Claws in digraphs. Amine El Sahili and Mekkia Kouider

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1 Claws in digraphs. Amine El Sahili and Mekkia Kouider Abstract. We study in this paper the existence of claws in digraphs. extend a result of Saks and Sós to the tournament-like digraphs. We 1. Introduction The digraphs considered here are all connected, they have no loops, multiple edges or circuits of length two. The chromatic number of a digraph is the chromatic number of its underlying graph. A block of a path in a digraph is a maximal directed subpath. We recall that the length of a path is the number of its edges. A rooted tree is a tree T with a specified vertex, called the root of T. A branching (resp. inbranching) is an orientation of a rooted tree in which every vertex but the root has indegree 1 (resp. outdegree 1). The level of a vertex in a branching is its distance from the root. A claw is an inbranching in which only the root may have an indegree more than 2. It will be denoted by P (s 1, s 2,, s k ) if it is formed by k directed paths ending at the root, of lengths s 1, s 2,, s k respectively. In particular, P (k, l) is a path with two blocs of lengths k and l. In [1] we proved Theorem 1. Every n + 1-chromatic digraph contains any path with two blocs of length n 1. Saks and Sós proved the following result about claws in tournaments: Theorem 2. [5] Any 2(n 1)-tournament contains any claw of order n. This result became a corollary of that one proved later by Havet and Thomasse[3] showing that a 2(n 1)-tournament contains any branching of order n. By forbidding some paths, we proved[1] the following: Theorem 3. Let D be an 2(n 1)-chromatic digraph without paths P (2n 3, 1). Then D contains any claw of order n. 1

2 2 AMINE EL SAHILI AND MEKKIA KOUIDER This gives a direct proof of Saks and Sós theorem by remarking that the path P (2n 3, 1) contains 2n 1 vertices. In order to generalize the Saks and Sós result to the arbitrary digraphs, we want in this paper to extend it to some type of digraphs, the tournament-like digraphs (defined below), We treat in particular the case when the claw is a path with two blocs. 2. Spanning branching forest. A branching forest is a digraph each of whose components is a branching. The level of a vertex v in a branching forest F, denoted by l F (v), is the order of the directed path in F ending at v from the root of the branching of F containing v. The set L i (F ) denotes the set of all vertices of level i in D. We denote by l(f ) the maximal integer i such that L i (F ) is non empty. Again if v is a vertex in F, we denote by T v (F ) the branching of F having v as root. Definition 1. A spanning branching forest F of a digraph D is said to be maximal if the sum l F (v) is maximal. Definition 2. A tournament-like digraph D is a digraph containing a maximal spanning branching forest F such that l(f ) = χ(d). Lemma 1. Let F be a maximal spanning branching forest of a digraph D. Then the sets L i (F ), i 0, are stable sets in D. Proof. Suppose to the contrary that there is an arc e = (x, y) in D with x, y L i (F ). Consider the spanning branching forest F = F + e f, where f denotes the arc of F with head y. Let T be the branching of root y in F. For each v D we have l F (v) = l F (v) + 1 if v T, and l F (v) = l F (v) otherwise. Thus l F (v) > l F (v) a contradiction By simply remarking that any vertex in L i (F ) is the terminus of a unique directed path of length i 1 in F, the Roy-Gallai theorem will be a simple consequence of lemma 1: Theorem 4 (Roy-Gallai). Every n-chromatic digraph contains a directed path of length n 1 (P (n 1, 0)). Proof. Let F be a maximal spanning branching forest of D. By lemma 1, the sets L i (F ), i 0, are stable sets in D. This implies that L n (F ) φ, we get the hoped path. Lemma 1 is also used in [1] to prove theorem1. The following property of a maximal spanning branching forest of a digraph D will be useful in the sequel: Lemma 2. Let F be a maximal spanning branching forest of a digraph D. Let e = (x, y) E(D) with l F (x) > l F (y), then F contains a directed yx path.

3 CLAWS IN DIGRAPHS 3 Proof. Suppose to the contrary that F contains no directed yx path, then F = F + e f, where f denotes the arc of F with head y, is a spanning branching forest of D. Let T be the branching of root y in F. For each v D we have l F (v) = l F (v) + (l F (x) l F (y)) if v T, and l F (v) = l F (v) otherwise. Thus l F (v) > l F (v). A contradiction with the definition of F. Consequently, we have the following property: Lemma 3. Let F be a maximal spanning branching forest of a digraph D. Set l(f ) = l. For every y L l (F ) and for every r < l, we have either N(y) L r (F ) = N + (y) L r (F ) or N(y) L r (F ) = N (y) L r (F ). Proof. Suppose to the contrary that F contains a vertex y L l (F ) and two vertices x, z L r (F ) for some r < l, such that e = (x, y), f = (y, z) E(D). Let us denote f the arc of F with head y and f the arc of F with head z if any. By the above lemma, there exists a directed zy path in F. Let F = F + e + f f f, if e = f, and, F = F + e + f f ) otherwise. Then F is a spanning branching forest of D. Let T be the branching of root z in F. For each v D we have l F (v) = l F (v) + 2 if v T y, l F (y) = l F (y) (l r) + 1 and l F (v) = l F (v) otherwise. Since v(t ) l r + 1, then l F (v) > l F (v) a contradiction We define now an important operation on maximal forests: Lemma 4. Let F be a maximal spanning branching forest of a digraph D with l(f ) = l. Let C = v r v r+1 v l 1 v l be a circuit of D such that v i L i (F ), 1 < r i l. Suppose that there is a vertex x L r 1 (F ) such that e = (x, v l ) E(D). Then the forest F obtained from F by rotating the cycle C in the negative sense, that is F = F + e + (v l, v r ) (v l 1, v l ) f, where f denotes the arc of F with head v r, is a maximal spanning branching forest and so T vr (F ) = v r v r+1 v l 1 v l and l(f ) = l. The forest F is called a rotating of F, it is denoted by I(F ) or Proof. Obviously, F is a spanning branching forest of D. For each v D we have l F (v) = l F (v) + 1 if v T vr (F ) v l, l F (v l ) = r and l F (v) = l F (v) otherwise. Since l F (u) + l 1 l F (v i ) = l F (u) + l 1 l F (v i ) then l F (v) l F (v). The i=s i=s equality holds since F is maximal. Thus T vr (F ) = v r v r+1 v l 1 v l and l(f ) = l(f ). Note that the above operation can be also defined even if r = 1 by simply rotating C, that is F = F + e + (v l, v r ) (v l 1, v l ). Consider a maximal spanning branching forest F of a digraph D with l(f ) = l. A vertex y L l (F ) is said to have a root r if N + (y) L r (F ) φ and N (y) L i (F ) φ for all i r 1, if any. We write r(y) = r. let us set U r (F ) = {y L l (F ), R(D) = F,r U r (F )

4 4 AMINE EL SAHILI AND MEKKIA KOUIDER and R(D) = {r(y), y R(D)}. All these sets may be empty. Lemma 5. Let F be a maximal spanning branching forest of a digraph D with l(f ) = l. Let u U r (F ) for some r < l, and let C = v r v r+1 v l 1 u be the circuit of D such that v i L i (F ), r i l 1. If F = I(F ) for some rotating of F using C, then we have: U t (F ) = U t (F ) for all t < r. and either U r (F ) = U r (F ) {u} or U r (F ) = (U r (F ) {u}) {v l 1 }. Proof. Since L i (F ) = L i (F ) for all i < r, then U t (F ) = U t (F ) for all t < r. Suppose that v l 1 / U r (F ) and let s prove that U r (F ) = U r (F ) {u}. Let v U r (F ) {u}, it has an outneighbor x L r (F ). Since v / T vr (F ) then x v r and so v U r (F ). We may use the same argument to prove the other inclusion. If R(D) φ, set R(D) = {r 0, r 1,, r s } with r 0 < r 1 < < r s, F = {F maximal,, l(f ) minimal} and define the following decreasing sequence of non empty classes of maximal forests: F 0 = {F F, U r0 (F ) is minimal} and F i+1 = {F F i, Uri+1 (F ) is minimal}, 1 i < s. Based on the above lemma we may remark easily that F k, 1 k s is closed by anytating operation: Lemma 6. Let F F k, 1 k s. Let u be a vertex in U rk (F ), let C = v rk v rk +1 v l 1 u be the circuit of D such that v i L i (F ), r k i l 1. If F = I(F ) for some rotating of F using C, then F F k and U rk (F ) = (U rk (F ) {u}) {v l 1 }. Proof. Remark first that U ri (F ) = U ri (F ) for all i < k, then F F k 1. The case U rk (F ) = U rk (F ) {u} is omitted due to the minimality of U rk (F ). Thus by lemma 5 U rk (F ) = (U rk (F ) {u}) {v l 1 } and so U rk (F ) = U rk (F ). Therefore F F k. Forests in F s have the following crucial property: Lemma 7. Let F F s. Let u be a vertex in U rk (F ), 1 k s, let C = v rk v rk +1 v l 1 u be the circuit of D such that v i L i (F ), r k i l 1. Then for all v C we have N (v) L j (F ) φ for j < r k, and N(v) (L j (F ) C) = φ for j r k Proof. Set F 1 = I(F ) for some rotating of F using C. Since F F k then by lemma 6 v l 1 U rk (F 1 ), and so we may apply I to F 1. By a simple induction and using lemma 6, we may define the periodic sequence of maximal forests in F s : F 0 = F, F i = I i (F ), i 1. By the above lemmas, v i U rk (F l i ), for r k i l 1. This gives N (v) L j (F ) φ for j < r k. On the other hand, suppose to the contrary that there is a vertex y (N(v i ) C), r k i l 1 such that l F (y) r k and let t be an integer such that i + t = l F (y) mod(r). We have l Ft (v i ) = l F (y). But l F (y) = l Fn (y) for all n 1 since y / T v (F n ) for any v C. There is then a level of F t which is not an independant set. A contradiction.

5 CLAWS IN DIGRAPHS 5 3. Claws Consider a tournament-like digraph D and let F be a maximal spanning branching forest F such that l(f ) = χ(d). Lemma 8. A tournament-like digraph D has a maximal forest F and a tournament T with V (T ) = {v r, v r+1,, v l 1, v l } such that v i L i (F ), N (v i ) L j (F ) φ for j < r < i < l. Proof. Let F F s. We have l = l(f ) = χ(d). We suppose that L l (F ) contains no vertex v such that N (v) L j (F ) φ for all j < l since otherwise we take T = {v}. There is a k {1, 2,, s} such that U rk (F ) φ since otherwise L l (F ) R(D) = φ and so, due to the first supposition, for all u L l (F ) there exists j < l such that N(u) L j (F ) = φ. Hence the vertices in L l (F ) may be distributed over the other levels and as l(f ) = χ(d), we obtain a (χ(d) 1)-coloring. A contradiction. Suppose to the contrary that the lemma does not hold. Consider a vertex u U rk (F ) and let C = v rk v rk +1 v l 1 v l = u be the circuit of D such that v i L i (F ), r k i l. Since D(V (C)) is not a tournament then there exist i and j, r k i, j l, such that v i v j / E(G(D)). By defining the periodic sequence of maximal forests in F s as in lemma 7, we may replace F by F l i which is always in F s. The vertex u is replaced by v i. Now r = l Fl i (v j ) r k. By lemma 7, N(v i ) (L r (F l i ) {v j } = φ. Then N(v i ) L r (F l i ) = φ as v i v j / E(G(D)). If we proceed similarly for all the vertices in s i=1 U ri (F ), we get a maximal forest F F s such that for all u L l (F ) there exists j < l such that N(u) L j (F ) = φ which leads to a contradiction. This completes the proof of the lemma. The following remark is a direct consequence of this lemma: Corollary 1. A tournament-like digraph D is strongly connected if and only if it is a strongly connected tournament. Proof. It is sufficient for the necessary condition to remark that in the proof of the above lemma and lemma 3, all the neighbors of the vertices in R(F ) which are in L i (F ) are inneighbors. This gives r = 1 and D is a strongly connected i<r tournament. We prove now our main result: Theorem 5. A tournament-like digraph D with χ(d) = 2n 2 contains any claw on n vertices. Proof. Let C = P (s 1, s 2,, s k ) be a claw of order n (s 0 + s s k = n 1) with s 0 = 0. Let T be a tournament on t vertices and F be a maximal forest as in the above lemma. Let i, 0 i k, be the maximal integer such that s 0 + s s i t 2. The case i = k corresponds to the case t = 2n 2 and so by the above lemma D is a tournament on 2n 2 vertices, it contains a copy of C by theorem 1. Then we suppose that i < k, we have s 0 + s s i + s = t 2 with

6 6 AMINE EL SAHILI AND MEKKIA KOUIDER 0 s < s i+1. Let s and r be such that s + s = s i+1 and r + t = 2n 2. We have s +s i+2 + +s k = r 2 < r. Thus T is a tournament on t > 2(s 0 +s 2 + +s i +s) 2 vertices, it contains a copy of the claw C = P (s 0, s 1,, s i, s). Call v 0,the root of C and let v be the tail of the leaf corresponding to the path of C of length s. By the above lemma, there is an inneighbor u of v in L r 1 (F ). Consider a directed w 1 u path P 1 of length s 1 in F where w 1 L r s (F ). The path P 1 together with w 1 uv v 0 form the (i+1)th branch of C. Suppose now that the (i+j)th branch of C is constructed without using vertices in L h (F ), h a = r (s +s i+2 + +s i+j ). if i + j < k then a s i+j+1 and so if we consider an in neighbor of v 0 in L a (F ), we may construct the (i + j + 1)th branch of C without using vertices in L h (F ), h a s i+j+1 = r (s + s i s i+j+1 ). This induction completes the construction of a copy of C in D. In the particular case when the claw is formed by two branches (path with two blocks), the bound 2n 2 may be improved to n : Theorem 6. A tournament-like digraph D with χ(d) = n contains any path with two blocks on n vertices. Proof. Consider a path with two blocs P (k, l) with k + l = n 1. let T be a tournament on t vertices and F be a maximal forest as in the above lemma. If t < l + 1 then a path P (k, l) is easily constructed. Otherwise T contains a path P (s, l) with s + l = t 1. let u be the end of this path corresponding to the bloc of length s and let v be an inneighbor of u in L n t 1 (F ). By remarking that v is the terminus of a directed path P of length n t 2 in F T, we get a path P (k, l). An example of a tournament-like digraph which is not a tournament can be obtained easily from any n-chromatic simple graph G : put V (G) = n i=1 S i where S i are stable sets in G. An edge x i x j E(G), x i S i, x j S j is oriented from x i to x j whenever i < j. The obtained acyclic digraph has no directed path of length n so it is a tournament-like digraph. An example with circuit can be formed by an odd oriented cycle C containing no directed paths of length 3, such that any vertex in C is joined to all the vertices of a strongly connected tournament on n 3 vertices. A natural problem is to ask about the characterization of tournament-like digraphs.

7 CLAWS IN DIGRAPHS 7 References [1] A. El Sahili and M. Kouider. About paths with two blocks. Submitted [2] T. Gallai. On directed paths and circuits. In Theory of Graphs, (P. Erdös and G. Katona, editors). Academic press, (1968), pp [3] F. Havet and S. Thomassé. Median order of tournaments: A tool for the second neighborhood problem and Sumner s conjecture. J. Graph Theory Ser B 35 (2000), [4] B. Roy. Nombre chromatique et plus longs chemins d un graphe. Rev. Française Automat. Informat. Recherche Opérationelle Sér. Rouge, 1 (1967), [5] M. Saks and V. T. Sós. On unavoidable subgraphs of tournaments in finite and infinite sets. Colloquia Mathematica Societatis János Bolyai, 37, Eger (Hungary), (1981), pp Amine El Sahili, Lebanese university, Faculty of sciences I, Hadas-Beyrutm Lebanon and Kouider Mekkia, U.M.R L.R.I. Bât. 490, Universite Paris-sud Orsay, France. address: Aminsahi@inco.com.lb and km@lri.fr