Gallai colorings and domination in multipartite digraphs
GGallai colorings and domination in multipartite digraphs
András Gyárfás ∗ Computer and Automation Research Institute,Hungarian Academy of Sciences,1518 Budapest, P.O. Box 63, Hungary, [email protected]
Gábor Simonyi † Alfréd Rényi Institute of Mathematics,Hungarian Academy of Sciences,1364 Budapest, P.O. Box 127, Hungary, [email protected]
Ágnes Tóth ‡ Department of Computer Science and Information Theory,Budapest University of Technology and Economics,1521 Budapest, P.O. Box 91, Hungary, [email protected]
Abstract
Assume that D is a digraph without cyclic triangles and its vertices are partitionedinto classes A , . . . , A t of independent vertices. A set U = ∪ i ∈ S A i is called a dominatingset of size | S | if for any vertex v ∈ ∪ i/ ∈ S A i there is a w ∈ U such that ( w, v ) ∈ E ( D ) .Let β ( D ) be the cardinality of the largest independent set of D whose vertices are fromdifferent partite classes of D . Our main result says that there exists a h = h ( β ( D )) such that D has a dominating set of size at most h . This result is applied to settlea problem related to generalized Gallai colorings, edge colorings of graphs without -colored triangles. ∗ Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA)No. K68322. † Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA)Nos. K76088 and NK78439. ‡ Research partially supported by the Hungarian Foundation for Scientific Research Grant and by theNational Office for Research and Technology (Grant number OTKA 67651). a r X i v : . [ m a t h . C O ] J un Introduction
Investigating comparability graphs Gallai [8] proved an interesting theorem about edge-colorings of complete graphs that contain no triangle for which all three of its edges receivedistinct colors. Such colorings turned out to be relevant and Gallai’s theorem proved to beuseful also in other contexts, see e.g., [3, 4, 5, 7, 9, 10, 13, 14].Honoring the above mentioned work of Gallai an edge-coloring of the complete graph iscalled a Gallai coloring if there is no completely multicolored triangle. Recently this notionwas extended to other (not necessarily complete) graphs in [11].A basic property of Gallai colored complete graphs is that at least one of the color classesspans a connected subgraph on the entire vertex set. In [11] it was proved that if we color theedges of a not necessarily complete graph G so that no -colored triangles appear then thereis still a large monochromatic connected component whose size is proportional to | V ( G ) | where the proportion depends on the independence number α ( G ) .In view of this result it is natural to ask whether one can also span the whole vertex setwith a constant number of connected monochromatic subgraphs where the constant dependsonly on α ( G ) . This question led to a problem about existence of dominating sets in directedgraphs that we believe to be interesting in itself. In this paper we solve this latter problemthereby giving an affirmative answer to the previous question.The paper is organized as follows. In Subsection 1.1 we describe our digraph problemand state our results on it. The connection with Gallai colorings will be explained in Sub-section 1.2. Section 2 contains the proofs of the results in Subsection 1.1. In Section 3 wefurther elaborate on a question the proofs give rise to. We consider multipartite digraphs, i.e., digraphs D whose vertices are partitioned into classes A , . . . , A t of independent vertices. Suppose that S ⊆ [ t ] . A set U = ∪ i ∈ S A i is calleda dominating set of size | S | if for any vertex v ∈ ∪ i/ ∈ S A i there is a w ∈ U such that ( w, v ) ∈ E ( D ) . The smallest | S | for which a multipartite digraph D has a dominating set U = ∪ i ∈ S A i is denoted by k ( D ) . Let β ( D ) be the cardinality of the largest independentset of D whose vertices are from different partite classes of D . (Such independent sets wesometimes refer to as transversal independent sets.) An important special case is when | A i | = 1 for each i ∈ [ t ] . In this case β ( D ) = α ( D ) and k ( D ) = γ ( D ) , the usual dominationnumber of D , the smallest number of vertices in D whose closed outneighborhoods cover V ( D ) . Our main result is the following theorem. Theorem 1.
For every integer β there exists an integer h = h ( β ) such that the followingholds. If D is a multipartite digraph such that D contains no cyclic triangle and β ( D ) = β ,then k ( D ) ≤ h . Notice that the condition forbidding cyclic triangles in D is important even when | A i | = 1 for all i and β ( D ) = 1 , i.e. for tournaments. It is well known that γ ( D ) can be arbitrarily2arge for tournaments (see, e.g., in [2]), so h (1) would not exist without excluding cyclictriangles.From the proof of Theorem 1 we will get a factorial upper bound for k ( D ) from therecurrence formula h ( β ) = 3 β + (2 β + 1) h ( β − . We have relatively small upper bounds on k only for β = 1 , . Theorem 2.
Suppose that D is a multipartite digraph such that D has no cyclic triangle. If β ( D ) = 1 then k ( D ) = 1 and if β ( D ) = 2 then k ( D ) ≤ . Though the upper bound on h ( β ) obtained from our proof of Theorem 1 is much weakerwe could not even rule out the existence of a bound that is linear in β . We cannot provea linear upper bound even in the special case when every partite class consists of only onevertex. Nevertheless, we treat this case also separately and provide a slightly better boundthan the one following from Theorem 1. The class of digraphs we have here, i.e., those withno directed triangles, is studied already and called the class of clique-acyclic digraphs , see[1]. Theorem 3.
Let f (1) = 1 and for α ≥ , f ( α ) = α + αf ( α − . If D is a clique-acyclicdigraph then γ ( D ) ≤ f ( α ( D )) . Apart from the obvious case α ( D ) = 1 (when D is a transitive tournament) we know thebest possible bound only for α ( D ) = 2 . Theorem 4. If D is a clique-acyclic digraph with α ( D ) = 2 , then γ ( D ) ≤ . Note that Theorem 4 is sharp as shown by the cyclically oriented pentagon. Moreover, theunion of t vertex disjoint cyclic pentagons shows that we can have α ( D ) = 2 t and γ ( D ) = 3 t .Thus in case a linear upper bound would be valid at least in the special case of clique-acyclicdigraphs, it could not be smaller than α ( D ) . There are some easy subcases though whenthe bound is simply α ( D ) . Proposition 5. If D has an acyclic orientation or D is a clique-acyclic perfect graph then γ ( D ) ≤ α ( D ) . Note that Proposition 5 is sharp in the sense that every graph G has a clique-acyclicorientation resulting in digraph D with γ ( D ) = α ( G ) = α ( D ) . Indeed, an acyclic orientationof G where every vertex of a fixed maximum independent set has indegree zero shows this.It is worth noting the interesting result of Aharoni and Holzman [1] stating that a clique-acyclic digraph always has a fractional kernel, i.e., a fractional independent set, which is alsofractionally dominating.We will see in Section 2 from the proof of Theorems 1 and 2 that the dominating sets wefind there contain two kinds of partite classes. The first kind could be substituted by justone vertex in it, while the second kind is chosen not so much to dominate others but becauseit is itself not dominated by others. That is, apart from a bounded number of exceptionalpartite classes we will dominate the rest of our digraph with a bounded number of vertices.In Section 3 we will prove another theorem showing that the exceptional classes are indeedneeded. 3 .2 Application to Gallai colorings Recall that Gallai colorings are originally defined as edge-colorings of complete graphs whereno triangle gets three different colors. As already mentioned earlier, one of the basic prop-erties of Gallai colorings is that at least one color spans a connected subgraph, i.e. formsa component covering all vertices of the underlying complete graph. In [11] the notion wasextended to arbitrary graphs and it was proved that in this setting there is still a largemonochromatic connected component. More precisely the following was proved.
Theorem 6. ([11])
Suppose that the edges of a graph G are colored so that no triangle iscolored with three distinct colors. Then there is a monochromatic component in G with atleast | V ( G ) | α ( G )+ α ( G ) − vertices. Another, in a sense stronger possible generalization of the above basic property of Gallaicolorings is also suggested by Theorem 6, this was asked first at a workshop at Fredericia inNovember, 2009.
Problem . Suppose that the edges of a graph G are colored so that no triangle is coloredwith three distinct colors. Is it true that the vertices of G can be covered by the vertices ofat most k monochromatic components where k depends only on α ( G ) ?We remark that an example in [11] shows that even if the k of Problem 1 exists, it must beat least cα ( G )log α ( G ) where c is a small constant.Theorem 1 implies an affirmative answer to Problem 1. Let g (1) = 1 and for α ≥ , let g ( α ) = g ( α −
1) + h ( α ) where h is the function given by Theorem 1.In the sequel we will use the notation G [ A ] that denotes the subgraph of graph G inducedby A ⊆ V ( G ) . Theorem 7.
Suppose that the edges of a graph G are colored so that no triangle is coloredwith three distinct colors. Then the vertices of G can be covered by the vertices of at most g ( α ( G )) monochromatic components. In case α ( G ) = 2 at most five components are enough. Note that the last statement of Theorem 7 generalizes Theorem 6 in the α ( G ) = 2 case. Proof.
For α ( G ) = 1 the result is obvious by Gallai’s theorem. For α ( G ) ≥ , supposethat v ∈ V ( G ) and let X be the set of vertices in G that are not adjacent to v . Byinduction, the subgraph G [ X ] can be covered by the vertices of g ( α ( G ) − monochromaticcomponents. Let t be the number of colors used on edges of G incident to v and let A i be the set of vertices incident to v in color i . Observe that the condition on the coloringimplies that edges of G between A i , A j are colored with either color i or color j whenever ≤ i < j ≤ t . Thus orienting all edges of color i outward from A i for every i , all edges of G between different classes A j are oriented. Moreover, in this orientation there are no cyclictriangles. Thus Theorem 1 is applicable to the oriented subgraph H spanned by the unionof the classes A j after the edges inside the A j ’s are removed. We obtain at most h ( α ( G )) dominating sets A i and each set v ∪ A i together with the vertices that A i dominates form4 connected subgraph of G in color i . Thus all vertices of G can be covered by at most g ( α ( G ) −
1) + h ( α ( G )) = g ( α ( G )) connected components. In case of α ( G ) = 2 we can useTheorem 2 to get a covering with at most five monochromatic components. We will use the following notation throughout. If D is a digraph and U ⊆ V ( D ) is a subset ofits vertex set then N + ( U ) = { v ∈ V ( D ) : ∃ u ∈ U ( u, v ) ∈ E ( D ) } is the outneighborhood of U .The closed outneighborhood ˆ N + ( U ) of U is meant to be the set U ∪ N + ( U ) . When U = { u } is a single vertex we also write N + ( u ) and ˆ N + ( u ) for N + ( U ) and ˆ N + ( U ) , respectively. When ( u, v ) ∈ E ( D ) , we will often say that u sends an edge to v .We first deal with the case β ( D ) = 1 and prove the first statement of Theorem 2. As itwill be used several times later, we state it separately as a lemma. Lemma 8.
Let D be a multipartite digraph with no cyclic triangle. If β ( D ) = 1 then k ( D ) = 1 .Proof. Let K be a partite class for which | ˆ N + ( K ) | is largest. We claim that K is a dominatingset. Suppose indirectly, that there is a vertex l in a partite class L (cid:54) = K , which is notdominated by K . Since all edges between distinct partite classes are present in D withsome orientation, l must send an edge to all vertices of K . Furthermore, if a vertex m ina partite class M (cid:54) = K, L is an outneighbor of some k ∈ K then it is also an outneighborof l , otherwise m , l and k would form a cyclic triangle. Thus ˆ N + ( K ) ⊆ ˆ N + ( L ) . Moreover, l ∈ ˆ N + ( L ) \ ˆ N + ( K ) , so | ˆ N + ( L ) | > | ˆ N + ( K ) | contradicting the choice of K . This completesthe proof of the lemma. K L Mlk m
In the following two subsections we prove Theorems 2 and 1, respectively. independent vertices To prove the second statement of Theorem 2 we will need the following stronger variant ofLemma 8. 5 emma 9.
Let D be a multipartite digraph with no cyclic triangle and β ( D ) = 1 . Thenthere is a partite class K which is a dominating set, and there is a vertex k ∈ K such that V ( D ) \ ( K ∪ L ) ⊆ N + ( k ) for some partite class L (cid:54) = K . Thus Lemma 9 states that the dominating partite class K has an element that alonedominates almost the whole of D , there may be only one exceptional partite class L whosevertices are not dominated by this single element of K .For proving Lemma 9, the following observations will be used, where X, Y, Z will denotepartite classes.
Observation 10.
Let D be a multipartite digraph with no cyclic triangle and β ( D ) = 1 .Suppose that for vertices x , x ∈ X and y ∈ Y the edges ( x , y ) and ( y, x ) are present in D . Then for every z ∈ Z (cid:54) = X, Y with ( x , z ) ∈ E ( D ) we also have ( x , z ) ∈ E ( D ) .Proof. Assume indirectly that for some z ∈ Z the orientation is such that we have ( x , z ) , ( z, x ) ∈ E ( D ) . Then the edge connecting z and y cannot be oriented either way: ( z, y ) ∈ E ( D ) would give a cyclic triangle on vertices z, y, x , while ( y, z ) ∈ E ( D ) would create oneon y, z, x . x x y zX Y Z Observation 11.
Let D be a multipartite digraph with no cyclic triangle and β ( D ) = 1 .Suppose that for vertices x , x ∈ X and y , y ∈ Y the edges ( x , y ) , ( y , x ) , ( x , y ) , ( y , x ) are present in D forming a cyclic quadrangle. Then in every partite class Z (cid:54) = X, Y theoutneighborhood of these four vertices is the same.Proof.
Let z be an element of Z ∩ N + ( x ) . By ( y , x ) ∈ E ( D ) we must have z ∈ Z ∩ N + ( y ) ,otherwise y , x , z would form a cyclic triangle. Thus we have Z ∩ N + ( x ) ⊆ Z ∩ N + ( y ) .Now shifting the role of vertices along the oriented quadrangle backwards we similarly get Z ∩ N + ( x ) ⊆ Z ∩ N + ( y ) ⊆ Z ∩ N + ( x ) ⊆ Z ∩ N + ( y ) ⊆ Z ∩ N + ( x ) proving that we haveequality everywhere. 6 x y y zX Y Z Note that in Observation 11, as β ( D ) = 1 , the inneighborhood of the vertices x , x , y , y is also the same, so these vertices split to out- and inneighborhood in the same way everypartite class Z (cid:54) = X, Y . Proof of Lemma 9.
We know from Lemma 8 that there is a partite class K which is a dom-inating set.Let k be an element of K for which | N + ( k ) | is maximal. If k itself dominates all thevertices not in K then we are done. (In that case we do not even need an exceptional class L .) Otherwise, there is a vertex l in a partite class L (cid:54) = K for which the edge between l and k is oriented towards k . As L ⊆ N + ( K ) , there must be a vertex k ∈ K which sends anedge to l .Using Observation 10 for the vertices k , k and l , we obtain that k sends an edge notjust to l but to every vertex in N + ( k ) \ L . By the choice of k this implies the existence ofa vertex l ∈ L for which ( k, l ) , ( l , k ) ∈ E ( D ) . Thus the vertices k, l , k , l form a cyclicquadrangle. Applying Observation 11 this implies that these four vertices have the sameoutneighborhood in V ( D ) \ ( K ∪ L ) .We claim that N + ( k ) contains all vertices of D \ ( K ∪ L ) . Assume indirectly, that thereis a vertex m in a partite class M (cid:54) = K, L which is not dominated by k . We can arguesimilarly as we did for l . Namely, since M ⊆ N + ( K ) there is some k ∈ K (perhapsidentical to k ) dominating m . Applying Observation 10 to the vertices k, m and k , weobtain ( N + ( k ) \ M ) ⊆ N + ( k ) . Then by the choice of k we must have a vertex m ∈ M forwhich ( k, m ) , ( m , k ) ∈ E ( D ) . So vertices k, m , k , m also form a cyclic quadrangle, andObservation 11 gives us that Z ∩ N + ( k ) = Z ∩ N + ( m ) = Z ∩ N + ( k ) = Z ∩ N + ( m ) for allpartite classes Z (cid:54) = K, M .The contradiction will be that the edge between l and m should be oriented both ways.Indeed, since ( l , k ) ∈ E ( D ) and in L the inneighbors of k and m are the same, we musthave ( l , m ) ∈ E ( D ) . However, ( m , k ) ∈ E ( D ) and the fact that k and l split M in thesame way implies ( m , l ) ∈ E ( D ) . This contradiction completes the proof of the lemma.7 L Mk l k l m k m Now we are ready to prove the second statement of Theorem 2.
Proof of Theorem 2.
We have already proven the first statement of the theorem. To provethe second part let D be a multipartite digraph with no cyclic triangle and β ( D ) = 2 . Weuse induction on the number of vertices. The base case is obvious. Let p be a vertex of D and consider the subdigraph ˆ D := D \ { p } .By induction k ( ˆ D ) ≤ . Let K , L , M and N be four partite classes of ˆ D that form adominating set in ˆ D . If p ∈ ˆ N + ( K ∪ L ∪ M ∪ N ) then we are done, the same four sets alsodominate D . If p / ∈ ˆ N + ( K ∪ L ∪ M ∪ N ) then we will choose four other partite classes thatwill dominate D . First we choose P , the class of p . We partition every other partite classinto three parts according to how it is connected to p . For any class Z , let Z denote the setof vertices in Z dominated by p , let Z be the set of vertices in Z nonadjacent to p , and let Z denote the set of remaining vertices of Z , i.e., those which send and edge to p . We willrefer to Z i as the i -th part of the partite class Z , where i = 1 , , . Note that K , L , M , N are all empty, otherwise we would have p ∈ ˆ N + ( K ∪ L ∪ M ∪ N ) .Let D be the subdigraph of D induced by the vertices in the second part of the partiteclasses of D \ P in their above partition. D is also a multipartite digraph with no cyclictriangle and β ( D ) = 1 . The latter follows from the fact that the vertices of D are allnonadjacent to p and β ( D ) = 2 . Thus by Lemma 8 the vertices of D can be dominated byone partite class Q , the second part of some partite class Q of D . We choose Q to be thesecond partite class in our dominating set. Observe that all vertices of D not dominated sofar, i.e., those not in ˆ N + ( P ∪ Q ) should belong to the third part of their partite classes. Let u be such a vertex. (If there is none, then we are done.) We know u / ∈ K ∪ L ∪ M ∪ N as noneof these four classes has a third part. Since K ∪ L ∪ M ∪ N is a dominating set in ˆ D thereis a vertex k in one of these four classes for which ( k, u ) is an edge of D . No vertex in thefirst part of a class can send an edge to a vertex lying in the third part of some other class,otherwise the latter two vertices would form a cyclic triangle with p . Thus, since K, L, M, N has no third parts, k must be in the second part of one of them.8 qk u PK L M N QQ R Lemma 9 implies that there is a vertex q ∈ Q with V ( D ) ∩ ˆ N + ( q ) containing V ( D ) except one exceptional class R . We choose R , the partite class of R , to be the third partiteclass in our dominating set. ( R may or may not be identical to one of K , L , M , N . It isnot difficult to see that we will really need R for the domination only if it is one of thesefour classes.) If u / ∈ ˆ N + ( R ) then k must be an outneighbor of q . Observe that ( u, q ) cannotbe an edge of D , otherwise q , k and u would form a cyclic triangle. But ( q, u ) cannot bean edge either, as u / ∈ N + ( Q ) . Thus u and every so far undominated vertex is nonadjacentto q . Thus the set U of undominated vertices induces a subgraph D [ U ] with β ( D [ U ]) = 1 ,otherwise adding q we would get β ( D ) ≥ . But then by Lemma 8 all vertices in U can bedominated by only one additional, fourth class. Surprisingly, our proof of Theorem 1 is not a direct generalization of the argument provingTheorem 2 in the previous subsection. In fact, in a way it is conceptually simpler.
Proof of Theorem 1.
We have seen that h (1) = 1 (and h (2) = 4 ) is an upper bound for k ( D ) if β ( D ) = 1 (and if β ( D ) = 2 ). Now we prove that h ( β ) = 3 β + (2 β + 1) h ( β − is anupper bound on k ( D ) if β ( D ) = β ≥ . Let D be a multipartite digraph with no cyclictriangle and β ( D ) = β . Let k , k , . . . , k β be vertices of D , each from a different partiteclass, such that | ˆ N + ( ∪ βi =1 { k i } ) | is maximal. Let the partite class of k i be K i for all i andlet K denote ∪ βi =1 { k i } . First we declare the β partite classes of these vertices k i to be partof our dominating set. Next we partition every other partite class into β + 2 parts. Foran arbitrary partite class Z (cid:54) = K i ( i = 1 , . . . , β ) we denote by Z the set Z ∩ N + ( K ) . For i = 1 , , . . . , β let Z i be the set of vertices in Z \ Z that are not sending an edge to k i , butsending an edge to k j for all j < i . Finally, we denote by Z β +1 , the remaining part of Z ,that is the set of those vertices of Z that send an edge to all vertices k , k , . . . , k β . (As inthe proof of Theorem 2 we will refer to the set Z i as the i -th part of Z .) The subgraph D i of D induced by the i -th parts of the partite classes of D \ ( ∪ βi =1 K i ) is also a multipartitedigraph with no cyclic triangle. For ≤ i ≤ β it satisfies β ( D i ) ≤ β − , since adding k i to any transversal independent set of D i we get a larger transversal independent set. So byinduction on β , each of these β digraphs D i can be dominated by at most h ( β − partiteclasses. We add the appropriate βh ( β − partite classes to our dominating set.If β ( D β +1 ) ≤ β − also holds then the whole graph can be dominated by choosingadditional h ( β − partite classes. Otherwise let L = { l , l , . . . , l β } be an independent set9f size β with all its vertices in V ( D β +1 ) belonging to distinct partite classes (of D ), that aredenoted by L , L , . . . , L β , respectively. We claim that in the remaining part of D β +1 , i.e.,in D β +1 \ ( ∪ βi =1 L i ) there is no other independent set of size β with all elements belongingto different partite classes. Assume indirectly that m ∈ M , m ∈ M , . . . , m β ∈ M β form such an independent set M . As L is a maximal transversal independent set, everyelement of a partite class different from L , . . . , L β is connected to at least one of the l i ’s.And since every element of L sends an edge to all the vertices k , . . . , k β , we must have N + ( K ) \ ( ∪ βi =1 L i ) ⊆ N + ( L ) otherwise a cyclic triangle would appear. (The latter is becauseif k i ( i ∈ { , , . . . , β } ) sends an edge to v , and l j ( j ∈ { , , . . . , β } ) sends an edge to k i ,moreover l j is connected with v then the edge between l j and v must be oriented towards v .) K K K β L L β M M β k k k β l l β m m β D D D β D β +1 Similarly, we have N + ( K ) \ ( ∪ βi =1 M i ) ⊆ N + ( M ) . Thus if such an M exists then ˆ N + ( K ) ⊆ N + ( L ∪ M ) while ˆ N + ( L ∪ M ) also contains the additional vertices belonging to L ∪ M . Thiscontradicts the choice of K . (Note that L∪M dominates also the vertices in ( K ∪· · ·∪ K β ) ∩ ( N + ( k ) ∪ · · · ∪ N + ( k β )) .) Thus if we add the classes L , . . . , L β to our dominating set, thestill not dominated part of D can be dominated by h ( β − further classes. So we constructeda dominating set of D containing at most β +2 βh ( β − β + h ( β −
1) = 3 β +(2 β +1) h ( β − partite classes. This proves the statement.Note that we have proved a little bit more than stated in Theorem 1. Namely, weshowed that there is a set of at most h ( β ) vertices of D which dominates the whole graphexcept perhaps their own partite classes and at most h ( β ) other exceptional classes. Fromthe proof we obtain the recursion formula h ( β ) ≤ β + (2 β + 1) h ( β − and h ( β ) ≤ β + (2 β + 1) h ( β − . For the proof of Theorem 3 we will use the following theorem due to Chvátal and Lovász [6].
Theorem CL ([6]).
Every directed graph D contains a semi-kernel, that is an independentset U satisfying that for every vertex v ∈ D there is an u ∈ U such that one can reach v from u via a directed path of at most two edges. roof of Theorem 3. The statement is trivial for α ( D ) = 1 , since a transitive tournament isdominated by its unique vertex of indegree . We use induction on α = α ( D ) . Assume thetheorem is already proven for α − . Consider D with α ( D ) = α and a semi-kernel U in D that exists by Theorem CL. We define a set S with | S | ≤ f ( α ) elements dominating eachvertex. Let U ⊆ S . Then S already dominates the neighborhood of U . Denote by T thesecond outneighborhood of U (i.e., the set of all vertices not in U and not yet dominated).Observe that for every vertex w ∈ T there is a vertex u ∈ U such that neither ( u, w ) nor ( w, u ) is an edge. Indeed, let u be the vertex of U from which w can be reached by traversingtwo directed edges. Then ( w, u ) / ∈ E ( D ) otherwise we would have a cyclic triangle. But ( u, w ) / ∈ E ( D ) is immediate from knowing that w is not in the first outneighborhood of U .Partition T into | U | ≤ α classes L u indexed by the elements of U where w ∈ L u means that u and w are nonadjacent. Thus all vertices in each class L u are independent from the samevertex in U implying that the induced subgraph D [ L u ] has independence number at most α − . Thus D [ L u ] can be dominated by at most f ( α − vertices. Add these to S forevery u ∈ U . So all vertices can be dominated by at most α + αf ( α −
1) = f ( α ) verticescompleting the proof. UuL u For α ( D ) = 2 the above theorem gives γ ( D ) ≤ f (2) = 4 . Compared to this the improve-ment of Theorem 4 is only , but as already mentioned, the cyclically oriented five-cycleshows that γ ( D ) ≤ is the best possible upper bound.The proof of Theorem 4 goes along similar lines as the proof we had for the secondstatement of Theorem 2. Proof of Theorem 4.
We use induction on the number of vertices in D . Let p be a vertex of D , and partition the remaining vertices of D into three parts. Let V be the set of verticesthat are dominated by p , V the set of vertices nonadjacent to p , and let V be the set ofvertices which send an edge to p . We assume by induction that D \ { p } can be dominated bythree vertices. (The base case is obvious.) If at least one of these is located in V then p isalso dominated by them and we are done. Otherwise we create a new dominating set. Firstwe choose p , and by p we dominate all the vertices in V . Observe that any two vertices in V must be connected, because two nonadjacent vertices of V and p would form an independentset of size . Thus D [ V ] is a transitive tournament and so it can be dominated by just one11ertex, let it be q ∈ V . Let U be the set of remaining undominated vertices. That is, U = V \ N + ( q ) . Consider an arbitrary element u ∈ U . We know that u is dominated bya vertex of the dominating set of D \ { p } . Let this vertex be k , clearly it cannot belong to V (then it would dominate p ). We also have k / ∈ V , otherwise there is a cyclic triangle onthe vertices p, k , and u . So k ∈ V , and thus q sends an edge to k . Since u is undominated, ( q, u ) is not an edge of D . With the edge ( u, q ) , we would get a cyclic triangle on u , q and k . So u and all the vertices in U are nonadjacent to q , therefore α ( D [ U ]) = 1 and thus U can be dominated by one vertex r . Thus all vertices of D are dominated by the -elementset { p, q, r } . This completes the proof. pq kur V V V To prove Proposition 5 we formulate the following simple observation. Let χ ( F ) denote thechromatic number of graph F . Observation 12.
Let D be a directed graph and ¯ D the complementary graph of the undirectedgraph underlying D . If D is clique-acyclic, then γ ( D ) ≤ χ ( ¯ D ) .Proof. It follows from the definition of χ ( ¯ D ) that the vertex set of D can be covered by χ ( ¯ D ) complete subgraphs of D . Since D is clique-acyclic, all these complete subgraphs canbe dominated by one of their vertices. Thus all vertices are dominated by these χ ( ¯ D ) chosenvertices. Proof of Proposition 5.
If the orientation of D is acyclic, then consider those vertices thathave indegree zero. Let these form the set U . Delete these vertices and all vertices theydominate. Let set U contain the indegree zero vertices of the remaining graph, and deletethe vertices in U ∪ N + ( U ) . Proceed this way to form the sets U , . . . , U s , where finallythere are no remaining vertices after U s and its neighbors are deleted. It follows from theconstruction that U ∪ U ∪ · · · ∪ U s is an independent set and dominates all vertices notcontained in it.The second statement immediately follows from Observation 12 and the fact that χ ( ¯ D ) = α ( D ) if D is perfect, an immediate consequence of the Perfect Graph Theorem [15].12 On the exceptional classes
As already mentioned in the Introduction and also after the proof of Theorem 1, the state-ment of Theorem 1 could be formulated in a somewhat stronger form. Namely, we do notonly dominate our multipartite digraph D by h ( β ) partite classes, we actually dominate al-most all of D by h ( β ) vertices, where “almost” means that there is only a bounded number h ( β ) of partite classes not dominated this way. The first appearance of this phenomenonis in Lemma 9 where we showed that if β ( D ) = 1 then a single vertex dominates the wholegraph except at most one class. To complement this statement we show below that thisexceptional class is indeed needed, we cannot expect to dominate the whole graph by a con-stant number of vertices. In other words, if we want to dominate with a constant numberof singletons (and not by simply taking a vertex from each partite class), then we do needexceptional classes already in the β ( D ) = 1 case.For a bipartite digraph D with partite classes A and B let γ A ( D ) denote the minimumnumber of vertices in A that dominate B and similarly let γ B ( D ) denote the minimumnumber of vertices in B dominating A . Let γ ( D ) = min { γ A ( D ) , γ B ( D ) } . Theorem 13.
There exists a sequence of oriented complete bipartite graphs { D k } ∞ k =1 satis-fying γ ( D k ) > k . We note that the existence of D k with n vertices in each partite class and satisfying γ ( D k ) > k follows by a standard probabilistic argument provided that (cid:0) nk (cid:1) (1 − − k ) n < .Our proof below is constructive, however. Proof.
We give a simple recursive construction for D k in which we blow up the vertices of acyclically oriented cycle C k +2 and connect the blown up versions of originally nonadjacentvertices that are an odd distance away from each other by copies of the already constructeddigraph D k − .Let D be a cyclic -cycle, i.e., a cyclically oriented K , . It is clear that neither partiteclass in this digraph can be dominated by a single element of the other partite class. Thus γ ( D ) > holds.Assume we have already constructed D k − satisfying γ ( D k − ) > k − . Let the twopartite classes of D k − be A k − = { a , . . . , a m } and B k − = { b , . . . , b m } . Now we construct D k as follows. Let the vertex set of D k be V ( D k ) = A k ∪ B k , where A k := { ( j, a i ) : 1 ≤ j ≤ k + 1 , ≤ i ≤ m } ,B k := { ( j, b i ) : 1 ≤ j ≤ k + 1 , ≤ i ≤ m } . There will be an oriented edge from vertex ( j, a i ) to ( r, b s ) if either j = r , or j (cid:54)≡ r + 1(mod k + 1) and ( a i , b s ) ∈ E ( D k − ) . All other edges between A k and B k are orientedtowards A k , i.e., this latter set of edges can be described as { (( r, b s ) , ( j, a i )) : j ≡ r + 1 (mod k + 1) or (( b s , a i ) ∈ E ( D k − ) and j (cid:54) = r ) } .
13t is only left to prove that γ ( D k ) > k . Let us use the notation A k ( j ) = { ( j, a i ) : 1 ≤ i ≤ m } , B k ( j ) = { ( j, b i ) : 1 ≤ i ≤ m } . Consider a set K of k vertices of A k , we show it cannotdominate B k . There must be an r ∈ { , . . . , k +1 } by pigeon-hole for which K ∩ A k ( r ) = ∅ and K ∩ A k ( r + 1) (cid:54) = ∅ . (Addition here is meant modulo ( k + 1) .) Fix this r . We claim that somevertex in B k ( r ) will not be dominated by K . Indeed, the vertex ( r +1 , a i ) ∈ K ∩ A k ( r +1) doesnot send any edge into B k ( r ) , so we have only at most k − vertices in K that can dominatevertices in B k ( r ) and all these vertices are in A k \ A k ( r ) . Notice that the induced subgraphof D k on B k ( r ) ∪ A k \ A k ( r ) admits a digraph homomorphism (that is an edge-preservingmap) into D k − . Indeed, the projection of each vertex to its second coordinate gives such amap by the definition of D k . So if the above mentioned k − vertices would dominate theentire set B k ( r ) , then their homomorphic images would dominate the homomorphic imageof B k ( r ) in D k − . The latter image is the entire set B k − and by our induction hypthesis itcannot be dominated by k − vertices of A k − . Thus we indeed have γ A k ( D k ) > k .The proof of γ B k ( D k ) > k is similar by symmetry. Thus we have γ ( D k ) > k as stated. References [1] R. Aharoni, R. Holzman, Fractional kernels in digraphs,
J. Combin. Theory Ser. B. , (1998), 1–6.[2] N. Alon, J. H. Spencer, The Probabilistic Method , Third edition, Wiley-InterscienceSeries in Discrete Mathematics and Optimization, John Wiley and Sons, Hoboken, NJ,2008.[3] R.N. Ball, A. Pultr and P. Vojtěchovský, Colored graphs without colorful cycles,
Com-binatorica , (4) (2007), 407–427.[4] K. Cameron, J. Edmonds, Lambda composition, J. Graph Theory , (1997), 9–16.[5] K. Cameron, J. Edmonds, L. Lovász, A note on perfect graphs, Period. Math. Hungar. , (3) (1986), 441–447.[6] V. Chvátal, L. Lovász, Every directed graph has a semi-kernel, Hypergraph Seminar (Proc. First Working Sem., Ohio State Univ., Columbus, Ohio, 1972; dedicated toArnold Ross), Lecture Notes in Math., Vol. 411, Springer, Berlin, 1974, pp. 175.[7] S. Fujita, C. Magnant, K. Ozeki, Rainbow Generalizations of Ramsey Theory: A Survey,
Graphs Combin. , (2010), 1–30.[8] T. Gallai, Transitiv orientierbare Graphen, Acta Math. Sci. Hungar. , (1967) 25–66.English translation by F. Maffray and M. Preissmann, in: J. L. Ramírez Alfonsín andB. A. Reed (editors), Perfect Graphs, John Wiley and Sons, 2001, 25–66.[9] V. Gurvich, Decomposing complete edge-chromatic graphs and hypergraphs. Revisited, Discrete Applied Math. , (14), 3069–3085.1410] A. Gyárfás, G. Simonyi, Edge colorings of complete graphs without tricolored triangles, J. Graph Theory , (2004), 211–216.[11] A. Gyárfás, G. N. Sárközy, Gallai colorings of non-complete graphs, Discrete Math. , (2010), 977–980.[12] A. Gyárfás, G. N. Sárközy, A. Sebő, S. Selkow, Ramsey-type results for Gallai colorings, J. Graph Theory , to appear.[13] J. Körner, G. Simonyi, Graph pairs and their entropies: Modularity problems,
Combi-natorica , (2000), 227–240.[14] J. Körner, G. Simonyi, Zs. Tuza, Perfect couples of graphs, Combinatorica , (1992),179–192.[15] L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Math. ,2