Let V = V1 ∪ · · · ∪Vτ(D) be a minimal partition in induced transitive subgraphs. As before, each outer-neighbourhood N+(Vi ) will receive a different colour and we use a common colourfor the interior arcs of all parts Vi , i = 1, . . . , τ (D), giving us a count of τ(D) + 1colours. To see that such a colouring does not admit a mp-kernel, observe that thereis a monochromatic uv-path if and only if uv ∈ A(D). Therefore, D has a mp-kernelif and only if it has a kernel.

An argument analogous to that in the previous propositions concludes the proof.��

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Lemma 8 A digraph D has no kernel if and only if π(D) < θ(D)+ max dc(Ki )+ 1.

Proof We need only to exhibit the colouring without mp-kernel and the rest of thearguments are as before, so we skip them. First, let us partition G in θ(G) disjointtournaments. To each tournament in the partition Ti , i = 1, . . . , θ(G), we assigna different colour and we give that colour to each of the arcs that goes from thattournament to another (giving us a count of at most θ(G) colours so far). Now, thearcs inside each tournament can be coloured with max dc(Ki )+1 since in a tournamenta partition in acyclic subgraphs is also a partition in transitive ones (cf. Lemma 7).Observe that in this colouring there is a monochromatic path if and only if there is anedge joining its ends; and therefore there is a mp-kernel if and only if there is a kernel.

��Proposition 10 Let D = ∑

v∈V (G) Dv be a Zykov sum over any digraph G ofdigraphs Dv without kernel. Then, D has no kernel if and only if π(D) < |V (G)| +max{π(Dv)} + 1.

Proof Let us suppose D has no kernel and let k = max{π(Dv)}+1. Since each Dv hasno kernel and π(Dv) ≤ k, there exist a k-colouring of Dv without an mp-kernel. Let usfix such a colouring in each term of the sum. We now colour the remaining arcs using|V (G)| different colours as follows: for each v ∈ V (G) and each u with vu ∈ A(G),we colour all arcs from Dv to Du with the colour v. We claim that such a colouringhas no mp-kernel. For, suppose it has an mp-kernel K ⊂ V (D). Let Kv := K ∩ Dv bethe intersection of the kernel with each of the terms. Let Kv be a non empty “piece”of the mp-kernel. Since Dv , has no mp-kernel, there should exist a vertex w ∈ Dv

which is not absorbed in Kv . Therefore there should exist a u ∈ V (G) such that Ku isnon-empty and it absorbs w. Due to the nature of this colouring, this implies that thereis an arc from w to a vertex in Du and therefore vu is an arc of G which contradicts theindependence of K . As in previous arguments, this colouring can easily be subdividedto prove that there are colourings for each k′ ≥ k without mp-kernel. ��

6 Remarks and an Open Problem

So far, in all examples we have dealt with, the panchromatic number is either 2 or themaximum possible (the number of arcs). This motivates us to propose the following

Conjecture ∀D : π(D) ∈ {2, |A(D)|}Acknowledgments We would like to thank the reviewers who carefully read the manuscript and helpedus to improve the presentation of this paper.

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On Panchromatic Digraphs and the Panchromatic Number

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