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Strong subtournaments containing a given vertex in regular multipartite tournaments
Discrete Mathematics 308 (2008) 5516–5521 www.elsevier.com/locate/disc
Strong subtournaments containing a given vertex in regular multipartite tournaments Lutz Volkmann, Stefan Winzen Lehrstuhl II f¨ur Mathematik, RWTH Aachen University, Germany Received 23 May 2007; received in revised form 2 October 2007; accepted 2 October 2007 Available online 19 November 2007
Abstract If x is a vertex of a digraph D, then we denote by d + (x) and d − (x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by i g (D) = max {d + (x), d − (x)} − min {d + (y), d − (y)}. x∈V (D)
y∈V (D)
If i g (D) = 0, then D is regular and if i g (D) ≤ 1, then D is called almost regular. A c-partite tournament is an orientation of a complete c-partite graph. Recently, Volkmann and Winzen [L. Volkmann, S. Winzen, Almost regular c-partite tournaments contain a strong subtournament of order c when c ≥ 5, Discrete Math. (2007), 10.1016/j.disc.2006.10.019] showed that every almost regular c-partite tournament D with c ≥ 5 contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c}. In this paper for the class of regular multipartite tournaments we will consider the more difficult question for the existence of strong subtournaments containing a given vertex. We will prove that each vertex of a regular multipartite tournament D with c ≥ 7 partite sets is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c − 4}. c 2007 Elsevier B.V. All rights reserved.
Keywords: Multipartite tournaments; Regular multipartite tournaments; Subtournaments
1. Terminology and introduction In this paper all digraphs are finite without loops and multiple arcs. The vertex set and arc set of a digraph D is denoted by V (D) and E(D), respectively. If x y is an arc of a digraph D, then we write x → y and say x dominates y, and if X and Y are two disjoint vertex sets or subdigraphs of D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X → Y . Furthermore, X ; Y denotes the fact that there is no arc leading from Y to X . By d(X, Y ) we denote the number of arcs from the set X to the set Y , + i.e., d(X, Y ) = |{x y ∈ E(D) : x ∈ X, y ∈ Y }|. If D is a digraph, then the out-neighborhood N D (x) = N + (x) − − of a vertex x is the set of vertices dominated by x and the in-neighborhood N D (x) = N (x) is the set of vertices dominating x. Therefore, if there is the arc x y ∈ E(D), then y is an outer neighbor of x and x is an inner neighbor
E-mail address: [email protected] (L. Volkmann). c 2007 Elsevier B.V. All rights reserved. 0012-365X/$ - see front matter doi:10.1016/j.disc.2007.10.006
L. Volkmann, S. Winzen / Discrete Mathematics 308 (2008) 5516–5521
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+ − of y. The numbers d D (x) = d + (x) = |N + (x)| and d D (x) = d − (x) = |N − (x)| are called the outdegree and indegree of x, respectively. For a vertex set X of D, we define D[X ] as the subdigraph induced by X . If we speak of a cycle (path), then we mean a directed cycle (directed path), and a cycle of length n is called an n-cycle. A cycle (path) of a digraph D is Hamiltonian, if it includes all the vertices of D. A digraph D is called vertex-pancyclic, if every vertex of D is contained in cycles of length n for all n ∈ {3, 4, . . . , |V (D)|}. A digraph D is said to be strongly connected or just strong, if for every pair x, y of vertices in D, there is a path from x to y. The digraph D is called k-strong, if for arbitrary k − 1 vertices x1 , x2 , . . . , xk−1 of D the digraph D[V (D) − {x1 , x2 , . . . , xk−1 }] is strong. The connectivity of D, denoted by κ(D), is then defined to be the largest value of k such that D is k-strong. If we replace in a digraph D every arc x y by yx, then we call the resulting digraph the converse of D, denoted by D −1 . There are several measures of how much a digraph differs from being regular. In [13], Yeo defines the global irregularity of a digraph D by
i g (D) = max {d + (x), d − (x)} − min {d + (y), d − (y)} x∈V (D)
y∈V (D)
and the local irregularity as il (D) = max |d + (x) − d − (x)| over all vertices x of D. Clearly, il (D) ≤ i g (D). If i g (D) = 0, then D is regular and if i g (D) ≤ 1, then D is called almost regular. A c-partite or multipartite tournament is an orientation of a complete c-partite graph. A tournament is a c-partite tournament with exactly c vertices. If V1 , V2 , . . . , Vc are the partite sets of a c-partite tournament D and the vertex x of D belongs to the partite set Vi , then we define V (x) = Vi . If D is a c-partite tournament with the partite sets V1 , V2 , . . . , Vc such that |V1 | ≤ |V2 | ≤ · · · ≤ |Vc |, then |Vc | = α(D) is the independence number of D, and we define γ (D) = |V1 |. Multipartite tournaments and especially cycles and paths in this class of digraphs are well-studied (see e.g. [1,2,5,7,8, 11]). In recent years, a special class of cycles have gained increased importance—cycles with at most one vertex from each partite set or equivalently strong subtournaments. In 1999, a first result was presented by Volkmann [6]. Theorem 1.1 (Volkmann [6]). Let D be an almost regular c-partite tournament with c ≥ 4. Then D contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c − 1}. Recently, Volkmann and Winzen [9] settled the conjecture of Volkmann [6] in affirmative that Theorem 1.1 also holds for p = c, if c ≥ 5. This yields the following theorem. Theorem 1.2 (Volkmann and Winzen [9]). Let D be an almost regular c-partite tournament with c ≥ 5. Then D contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c}. One possibility to generalize this result is to look for multipartite tournaments of a higher irregularity. A step in this direction was made by Winzen in 2004. Theorem 1.3 (Winzen [10]). Let D be a c-partite tournament with at least 3 vertices in each partite set, i g (D) ≤ l, c ≥ l + 2 and l ≥ 2. Then D contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c − l + 1}. This paper deals with another interesting generalization of Theorem 1.2. In the class of regular multipartite tournaments we look for strong subtournaments containing a given vertex. Since every 3-cycle is a strong subtournament of order 3, the following theorem of Yeo presents a first result in this direction. Theorem 1.4 (Yeo [12]). Every regular multipartite tournament with at least 5 partite sets is vertex-pancyclic. Using this theorem as the basis of induction we will prove that each vertex of a regular c-partite tournament D with c ≥ 7 is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c − 4}. 2. Preliminary results The following results play an important role in our investigations. Theorem 2.1 (Moon [4]). Every vertex of a strongly connected tournament T is contained in a cycle of order m for all 3 ≤ m ≤ |V (T )|.
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The next well-known proposition follows directly from Theorem 2.1. It was formulated and proved by Korvin [3] in 1967. Corollary 2.2. Every strongly connected tournament T with at least 4 vertices contains two different vertices v1 and v2 such that T − {vi } is still strongly connected for i = 1, 2. Theorem 2.3 (Yeo [11]). If D is a multipartite tournament, then κ(D) ≥
|V (D)| − α(D) − 2il (D) . 3
Lemma 2.4 (Yeo [11]). If X is a non-empty vertex set of a digraph D, then il (D) ≥
|d(X, V (D) − X ) − d(V (D) − X, X )| , |X |
which means that if D is regular, then it follows that |d(X, V (D) − X ) − d(V (D) − X, X )| = 0. Lemma 2.5 (Volkmann [6]). If T is a strongly connected tournament of order |V (T )| ≥ 4, then there exists a vertex u ∈ V (T ) of maximum outdegree such that for all x ∈ V (T ) − {u}, the subtournament T − x has a Hamiltonian path with the initial vertex u. Remark 2.6. If V1 , V2 , . . . , Vc are the partite sets of a regular c-partite tournament D, then it follows that r = |V1 | = |V2 | = . . . = |Vc | and (c − 1)r 2 for all x ∈ V (D). That means especially that c is odd or r is even. d + (x), d − (x) =
3. Main result Theorem 3.1. If D is a regular c-partite tournament with c ≥ 7, then each vertex of D is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c − 4}. Proof. Let V1 , V2 , . . . , Vc the partite sets of D such that |V1 | = |V2 | = . . . = |Vc | = r . According to Theorem 2.3, we have that |V (D)| − α(D) (c − 1)r κ(D) ≥ = ≥ 2r > 1, 3 3 which means that D is strongly connected. Let v1 ∈ V (D) be arbitrary. Assume, without loss of generality, that v1 ∈ V1 . We proceed the proof by induction on the order p of the strongly connected subtournaments containing v1 . Theorem 1.4 ensures the existence of a 3-cycle through the vertex v1 and hence the desired strong subtournament of order 3. Now let c ≥ 8 and let T p be a strong subtournament of order p containing v1 with 3 ≤ p ≤ c − 5. Without loss of generality, we assume that T p = D[{v1 , v2 , . . . , v p }] such that vi ∈ Vi for i = 1, 2, . . . , p. Suppose that v1 is not contained in any strong subtournament of order p + 1 of D. If there is a vertex z ∈ V p+1 ∪ V p+2 ∪ . . . Vc such that z has an out-neighbor as well as an in-neighbor in T p , then it is straightforward to verify that D[{z, v1 , v2 , . . . , v p }] is a strong subtournament of order p + 1, a contradiction. If such a vertex does not exist, then let Vi0 ⊆ Vi and Vi00 = Vi − Vi0 such that V (T p ) → Vi0 when Vi0 6= ∅ and Vi00 → V (T p ) when Vi00 6= ∅ for i = p + 1, p + 2, . . . , c. 0 0 00 00 In addition, we define V 0 = V p+1 ∪ V p+2 ∪ · · · ∪ Vc0 and V 00 = V p+1 ∪ V p+2 ∪ · · · ∪ Vc00 . Let U contain all the 0 00 vertices of V (D) − (V ∪ V ∪ V (T p )), which are dominated by a vertex from V 0 , and let W be the set of vertices from V (D) − (V 0 ∪ V 00 ∪ V (T p )), which are not dominated by any vertex from V 0 . Thus, W → V 0 , if W 6= ∅. Now we distinguish two cases.
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Case 1. Let V 0 = ∅ or V 00 = ∅. Without loss of generality, we discuss the case V 00 = ∅. Then Vi0 = Vi for p + 1 ≤ i ≤ c, and we write V instead of V 0 . By the definition of the sets U and W it follows that d(V, V (D) − V ) ≤ |U ||V | and d(V (D) − V, V ) ≥ |V ||V (D) − (U ∪ V )|. Consequently, Lemma 2.4 implies 0 = d(V (D) − V, V ) − d(V, V (D) − V ) ≥ |V |(|V (D)| − |V | − 2|U |), and this yields to |V (D)| − |V | . 2 We now consider the following two subcases. |U | ≥
(1)
Subcase 1.1. Let p = 3. If there exists any vertex u ∈ U − V (v1 ) such that u dominates two vertices from T p , then u, a vertex z ∈ V such that z → u, and the two vertices from T p , which are not in the same partite set as u, induce a strong subtournament of order 4 containing v1 , a contradiction. If such a vertex does not exist, then, since every vertex of U has exactly two neighbors in T p , we deduce that d(U − V (v1 ), V (T p )) ≤ d(V (T p ), U − V (v1 )). If w ∈ W , then w also has exactly two neighbors in T p , and hence it follows that d(W, V (T p )) ≤ 2|W |. In view of Lemma 2.4, we now obtain 0 = d(V (T p ), V (D) − V (T p )) − d(V (D) − V (T p ), V (T p )) = d(V (T p ), V ) + d(V (T p ), U ) + d(V (T p ), W ) − d(V, V (T p )) − d(U, V (T p )) − d(W, V (T p )) ≥ |V (T p )||V | + d(V (T p ), U ) − d(U, V (T p )) − d(W, V (T p )) ≥ 3|V | − 2|W | + d(V (T p ), U ∩ V (v1 )) − d(U ∩ V (v1 ), V (T p )) ≥ 3|V | − 2|W | − 2(r − 1) = 3|V | − 2(|V (D)| − |V | − |U | − |V (T p )|) − 2r + 2 = 5|V | − 2|V (D)| + 2|U | + 8 − 2r. Combining this estimation with (1), we find |V (D)| − |V | ≤ 2|U | ≤ 2|V (D)| − 5|V | − 8 + 2r, and this implies |V (D)| = |V | + 3r ≥ 4|V | + 8 − 2r ⇒ 5r ≥ 3|V | + 8, a contradiction to |V | = (c − p)r ≥ 5r and r ≥ 1. Subcase 1.2. Let p ≥ 4. According to Lemma 2.5, there is a vertex v ∈ V (T p ) such that for all y ∈ V (T p ) − {v}, the subtournament T p − y has a Hamiltonian path with the initial vertex v. If there is a vertex u ∈ (U − V (v1 )) with u → v, then let w ∈ V such that w → u. If u ∈ Vt , then the vertices w, u and v j with 1 ≤ j ≤ p and j 6= t induce a strongly connected subtournament of order p + 1 and v1 is a vertex of this subtournament, a contradiction. If otherwise, there is no such vertex u, then clearly, v ; (U − V (v1 )). By Lemma 2.5, the vertex v has maximum outdegree in T p , and thus, d + (v, T p ) ≥ 2. If v ∈ Vi , then, because of |V | = (c − p)r ≥ 5r , it follows from (1) that d + (v) ≥ |V | + |U − (Vi − {v}) − (V1 − {v1 })| + d + (v, T p ) |V (D)| − |V | ≥ |V | + − 2r + 2 + 2 2 |V (D)| + |V | − 4r + 8 = 2 |V (D)| + r + 8 ≥ 2 (c − 1)r + 2r + 8 = , 2 a contradiction to Remark 2.6.
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Case 2. Let V 0 6= ∅ and V 00 6= ∅. Without loss of generality, let |V 00 | ≥ |V 0 |. Let Uˆ = U − V (v1 ) and ˆ W = W − V (v1 ). For a vertex v 00 ∈ V 00 we define the set Uv 00 ⊆ Uˆ by Uv 00 = {u ∈ Uˆ | N − (u) ∩ V 0 ⊂ V (v 00 )}. If there are vertices v 00 ∈ V 00 and u ∈ Uˆ − Uv 00 such that u → v 00 , then let v 0 ∈ ((N − (u) ∩ V 0 ) − V (v 00 )). If u ∈ V (v j ) with v j ∈ (V (T p ) − V (v1 )), then we see that D[(V (T p ) − {v j }) ∪ {v 0 , u, v 00 }] is a strongly connected tournament of order p + 2. According to Corollary 2.2, this tournament has a strongly connected subtournament of order p + 1 containing the vertex v1 , a contradiction. Combining our results, for any vertex v 00 ∈ V 00 we observe that v 00 ; (V 0 ∪ (N + (v 00 ) ∩ V 00 ) ∪ V (T p ) ∪ (Uˆ − Uv 00 )).
(2)
Subcase 2.1. Assume that V 0 consists of vertices of only one partite set V j . This implies that |V 00 | = (c − p)r − |V j ∩ V 0 |. Because of c − p ≥ 5, there is a vertex v 00 ∈ V 00 − V j such that |V 00 | − |V j ∩ V 00 | − |V (v 00 ) ∩ V 00 | 2 (c − p)r − |V j ∩ V 0 | − |V j ∩ V 00 | − |V (v 00 ) ∩ V 00 | = 2 (c − p − 2)r ≥ . 2
+ 00 d D[V 00 ] (v ) ≥
The choice of the vertex v 00 implies that Uv 00 = ∅ and with (2) it follows that v 00 → Uˆ . Altogether we deduce that d + (v 00 )
≥ c− p≥5
≥
d + (v 0 ) − |V (v1 ) − {v1 }| + d + (v 0 ) − r + 1 +
(c − p − 2)r + |{v 0 }| + |V (T p )| 2
3r +1+ p 2
r + p + 2, 2 because of r ≥ 1 and p ≥ 3 a contradiction to the regularity of D. =
d + (v 0 ) +
Subcase 2.2. Assume that V 0 contains vertices from at least two partite sets. Let v 00 ∈ V 00 such that + 00 d D[V 00 ] (v ) ≥
|V 00 | − |V (v 00 ) ∩ V 00 | . 2
Then there exists a vertex v 0 ∈ V 0 − V (v 00 ) such that |V 0 − V (v 0 ) − V (v 00 )| . 2 According to (2) we observe that − 0 d D[V 0 −V (v 00 )] (v ) ≥
d + (v 00 ) ≥ |Uˆ − Uv 00 | + |V 0 | − |V (v 00 ) ∩ V 0 | + |V (T p )| + |N + (v 00 ) ∩ V 00 |, whereas − 0 0 0 ˆ d + (v 0 ) ≤ |V 0 | − d D[V 0 −V (v 00 )] (v ) − |V (v ) ∩ V | + |U − Uv 00 | + |V (v1 ) − {v1 }|.
Combining these results we arrive at d + (v 00 )
≥
≥
= ≥
− 00 0 0 0 0 d + (v 0 ) + d D[V 0 −V (v 00 )] (v ) + |V (v ) ∩ V | − |V (v1 ) − {v1 }| − |V (v ) ∩ V |
+ p + |N + (v 00 ) ∩ V 00 | |V 0 | − |V (v 0 ) ∩ V 0 | − |V (v 00 ) ∩ V 0 | |V 00 | − |V (v 00 ) ∩ V 00 | + d + (v 0 ) + 2 2 + |V (v 0 ) ∩ V 0 | − r + 1 + p − |V (v 00 ) ∩ V 0 | |V 0 | + |V 00 | + |V (v 0 ) ∩ V 0 | − r − 2|V (v 00 ) ∩ V 0 | d + (v 0 ) + −r +1+ p 2 (c − p − 3)r + 3 d + (v 0 ) + −r +1+ p 2
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2r + 3 −r +1+ p 2 5 = d + (v 0 ) + p + , 2 a contradiction to the regularity of D. This completes the proof of this theorem. c− p≥5
≥
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d + (v 0 ) +
The relatively strong contradictions let us assume that Theorem 3.1 is not the best possible. Conjecture 3.2. Let D be a regular c-partite tournament with c ≥ 5. Then each vertex of D is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c}. Another interesting question is what happens, if the global irregularity of the multipartite increases. Problem 3.3. Let D be a c-partite tournament of global irregularity i g (D) = i ≥ 1. Find optimal values l(i) such that each vertex of D is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , l(i)}. References [1] Y. Guo, Semicomplete multipartite digraphs: A generalization of tournaments, Habilitation Thesis, RWTH Aachen, 1998. [2] G. Gutin, Cycles and paths in semicomplete multipartite digraphs, theorems and algorithms: A survey, J. Graph Theory 19 (1995) 481–505. [3] G. Korvin, Some combinatorial problems on complete directed graphs, in: Theory of Graphs, International Symposium, Rom, 1966, Dunod Paris, Gordon and Beach, New York, 1967, pp. 197–213. [4] J.W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9 (1996) 297–301. [5] M. Tewes, In-tournaments and semicomplete multipartite digraphs, Ph.D. Thesis, RWTH Aachen, Germany, 1999. [6] L. Volkmann, Strong subtournaments of multipartite tournaments, Australas J. Combin. 20 (1999) 189–196. [7] L. Volkmann, Cycles in multipartite tournaments: Results and problems, Discrete Math. 245 (2002) 19–53. [8] L. Volkmann, Multipartite tournaments: A survey, Discrete Math. (2007), doi:10.1016/j.disc.2007.03.053. [9] L. Volkmann, S. Winzen, Almost regular c-partite tournaments contain a strong subtournament of order c when c ≥ 5, Discrete Math. (2007), doi:10.1016/j.disc.2006.10.019. [10] S. Winzen, Strong subtournaments of close to regular multipartite tournaments, Australas. J. Combin. 29 (2004) 49–57. [11] A. Yeo, Semicomplete multipartite digraphs, Ph. D. Thesis, Odense University, 1998. [12] A. Yeo, Diregular c-partite tournaments are vertex-pancyclic when c ≥ 5, J. Graph Theory 32 (1999) 137–152. [13] A. Yeo, How close to regular must a semicomplete multipartite digraph be to secure Hamiltonicity? Graphs Combin. 15 (1999) 481–493.
Strong subtournaments containing a given vertex in regular multipartite tournaments Lutz Volkmann, Stefan Winzen Lehrstuhl II f¨ur Mathematik, RWTH Aachen University, Germany Received 23 May 2007; received in revised form 2 October 2007; accepted 2 October 2007 Available online 19 November 2007
Abstract If x is a vertex of a digraph D, then we denote by d + (x) and d − (x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by i g (D) = max {d + (x), d − (x)} − min {d + (y), d − (y)}. x∈V (D)
y∈V (D)
If i g (D) = 0, then D is regular and if i g (D) ≤ 1, then D is called almost regular. A c-partite tournament is an orientation of a complete c-partite graph. Recently, Volkmann and Winzen [L. Volkmann, S. Winzen, Almost regular c-partite tournaments contain a strong subtournament of order c when c ≥ 5, Discrete Math. (2007), 10.1016/j.disc.2006.10.019] showed that every almost regular c-partite tournament D with c ≥ 5 contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c}. In this paper for the class of regular multipartite tournaments we will consider the more difficult question for the existence of strong subtournaments containing a given vertex. We will prove that each vertex of a regular multipartite tournament D with c ≥ 7 partite sets is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c − 4}. c 2007 Elsevier B.V. All rights reserved.
Keywords: Multipartite tournaments; Regular multipartite tournaments; Subtournaments
1. Terminology and introduction In this paper all digraphs are finite without loops and multiple arcs. The vertex set and arc set of a digraph D is denoted by V (D) and E(D), respectively. If x y is an arc of a digraph D, then we write x → y and say x dominates y, and if X and Y are two disjoint vertex sets or subdigraphs of D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X → Y . Furthermore, X ; Y denotes the fact that there is no arc leading from Y to X . By d(X, Y ) we denote the number of arcs from the set X to the set Y , + i.e., d(X, Y ) = |{x y ∈ E(D) : x ∈ X, y ∈ Y }|. If D is a digraph, then the out-neighborhood N D (x) = N + (x) − − of a vertex x is the set of vertices dominated by x and the in-neighborhood N D (x) = N (x) is the set of vertices dominating x. Therefore, if there is the arc x y ∈ E(D), then y is an outer neighbor of x and x is an inner neighbor
E-mail address: [email protected] (L. Volkmann). c 2007 Elsevier B.V. All rights reserved. 0012-365X/$ - see front matter doi:10.1016/j.disc.2007.10.006
L. Volkmann, S. Winzen / Discrete Mathematics 308 (2008) 5516–5521
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+ − of y. The numbers d D (x) = d + (x) = |N + (x)| and d D (x) = d − (x) = |N − (x)| are called the outdegree and indegree of x, respectively. For a vertex set X of D, we define D[X ] as the subdigraph induced by X . If we speak of a cycle (path), then we mean a directed cycle (directed path), and a cycle of length n is called an n-cycle. A cycle (path) of a digraph D is Hamiltonian, if it includes all the vertices of D. A digraph D is called vertex-pancyclic, if every vertex of D is contained in cycles of length n for all n ∈ {3, 4, . . . , |V (D)|}. A digraph D is said to be strongly connected or just strong, if for every pair x, y of vertices in D, there is a path from x to y. The digraph D is called k-strong, if for arbitrary k − 1 vertices x1 , x2 , . . . , xk−1 of D the digraph D[V (D) − {x1 , x2 , . . . , xk−1 }] is strong. The connectivity of D, denoted by κ(D), is then defined to be the largest value of k such that D is k-strong. If we replace in a digraph D every arc x y by yx, then we call the resulting digraph the converse of D, denoted by D −1 . There are several measures of how much a digraph differs from being regular. In [13], Yeo defines the global irregularity of a digraph D by
i g (D) = max {d + (x), d − (x)} − min {d + (y), d − (y)} x∈V (D)
y∈V (D)
and the local irregularity as il (D) = max |d + (x) − d − (x)| over all vertices x of D. Clearly, il (D) ≤ i g (D). If i g (D) = 0, then D is regular and if i g (D) ≤ 1, then D is called almost regular. A c-partite or multipartite tournament is an orientation of a complete c-partite graph. A tournament is a c-partite tournament with exactly c vertices. If V1 , V2 , . . . , Vc are the partite sets of a c-partite tournament D and the vertex x of D belongs to the partite set Vi , then we define V (x) = Vi . If D is a c-partite tournament with the partite sets V1 , V2 , . . . , Vc such that |V1 | ≤ |V2 | ≤ · · · ≤ |Vc |, then |Vc | = α(D) is the independence number of D, and we define γ (D) = |V1 |. Multipartite tournaments and especially cycles and paths in this class of digraphs are well-studied (see e.g. [1,2,5,7,8, 11]). In recent years, a special class of cycles have gained increased importance—cycles with at most one vertex from each partite set or equivalently strong subtournaments. In 1999, a first result was presented by Volkmann [6]. Theorem 1.1 (Volkmann [6]). Let D be an almost regular c-partite tournament with c ≥ 4. Then D contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c − 1}. Recently, Volkmann and Winzen [9] settled the conjecture of Volkmann [6] in affirmative that Theorem 1.1 also holds for p = c, if c ≥ 5. This yields the following theorem. Theorem 1.2 (Volkmann and Winzen [9]). Let D be an almost regular c-partite tournament with c ≥ 5. Then D contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c}. One possibility to generalize this result is to look for multipartite tournaments of a higher irregularity. A step in this direction was made by Winzen in 2004. Theorem 1.3 (Winzen [10]). Let D be a c-partite tournament with at least 3 vertices in each partite set, i g (D) ≤ l, c ≥ l + 2 and l ≥ 2. Then D contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c − l + 1}. This paper deals with another interesting generalization of Theorem 1.2. In the class of regular multipartite tournaments we look for strong subtournaments containing a given vertex. Since every 3-cycle is a strong subtournament of order 3, the following theorem of Yeo presents a first result in this direction. Theorem 1.4 (Yeo [12]). Every regular multipartite tournament with at least 5 partite sets is vertex-pancyclic. Using this theorem as the basis of induction we will prove that each vertex of a regular c-partite tournament D with c ≥ 7 is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c − 4}. 2. Preliminary results The following results play an important role in our investigations. Theorem 2.1 (Moon [4]). Every vertex of a strongly connected tournament T is contained in a cycle of order m for all 3 ≤ m ≤ |V (T )|.
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The next well-known proposition follows directly from Theorem 2.1. It was formulated and proved by Korvin [3] in 1967. Corollary 2.2. Every strongly connected tournament T with at least 4 vertices contains two different vertices v1 and v2 such that T − {vi } is still strongly connected for i = 1, 2. Theorem 2.3 (Yeo [11]). If D is a multipartite tournament, then κ(D) ≥
|V (D)| − α(D) − 2il (D) . 3
Lemma 2.4 (Yeo [11]). If X is a non-empty vertex set of a digraph D, then il (D) ≥
|d(X, V (D) − X ) − d(V (D) − X, X )| , |X |
which means that if D is regular, then it follows that |d(X, V (D) − X ) − d(V (D) − X, X )| = 0. Lemma 2.5 (Volkmann [6]). If T is a strongly connected tournament of order |V (T )| ≥ 4, then there exists a vertex u ∈ V (T ) of maximum outdegree such that for all x ∈ V (T ) − {u}, the subtournament T − x has a Hamiltonian path with the initial vertex u. Remark 2.6. If V1 , V2 , . . . , Vc are the partite sets of a regular c-partite tournament D, then it follows that r = |V1 | = |V2 | = . . . = |Vc | and (c − 1)r 2 for all x ∈ V (D). That means especially that c is odd or r is even. d + (x), d − (x) =
3. Main result Theorem 3.1. If D is a regular c-partite tournament with c ≥ 7, then each vertex of D is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c − 4}. Proof. Let V1 , V2 , . . . , Vc the partite sets of D such that |V1 | = |V2 | = . . . = |Vc | = r . According to Theorem 2.3, we have that |V (D)| − α(D) (c − 1)r κ(D) ≥ = ≥ 2r > 1, 3 3 which means that D is strongly connected. Let v1 ∈ V (D) be arbitrary. Assume, without loss of generality, that v1 ∈ V1 . We proceed the proof by induction on the order p of the strongly connected subtournaments containing v1 . Theorem 1.4 ensures the existence of a 3-cycle through the vertex v1 and hence the desired strong subtournament of order 3. Now let c ≥ 8 and let T p be a strong subtournament of order p containing v1 with 3 ≤ p ≤ c − 5. Without loss of generality, we assume that T p = D[{v1 , v2 , . . . , v p }] such that vi ∈ Vi for i = 1, 2, . . . , p. Suppose that v1 is not contained in any strong subtournament of order p + 1 of D. If there is a vertex z ∈ V p+1 ∪ V p+2 ∪ . . . Vc such that z has an out-neighbor as well as an in-neighbor in T p , then it is straightforward to verify that D[{z, v1 , v2 , . . . , v p }] is a strong subtournament of order p + 1, a contradiction. If such a vertex does not exist, then let Vi0 ⊆ Vi and Vi00 = Vi − Vi0 such that V (T p ) → Vi0 when Vi0 6= ∅ and Vi00 → V (T p ) when Vi00 6= ∅ for i = p + 1, p + 2, . . . , c. 0 0 00 00 In addition, we define V 0 = V p+1 ∪ V p+2 ∪ · · · ∪ Vc0 and V 00 = V p+1 ∪ V p+2 ∪ · · · ∪ Vc00 . Let U contain all the 0 00 vertices of V (D) − (V ∪ V ∪ V (T p )), which are dominated by a vertex from V 0 , and let W be the set of vertices from V (D) − (V 0 ∪ V 00 ∪ V (T p )), which are not dominated by any vertex from V 0 . Thus, W → V 0 , if W 6= ∅. Now we distinguish two cases.
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Case 1. Let V 0 = ∅ or V 00 = ∅. Without loss of generality, we discuss the case V 00 = ∅. Then Vi0 = Vi for p + 1 ≤ i ≤ c, and we write V instead of V 0 . By the definition of the sets U and W it follows that d(V, V (D) − V ) ≤ |U ||V | and d(V (D) − V, V ) ≥ |V ||V (D) − (U ∪ V )|. Consequently, Lemma 2.4 implies 0 = d(V (D) − V, V ) − d(V, V (D) − V ) ≥ |V |(|V (D)| − |V | − 2|U |), and this yields to |V (D)| − |V | . 2 We now consider the following two subcases. |U | ≥
(1)
Subcase 1.1. Let p = 3. If there exists any vertex u ∈ U − V (v1 ) such that u dominates two vertices from T p , then u, a vertex z ∈ V such that z → u, and the two vertices from T p , which are not in the same partite set as u, induce a strong subtournament of order 4 containing v1 , a contradiction. If such a vertex does not exist, then, since every vertex of U has exactly two neighbors in T p , we deduce that d(U − V (v1 ), V (T p )) ≤ d(V (T p ), U − V (v1 )). If w ∈ W , then w also has exactly two neighbors in T p , and hence it follows that d(W, V (T p )) ≤ 2|W |. In view of Lemma 2.4, we now obtain 0 = d(V (T p ), V (D) − V (T p )) − d(V (D) − V (T p ), V (T p )) = d(V (T p ), V ) + d(V (T p ), U ) + d(V (T p ), W ) − d(V, V (T p )) − d(U, V (T p )) − d(W, V (T p )) ≥ |V (T p )||V | + d(V (T p ), U ) − d(U, V (T p )) − d(W, V (T p )) ≥ 3|V | − 2|W | + d(V (T p ), U ∩ V (v1 )) − d(U ∩ V (v1 ), V (T p )) ≥ 3|V | − 2|W | − 2(r − 1) = 3|V | − 2(|V (D)| − |V | − |U | − |V (T p )|) − 2r + 2 = 5|V | − 2|V (D)| + 2|U | + 8 − 2r. Combining this estimation with (1), we find |V (D)| − |V | ≤ 2|U | ≤ 2|V (D)| − 5|V | − 8 + 2r, and this implies |V (D)| = |V | + 3r ≥ 4|V | + 8 − 2r ⇒ 5r ≥ 3|V | + 8, a contradiction to |V | = (c − p)r ≥ 5r and r ≥ 1. Subcase 1.2. Let p ≥ 4. According to Lemma 2.5, there is a vertex v ∈ V (T p ) such that for all y ∈ V (T p ) − {v}, the subtournament T p − y has a Hamiltonian path with the initial vertex v. If there is a vertex u ∈ (U − V (v1 )) with u → v, then let w ∈ V such that w → u. If u ∈ Vt , then the vertices w, u and v j with 1 ≤ j ≤ p and j 6= t induce a strongly connected subtournament of order p + 1 and v1 is a vertex of this subtournament, a contradiction. If otherwise, there is no such vertex u, then clearly, v ; (U − V (v1 )). By Lemma 2.5, the vertex v has maximum outdegree in T p , and thus, d + (v, T p ) ≥ 2. If v ∈ Vi , then, because of |V | = (c − p)r ≥ 5r , it follows from (1) that d + (v) ≥ |V | + |U − (Vi − {v}) − (V1 − {v1 })| + d + (v, T p ) |V (D)| − |V | ≥ |V | + − 2r + 2 + 2 2 |V (D)| + |V | − 4r + 8 = 2 |V (D)| + r + 8 ≥ 2 (c − 1)r + 2r + 8 = , 2 a contradiction to Remark 2.6.
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Case 2. Let V 0 6= ∅ and V 00 6= ∅. Without loss of generality, let |V 00 | ≥ |V 0 |. Let Uˆ = U − V (v1 ) and ˆ W = W − V (v1 ). For a vertex v 00 ∈ V 00 we define the set Uv 00 ⊆ Uˆ by Uv 00 = {u ∈ Uˆ | N − (u) ∩ V 0 ⊂ V (v 00 )}. If there are vertices v 00 ∈ V 00 and u ∈ Uˆ − Uv 00 such that u → v 00 , then let v 0 ∈ ((N − (u) ∩ V 0 ) − V (v 00 )). If u ∈ V (v j ) with v j ∈ (V (T p ) − V (v1 )), then we see that D[(V (T p ) − {v j }) ∪ {v 0 , u, v 00 }] is a strongly connected tournament of order p + 2. According to Corollary 2.2, this tournament has a strongly connected subtournament of order p + 1 containing the vertex v1 , a contradiction. Combining our results, for any vertex v 00 ∈ V 00 we observe that v 00 ; (V 0 ∪ (N + (v 00 ) ∩ V 00 ) ∪ V (T p ) ∪ (Uˆ − Uv 00 )).
(2)
Subcase 2.1. Assume that V 0 consists of vertices of only one partite set V j . This implies that |V 00 | = (c − p)r − |V j ∩ V 0 |. Because of c − p ≥ 5, there is a vertex v 00 ∈ V 00 − V j such that |V 00 | − |V j ∩ V 00 | − |V (v 00 ) ∩ V 00 | 2 (c − p)r − |V j ∩ V 0 | − |V j ∩ V 00 | − |V (v 00 ) ∩ V 00 | = 2 (c − p − 2)r ≥ . 2
+ 00 d D[V 00 ] (v ) ≥
The choice of the vertex v 00 implies that Uv 00 = ∅ and with (2) it follows that v 00 → Uˆ . Altogether we deduce that d + (v 00 )
≥ c− p≥5
≥
d + (v 0 ) − |V (v1 ) − {v1 }| + d + (v 0 ) − r + 1 +
(c − p − 2)r + |{v 0 }| + |V (T p )| 2
3r +1+ p 2
r + p + 2, 2 because of r ≥ 1 and p ≥ 3 a contradiction to the regularity of D. =
d + (v 0 ) +
Subcase 2.2. Assume that V 0 contains vertices from at least two partite sets. Let v 00 ∈ V 00 such that + 00 d D[V 00 ] (v ) ≥
|V 00 | − |V (v 00 ) ∩ V 00 | . 2
Then there exists a vertex v 0 ∈ V 0 − V (v 00 ) such that |V 0 − V (v 0 ) − V (v 00 )| . 2 According to (2) we observe that − 0 d D[V 0 −V (v 00 )] (v ) ≥
d + (v 00 ) ≥ |Uˆ − Uv 00 | + |V 0 | − |V (v 00 ) ∩ V 0 | + |V (T p )| + |N + (v 00 ) ∩ V 00 |, whereas − 0 0 0 ˆ d + (v 0 ) ≤ |V 0 | − d D[V 0 −V (v 00 )] (v ) − |V (v ) ∩ V | + |U − Uv 00 | + |V (v1 ) − {v1 }|.
Combining these results we arrive at d + (v 00 )
≥
≥
= ≥
− 00 0 0 0 0 d + (v 0 ) + d D[V 0 −V (v 00 )] (v ) + |V (v ) ∩ V | − |V (v1 ) − {v1 }| − |V (v ) ∩ V |
+ p + |N + (v 00 ) ∩ V 00 | |V 0 | − |V (v 0 ) ∩ V 0 | − |V (v 00 ) ∩ V 0 | |V 00 | − |V (v 00 ) ∩ V 00 | + d + (v 0 ) + 2 2 + |V (v 0 ) ∩ V 0 | − r + 1 + p − |V (v 00 ) ∩ V 0 | |V 0 | + |V 00 | + |V (v 0 ) ∩ V 0 | − r − 2|V (v 00 ) ∩ V 0 | d + (v 0 ) + −r +1+ p 2 (c − p − 3)r + 3 d + (v 0 ) + −r +1+ p 2
L. Volkmann, S. Winzen / Discrete Mathematics 308 (2008) 5516–5521
2r + 3 −r +1+ p 2 5 = d + (v 0 ) + p + , 2 a contradiction to the regularity of D. This completes the proof of this theorem. c− p≥5
≥
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d + (v 0 ) +
The relatively strong contradictions let us assume that Theorem 3.1 is not the best possible. Conjecture 3.2. Let D be a regular c-partite tournament with c ≥ 5. Then each vertex of D is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c}. Another interesting question is what happens, if the global irregularity of the multipartite increases. Problem 3.3. Let D be a c-partite tournament of global irregularity i g (D) = i ≥ 1. Find optimal values l(i) such that each vertex of D is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , l(i)}. References [1] Y. Guo, Semicomplete multipartite digraphs: A generalization of tournaments, Habilitation Thesis, RWTH Aachen, 1998. [2] G. Gutin, Cycles and paths in semicomplete multipartite digraphs, theorems and algorithms: A survey, J. Graph Theory 19 (1995) 481–505. [3] G. Korvin, Some combinatorial problems on complete directed graphs, in: Theory of Graphs, International Symposium, Rom, 1966, Dunod Paris, Gordon and Beach, New York, 1967, pp. 197–213. [4] J.W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9 (1996) 297–301. [5] M. Tewes, In-tournaments and semicomplete multipartite digraphs, Ph.D. Thesis, RWTH Aachen, Germany, 1999. [6] L. Volkmann, Strong subtournaments of multipartite tournaments, Australas J. Combin. 20 (1999) 189–196. [7] L. Volkmann, Cycles in multipartite tournaments: Results and problems, Discrete Math. 245 (2002) 19–53. [8] L. Volkmann, Multipartite tournaments: A survey, Discrete Math. (2007), doi:10.1016/j.disc.2007.03.053. [9] L. Volkmann, S. Winzen, Almost regular c-partite tournaments contain a strong subtournament of order c when c ≥ 5, Discrete Math. (2007), doi:10.1016/j.disc.2006.10.019. [10] S. Winzen, Strong subtournaments of close to regular multipartite tournaments, Australas. J. Combin. 29 (2004) 49–57. [11] A. Yeo, Semicomplete multipartite digraphs, Ph. D. Thesis, Odense University, 1998. [12] A. Yeo, Diregular c-partite tournaments are vertex-pancyclic when c ≥ 5, J. Graph Theory 32 (1999) 137–152. [13] A. Yeo, How close to regular must a semicomplete multipartite digraph be to secure Hamiltonicity? Graphs Combin. 15 (1999) 481–493.