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The Caesarea Edmond Benjamin de Rothschild Foundation Institute ..... and Eji. ∈. ),(. Consider the following linear program: I. minimize ∑. = n ji jiji xc. 0,.
An Extension of the Greene and Greene-Kleitman Theorems to all Digraphs Irith Ben-Arroyo Hartman The Caesarea Edmond Benjamin de Rothschild Foundation Institute for Interdisciplinary Applications of Computer Science, University of Haifa Haifa 31905, Israel e-mail:[email protected]

Abstract Let G be a directed graph, and k a positive integer. We prove that there exists a kcolouring that is orthogonal to every k-optimal partition of V(G) into paths and cycles. This extends the Greene-Kleitman Theorem to all digraphs, and relates to Berge’s conjecture on path partitions and k-colourings. We also show that there exists a colouring that is orthogonal to every optimal collection of k disjoint paths and an arbitrary number of cycles, thus extending Greene’s Theorem to all digraphs. We conclude with some conjectures.

1. Introduction Let G = (V , E) be a directed graph. A path P in G is a sequence of distinct vertices ( v1 , v 2 ,K , v l ) such that ( vi , vi +1 ) ∈ E , for i = 1,2,K , l − 1 . The set of vertices {v1 , v 2 ,K , v l } of a path P = (v1 , v 2 , K, vl ) is denoted by V (P) . The cardinality of P, denoted by P , is the number of vertices in P, i.e. V (P) . A family P of paths is called a path partition of G if its members are vertex disjoint and ∪ {V ( P ) : P ∈ P } = V . For each positive integer k, the k-norm P k of a path partition P = {P1 , P2 ,K , Pm } is defined by m

P k = ∑ min{ Pi , k } . i =1

A partition which minimizes P

k

is called k-optimal. For example, a 1-optimal path

partition is a partition which contains a minimum number of paths. Denote by π k (G ) the k-norm of a k-optimal path partition in G. A k-colouring is a family C = {C1 , C2 ,K, Ck } of k disjoint independent sets C i called colour classes. (Some of the colour classes may be empty). The cardinality of a k

1

k-colouring C = {C1 ,C 2 ,K, Ck } is C k = ∑i =1 C i and C is said to be optimal if k

k

k

Ck

is as large as possible. Denote by α k (G ) the cardinality of an optimal kcolouring in G. The following theorem by Greene and Kleitman [10] refers to partially ordered sets (posets), or equivalently, to transitive and acyclic digraphs. Theorem 1.1 (Greene-Kleitman[10]): Let G be a transitive acyclic digraph, and let k be a positive integer. Then α k (G ) = π k (G) . For k = 1 Theorem 1.1 is known as Dilworth's Theorem [5]. If G is an acyclic digraph (and not necessarily transitive) then α k (G) ≥ π k (G ) as was shown in [13]. In fact, stronger theorems hold in this case. For k = 1 Theorem 1.1 is generalized for all digraphs by the Gallai-Milgram Theorem [8] which states that α 1 (G) ≥ π 1 (G) . For k > 1 and general digraphs only few partial results are known. We suggest in Section 2 a possible generalization of Theorem 1.1 for all digraphs and all k which implies the known existing results. The notions of path partition and k-colouring can be 'dualized' by interchanging the roles of 'path' and 'independent set' in the definitions above, in the following way: A colouring C is a partition of V into disjoint independent sets. For each positive integer k, the k-norm C k of a colouring C = {C1 , C 2 ,K, C m } is defined as

= ∑ min {Ci , k } m

C A colouring which minimizes C

k

k

i =1

is called k-optimal, and the k-norm of a k-optimal

colouring is denoted by χ k . A k-path is a family P k = {P1 , P2 , K, Pt } of at most k disjoint paths Pi . The cardinality of a k-path P k = {P1 , P2 , K, Pt } is P k = ∑i=1 Pi , and P t

Pk

k

is optimal if

is as large as possible. Denote by λ k (G ) the cardinality of an optimal k-path

in G. The following result is analogous to theorem 1.1, though its proof is unrelated to the proof of Theorem 1.1. Theorem 1.2 (Greene[9]): Let G be a transitive acyclic digraph, and let k be a positive integer. Then λ k (G ) = χ k ( G ) . If G is an acyclic digraph then λ k (G ) ≥ χ k (G ) as was shown in [12] and [13] , and in fact, stronger theorems are known in this case (see [1]). For k = 1 the Gallai-Roy Theorem [7], [14] states that λ1 (G ) ≥ χ 1 (G ) for all digraphs G, and only few partial

results are known for k ≥ 2 and general digraphs (see [11]). We propose in Section 3 an extension of Greene's Theorem for all digraphs, which implies Greene's Theorem as well as other known results on this problem.

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2. Extending Greene-Kleitman’s Theorem Linial [13] proposed to generalize the Greene-Kleitman Theorem to all digraph by the following conjecture: Conjecture 2.1 (Linial [13]): Let G be a directed graph and let k be a positive integer. Then α k (G) ≥ π k (G ) The conjecture holds, of-course, for transitive graphs (by the Greene-Kleitman Theorem) and for all acyclic graphs, as was shown in [13]. For general graphs little is known. It holds for k=1 by the Gallai-Milgram Theorem [8], and for few other cases which we will mention later. A natural extension of Greene-Kleitman’s theorem to all digraphs was suggested by a conjecture made by Berge[2]. Berge introduced the concept of orthogonality, (thou he used a different term). Definition 2.1 A path partition P = {P1 , P2 ,K , Pm } and a partial k-colouring C

k

are

k

orthogonal if every path Pi in P meets min{ Pi , k } different colour classes of C . Berge conjectured the following generalization of the Greene-Kleitman Theorem to all digraphs. Conjecture 2.2(Berge [2]): Given a graph G and a positive integer k, then for every koptimal path partition P there exists a partial k-colouring orthogonal to it. Conjecture 2.2 implies Conjecture 2.1 as follows: Every path Pi in a k-optimal path k

partition meets at least min{ Pi , k } vertices of some partial k-colouring C . Since a path partiton is a partition of the vertex set into disjoint classes, we have

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α k (G) = ∑P∈P C k ∩ P ≥ ∑ P∈P min {P , k } = π k (G ) Conjecture 2.2 holds for all acyclic digraphs as was shown in [1], [13], and [15]. We propose here a different generalization of the Greene-Kleitman Theorem. We shall redefine now the notions of a path partition, and its k-norm. Let G = (V , E ) be an arbitrary digraph. A cycle in G is a sequence of distinct vertices ( v0 , v1 ,K , v l−1 ) such that ( vi , v i+1 ) ∈ E for i = 0,1,2,K, l − 1 , where all indices are taken modulo l. Definition 2.2 : For a positive integer k, the k -norm of a path or cycle P, denoted by P k , is defined by

 min{ P, k } if P is a path Pk = 0 if P is a cycle  Definition 2.3: A path-cycle partition P c = {P1 , P2 ,K , Pm } is a set of vertex disjoint paths and cycles which cover V(G). The k-norm of a path-cycle partition is defined by

Pc

= ∑i=1 Pi k . Let π kc (G ) be the minimum k-norm of any path-cycle partition in m

k

G. The definitions above imply that cycles in P c do not contribute to its k- norm. c Furthermore, π k (G ) ≤ π k (G) for all G, and equality holds if G is acyclic. The following theorem extends Theorem 1.1 for all digraphs. Theorem 2.1: For all digraphs G, and positive integers k, α k (G) ≥ π k (G ) . c

We shall soon prove a stronger theorem, but first, we shall extend the concept of orthgonality to path-cycle partitions: Definition 2.4: A path-cycle partition P c and a partial k-colouring C are k

orthogonal if each P ∈ P c meets at least P k different colour classes of C . k

i

Again, if the graph is acyclic, then definition 2.4 coincides with definition 2.3 of orthogonality. The following theorem strengthens Theorem 2.1 for all digraphs. Theorem 2.2: Let G be a digraph, and let k be a positive integer. Then there exists a k

k-colouring C which is orthogonal to every k-optimal path-cycle partition of G. Theorem 2.2 implies Theorem 2.1 for the following reason:

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Let P c = {P1 , P2 ,K , Pm } be a k-optimal path-cycle partition, and let C colouring which is orthogonal to it. Then m

m

i =1

i =1

k

be a k-

α k (G) ≥ C k = ∑ Pi ∩ C k ≥∑ Pi k = P c k = π kc (G ) In the following section we prove Theorem 2.2, and hence also Theorem 2.1. Proof of Theorem 2.2: We shall use a linear program similar to the one defined in [1]. Let V = n , and for simplicity, assume that the vertices of G are labeled by 1,2,K, n . The variable set of the LP is {xij | (i, j ) ∈ I } where

I = {(i , j ) | 0 ≤ i , j ≤ n, i = 0 or j = 0 or (i, j ) ∈ E} Let C = ( ci , j ), (i , j ) ∈ I be defined by

ci , 0 = 0 for all 0 ≤ i ≤ n ; c 0, j = k for all 1 ≤ j ≤ n ;

c i,i = 1 for all 1 ≤ i ≤ n ; ci , j = 0 for i > 0 , j > 0 , i ≠ j and (i , j ) ∈ E Consider the following linear program: I. n

minimize n

∑x j= 0

n

∑x j= 0

i, j

=1

∑c

i , j =0 n

0, j

i, j

xi , j

= ∑ x i, 0 = n i= 0 n

∑x

for all i > 0;

i =0

i,j

= 1 for all j > 0

x i, j ≥ 0 for all pairs (i , j ) ∈ I

5

(1) (2) (3)

For a path-cycle partitionP c , let P 0 and P + denote the sets of paths in P of cardinality at most k, and at least k, respectively. Paths of cardinality k can be c

c

c

c

c

c

c

assigned arbitrarily to either P 0 or P + . Let Pc denote the set of cycles in P . For a set of paths P , let in[P ] (ter [P ] ) denote the set of initial (terminal) vertices of the paths in P . We are now ready to define the following matrix:

X (P c ) = ( x ij ) corresponding to P c :

x 0, 0 = n − P +c for j > 0 , x 0, j

1 if j is in in[P +c ] = otherwise 0

1 0

if i is in ter [P +c ] otherwise

for i > 0 , xi , 0 = 

1 i > 0, xi ,i =  0 1 i, j > 0, xi , j =  0

if i belongs to some path in P 0c otherwise if (i , j ) is in some P ∈ P +c ∪ P cc otherwise

Stated different ly, every path P = ( x1 ,K , x l ) of cardinality at least k in P

c

corresponds to the solution x 0,1 = x1, 2 = x 2, 3 = K = xl −1,l = xl , 0 = 1 . A path

P = ( x1 ,K , x t ) of cardinality t < k corresponds to the solution x1, 1 = x 2, 2 = x 3, 3 = K = x t ,t = 0 , and a cycle C = ( x1 , x 2 ,K , xl , x1 ) corresponds to the solution x1, 2 = x 2, 3 = K = xl −1,l = xl ,1 = 1. Since P is a path-cycle partition, the corresponding solution X defined above is indeed a feasible solution of I. c

Furthermore, the objective function of I computes P

c

. This follows from the fact

k

that each cycle contributes zero to the objective function, each path of length at least k contributes k to the objective function, and each path P of cardinality at most k contributes P to the objective function. Since the matrix of Equations (1) and (2) is totally unimodular, there exist integral optimal solutions of I. Hence, the converse holds as well, namely, every integral optimal solution of I corresponds to a k-optimal path-cycle partition of G. Consider the dual problem:

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II. n

n

i =1

j =1

maximize ∑ u i + ∑ v j + n (u 0 + v 0 ) where

u i + v j ≤ ci , j for pairs (i , j ) ∈ I

(4)

We may assume that there exists an integral optimal solution of II satisfying

u 0 = v 0 = 0, u i ≤ 0 and 0 ≤ v i ≤ k . We associate a partial k-colouring C following way. Let

k

(5)

= {C1 , C2 ,K, Ck } to such a solution in the

C r = {i > 0; u i + vi = 1 and vi = r} Note that for vertices i, j ∈ C r , 1 − u j = 1 − u i = v i = v j = r , implying that

u i + v j = 1. But if (i , j ) ∈ E then by Equation (4), u i + v j ≤ 0, hence Cr is an independent set for each r = 1, 2, K, k . , and thus C We shall now show that C

k

k

is indeed a partial k-colouring.

is orthogonal to every k-optimal path-cycle partition

P c in G. We shall use the complementary slackness conditions for I and II:

x i, j > 0 ⇒ u i + v j = ci , j

(6)

Let P be a “short” path, i.e. a path in P0 . Then for each i ∈ P, x i, i = 1, implying by c

(6) that u i + v i = 1, hence vertex i belongs to some Cr ∈ C . Furthermore, k

if (i , j ) ∈ E ( P ) , then by (4), u i + v j ≤ 0 . But vi + u i = 1 , hence v i ≥ v j + 1 , and the colours in P are strictly decreasing. c

Assume now that P is a “long” path, i.e. a path in P + . To simplify the notation, assume the path is P = (1,2,K, l ). Then

x 0,1 = x1, 2 = x 2, 3 = K = xl −1,l = x l, 0 = 1 (7) By (6) u l + v 0 = 0 , so by (5) u l = 0 . Similarly, by (6) u 0 + v1 = k , implying by (5) that v1 = k . For each i = 1,2,K, l − 1 by (6) u i + v i+1 = 0 and for each

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i = 1,2,K, l − 1 by (4), vi + u i ≤ 1 . It follows that for each r = 1,2,K, k there exist at least one vertex i on P such that u i + v i = 1, and vi = r , in other words, P meets k

all k colours in C . k

This completes the proof tha t C is orthogonal to every k-optimal path-cycle partition

Pc.

3. Extending Greene’s Theorem

Greene’s theorem states that for graphs of partially ordered sets and any given positive integer k, λ k (G ) = χ k (G ) . Linial [13]. made the following conjecture: Conjecture 3.1 (Linial [13]): For all digraphs G and positive integers k

λ k (G ) ≥ χ k (G ) . The conjecture holds for k=1 by the Gallai-Roy Theorem [7][14] , and for all acyclic digraphs ( see [13],[12]) Aharoni Hoffman and Hartman [1] suggested the concept of orthogonality between k-paths and colourings. Definition 3.1: A colouring C = {C1 , C2 ,K, Cm } and a k-path P

k

are orthogonal

if every colour class C i ∈ C meets at least min{ Ci , k} different paths of P

k

.

The following conjecture strengthens Linial’s conjecture: Conjecture 3.2 (Aharoni Hartman and Hoffman[1]) : Let G be a directed graph and let k be a positive integer. Then for every optimal k-path P orthogonal to P

k

k

there exists a colouring C

.

Conjecture 3.1 was proved in [1] for all acyclic digraphs, and in [11] for all bipartite graphs. For general graphs the conjecture is still open. We propose here a different extension of Greene’s Theorem, where instead of k-paths, a set of k paths and an arbitrary number of cycles is considered. k

Defintion 3.2 A k-cycle-path (or for brevity a k-c-path), denoted by Pc , is a disjoint k

collection of at most k paths, and an arbitrary number of cycles. Pc is optimal if

8

Pc = ∑i Pi is as large as possible, where the sum is taken over all members (paths k

and cycles) of Pc . Denote by λ k (G) the norm of an optimal k -c-path. k

c

k

Definition 3.3 A colouring C is orthogonal to a k-c-path Pc if each colour class

C i ∈C meets at least min{ Ci , k} paths in Pc . k

Note that λ k (G) ≥ λ k (G ) with equality when G is acyclic. The following theorems extend Theorem 1.2 for all digraphs. c

Theorem 3.1 For all directed graphs G and positive integers k, λ k (G) ≥ χ k (G) . c

Theorem 3.2 Let G be a digraph, and let k be a positive integer. Then there exists a colouring which is orthogonal to every optimal k-c-path. . k Theorem 3.2 implies Theorem 3.1 as is shown below: Let Pc be an optimal k-c-path and let C = {C1 , C2 ,K, Cm } be a colouring orthogonal to it. Then

λck = P c = ∑ Ci ∩ P c ≥ ∑ min {Ci , k } = C k ≥ χ k k

m

k

i =1

m

i =1

thus proving Theorem 3.1.

Outline of proof of Theorem 3.2 If V can be covered by an arbitrary number of disjoint cycles and k or fewer disjoint paths then by defining each vertex to be a distinct colour class we satisfy Theorem 3.2. So assume otherwise. Let V = n , and label the vertices 1,2,…,n. We shall use the linear program defined in [1] and [12]. The variable set of the LP is {xi , j | ( i, j ) ∈ I } where

I = {(i , j ) | 0 ≤ i , j ≤ n, i = 0 or j = 0 or (i, j ) ∈ E} Let C = ( ci , j ), (i , j ) ∈ I be defined by ci , 0 = 0 for all 0 ≤ i ≤ n ; c 0, j = 1 for all 1 ≤ j ≤ n ; c i ,i = 0 for all 0 ≤ i ≤ n ;

c i, j = 1 for i > 0 , j > 0 , i ≠ j and (i , j ) ∈ E

9

Consider the following linear program: I’. n

maximize n

∑x j= 0

n

∑x j= 0

i, j

0, j

i, j

xi , j

= ∑ x i, 0 = k i= 0 n

∑x

for all i > 0;

=1

∑c

i, j= 0 n

i= 0

i, j

(8)

= 1 for all j > 0

x i, j ≥ 0 for all pair (i , j ) ∈ I

(9) (10)

Every k-c-path Pc = {P1 , P2 ,K, Pm } where P1 , P2 ,K, Pt are paths (t ≤ k ) and k

Pt+1 , Pt + 2 ,K , Pm are cycles, corresponds to a feasible solution of I’, X = ( x i, j ) , defined in the following way:

for j > 0 , x 0, j

x 0, 0 = k − t 1 if j is in in[ P1 ,..., Pt ] = otherwise 0 1 0

for i > 0, xi , 0 = 

1 0

for i > 0 , xi ,i = 

1 0

for i, j > 0, i ≠ j , x i, j = 

if i is in ter [ P1 ,..., Pt ] otherwise if i ∉V ( P1 ) ∪ K ∪ V ( Pm ) otherwise

if (i, j ) is an edge of Pr for some r = 1,K, m otherwise

Stated differently, every path P = (1,K, l ) in Pc of cardinality l ≥ 1 corresponds to k

the solution x 0,1 = x1, 2 = x 2, 3 = K = xl −1,l = x l, 0 = 1 . A cycle C = (1,K, l ) corresponds to the solution x1, 2 = x2 , 3 = K = x l−1, l = xl ,1 = 1 .

( )

It is easy to see tha t X = x i, j is a feasible solution of I’, and the objective function k

computes Pc . The converse also holds: every integral optimal solution of I’ corresponds to an optimal k-c-path in G. Consider the dual problem.

10

II’. n

n

i =1

j =1

minimize k (u 0 + v0 ) + ∑ ui + ∑ v j where u i + v j ≥ ci, j for all (i, j ) ∈ I

(11)

We may assume that there exists an integral optimal solution of II’ satisfying u 0 = 0. To define the colouring C we define the following classes: Let

and

S r = {i > 0 | v i = − ui = r }

(12)

T j = { j} where v j ≠ − u j

(13)

Note that the classes S r receive their names from the values of the variables. Let s = max{ vi | u i + v i = 0} . Then we define the colouring to be:

C = {S1 , S 2 ,K , S s ,T1 , T2 ,K ,Tn }. To show that C is a colouring we need to prove that each S r is an independent set. If, by contradiction, there exist i, j ∈ S r , (i , j ) ∈ E , then u i + v i = 0, and u i + v j ≥ 1 , implying that v j − vi ≥ 1 , so

v i = v j = r is impossible. Using complementary slackness, it can be shown as was k

shown in [1], that each Ti is in some path of Pc and each S r meets all paths of k

Pc , proving the theorem.

4. Related Conjectures

Theorems 2.2 and 3.2 do indeed extend Greene and Greene-Kleitman’s Theorems, but Conjectures 2.1, 2.2, 3.1 and 3.2 are, unfortunately, still open for general digraphs. We propose here two other conjectures which may shed a new light on these problems. Let G be an undirected graph. We define π k (G ) as the maximum π k (G) among all possible orientations G of G. Conjecture 4.1: Let G be an undirected graph, and let k be a positive integer. Then there exists an acyclic orientation G of G for which

π k ( G) = π k (G) It easy to see that Conjecture 4.1 implies Conjecture 2.1: Let D be a directed graph, k a positive integer, and let G be the underlying undirected graph of D. Let G be an

11

acyclic orientation of G for which π k ( G) = π k (G) . Since Conjecture 2.1 holds for all acyclic digraphs we have,

α k ( D) = α k (G ) ≥ π k (G) ≥ π k ( D) implying Conjecture 2.1. We will now show that Conjecture 2.1 implies Conjecture 4.1, hence they are equivalent. Let

C k = {C1 ,C 2 ,K, Ck } be an optimal partial k-colouring of an undirected graph G.

We extend C into a complete colouring C = {C1 , C 2 ,K , C s } of G. We define now the k

acyclic orientation D(G) on G by orienting all edges in G from colour classes of large indices to smaller ones. That is, if x ∈ C i , y ∈ C j , ( x, y ) ∈ G and i > j then orient the edge x → y in D(G). Note that this orientation is acyclic, and each directed path P

{

}

in D meets at most min P , k vertices of C . Thus we have for a k-optimal path partition P = {P1 , P2 ,K , Pm } the following inequality:

α k = C k = ∑ ( Pi ∩ C k ) ≤ ∑ min {Pi , k }= π k m

m

i =1

i =1

(1)

But by Conjecture 2.1 α k ≥ π k , hence there is equality in (1), and the acyclic orientation D defined above yields a path partition with π k ( D ) = π k (G ) , implying Conjecture 4.1. In a similar manner we can attempt to strengthen Conjecture 3.1. For an undirected graph G, and a positive integer k we define:

λ k (G) = min λ k (G ) G

i.e. λ k (G ) is defined to be the minimum of λ k (G) among all orientations G of G . Conjecture 4.2 (I. B-A. Hartman, F. Salleh [11]): Let G be an undirected graph, and let k be a positive integer. Then there exists an acyclic orientation G of G for which

λ k (G) = λk (G) . Conjecture 4.2 is, in fact, equivalent to Conjecture 3.1, as was shown in [11].

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References [1]

R.Aharoni, I. Ben-Arroyo Hartman and A.J.Hoffman, Path Partitions and Packs of Acyclic Digraphs, Pacific Journal of Mathematics (2) vol.118 (1985), 249259.

[2]

C.Berge, k-optimal Partitions of a Directed Graph, Europ. J. Combinatorics (1982) 3,97-101.

[3]

K. Cameron, Polyhedral and Algorithmic Ramifications of Antichains, Ph.D. thesis, University of Waterloo, 1982.

[4]

K. Cameron, On k-optimal Dipath Partitions and Partial k-colourings of Acyclic Digraphs, Europ. J. Combinatorics(1986) 7, 115-118.

[5]

P. Dilworth, A Decomposition Theorem for Partially Ordered Sets, Ann. of Math.(1), vol. 51(1950),161-166.

[6]

S. Felsner, Orthogonal Structures in Directed Graphs, J. Comb. Theory, Ser B 57 (1993), 309-321.

[7]

T. Gallai, On Directed Paths and Circuits, in: Theory of Graphs,( eds. P. Erdos and G. Katona), Academic Press, New York, 1968, 115-118.

[8]

T. Gallai and A.N. Milgram, Verallgemeinerung eines Graphentheoretischen Satzes von Redei, Acta Sc. Math. 21 (1960), 181-186.

[9]

C. Greene, Some Partitions Associated with a Partially Ordered Set, J. Combinatorial Theory, Ser.A 20 (1976),69- 79.

[10] C. Greene, D.J. Kleitman, The Structure of Sperner k-Families, J. Combinatorial Theory, Ser.A 20 (1976),41-68. [11] I. Ben-Arroyo Hartman, F. Salleh, and D. Hershkowitz, On Greene's Theorem for General Digaphs, Journal of Graph-Theory 18 (1994),169-175,. [12] A.J. Hoffman, Extending Greene’s Theorem to Directed Graphs, J. Combinatorial Theory Ser A. 34 (1983), 102-107. [13] N. Linial, Extending the Greene-Kleitman Theorem to Directed Graphs, J. Combinatorial Theory, Ser.A 30 (1981),331-334. . [14] B. Roy, Nombre Chromatique et Plus Longs Chemins, Rev. F1: Automat. Informat. 1 (1976), 127-132

[15] M. Saks, A Short Proof of the k-saturated Partitions, Adv. In Math. 33 (1979), 207-211

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