Unavoidable trees in tournaments

An oriented tree T on n vertices is unavoidable if every tournament on n vertices contains a copy of T. In this paper, we give a sufficient condition for T to be unavoidable, and use this to prove that almost all labeled oriented trees are unavoidable, verifying a conjecture of Bender and Wormald. We additionally prove that every tournament on vertices contains a copy of every oriented tree T on n vertices with polylogarithmic maximum degree, improving a result of Kühn, Mycroft, and Osthus.     

Spanning k‐arc‐strong subdigraphs with few arcs in k‐arc‐strong tournaments

Given a k‐arc‐strong tournament T, we estimate the minimum number of arcs possible in a k‐arc‐strong spanning subdigraph of T. We give a construction which shows that for each k ≥ 2, there are tournaments T on n vertices such that every k‐arc‐strong spanning subdigraph of T contains at least arcs. In fact, the tournaments in our construction have the property that every spanning subdigraph with minimum in‐ and out‐degree at least k has arcs. This is best possible since it can be shown that every k‐arc‐strong tournament contains a spanning subdigraph with minimum in‐ and out‐degree at least k and no more than arcs. As our main result we prove that every k‐arc‐strong tournament contains a spanning k‐arc‐strong subdigraph with no more than arcs. We conjecture that for every k‐arc‐strong tournament T, the minimum number of arcs in a k‐arc‐strong spanning subdigraph of T is equal to the minimum number of arcs in a spanning subdigraph of T with the property that every vertex has in‐ and out‐degree at least k. We also discuss the implications of our results on related problems and conjectures. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 265–284, 2004     

A reversal index for tournament matrices *

Given a tournament matrix T, its reversal indexi_R (T), is the minimum k such that the reversal of the orientation of k arcs in the directed graph associated with T results in a reducible matrix. We give a formula for i_R (T) in terms of the score vector of T which generalizes a simple criterion for a tournament matrix to be irreducible. We show that i_R (T)≤[(n?1)/2] for any tournament matrix T of order n, with equality holding if and only if T is regular or almost regular, according as n is odd or even. We construct, for each k between 1 and [(n?1)/2], a tournament matrix of order n whose reversal index is k. Finally, we suggest a few problems.     

Hamiltonian paths and cycles in hypertournaments

Given two integers n and k, n ≥ k > 1, a k-hypertournament T on n vertices is a pair (V, A), where V is a set of vertices, |V| = n and A is a set of k-tuples of vertices, called arcs, so that for any k-subset S of V, A$ contains exactly one of the k! k-tuples whose entries belong to S. A 2-hypertournament is merely an (ordinary) tournament. A path is a sequence v₁a₁v₂v₃···v_t−1v_t of distinct vertices v₁, v₂,⋖, v_t and distinct arcs a₁, ⋖, a_t−1 such that v_i precedes v_t−1 in a, 1 ≤ i ≤ t − 1. A cycle can be defined analogously. A path or cycle containing all vertices of T (as v_i's) is Hamiltonian. T is strong if T has a path from x to y for every choice of distinct x, y ≡ V. We prove that every k-hypertournament on n (k) vertices has a Hamiltonian path (an extension of Redeis theorem on tournaments) and every strong k-hypertournament with n (k + 1) vertices has a Hamiltonian cycle (an extension of Camions theorem on tournaments). Despite the last result, it is shown that the Hamiltonian cycle problem remains polynomial time solvable only for k ≤ 3 and becomes NP-complete for every fixed integer k ≥ 4. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 277–286, 1997     

The domination and competition graphs of a tournament

Vertices x and y dominate a tournament T if for all vertices z ≠ x, y, either x beats z or y beats z. Let dom(T) be the graph on the vertices of T with edges between pairs of vertices that dominate T. We show that dom(T) is either an odd cycle with possible pendant vertices or a forest of caterpillars. While this is not a characterization, it does lead to considerable information about dom(T). Since dom(T) is the complement of the competition graph of the tournament formed by reversing the arcs of T, complementary results are obtained for the competition graph of a tournament. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 103–110, 1998     

Rooted Spanning Trees in Tournaments

A tournament of order n is an orientation of a complete graph with n vertices, and is specified by its vertex set V(T) and edge set E(T). A rooted tree is a directed tree such that every vertex except the root has in-degree 1, while the root has in-degree 0. A rooted k-tree is a rooted tree such that every vertex except the root has out-degree at most k; the out-degree of the root can be larger than k. It is well-known that every tournament contains a rooted spanning tree of depth at most 2; and the root of such a tree is also called a king in the literature. This result was strengthened to the following one: Every tournament contains a rooted spanning 2-tree of depth at most 2. We prove that every tournament of order n≥800 contains a spanning rooted special 2-tree in this paper, where a rooted special 2-tree is a rooted 2-tree of depth 2 such that all except possibly one, non-root, non-leaf vertices, have out-degree 2 in the tree. Revised: November 9, 1998     

Pancyclic arcs and connectivity in tournaments

A tournament is an orientation of the edges of a complete graph. An arc is pancyclic in a tournament T if it is contained in a cycle of length l, for every 3 ≤ l ≤ |T|. Let p(T) denote the number of pancyclic arcs in a tournament T. In 4 , Moon showed that for every non‐trivial strong tournament T, p(T) ≥ 3. Actually, he proved a somewhat stronger result: for any non‐trivial strong tournament h(T) ≥ 3 where h(T) is the maximum number of pancyclic arcs contained in the same hamiltonian cycle of T. Moreover, Moon characterized the tournaments with h(T) = 3. All these tournaments are not 2‐strong. In this paper, we investigate relationship between the functions p(T) and h(T) and the connectivity of the tournament T. Let p_k(n) := min {p(T), T k‐strong tournament of order n} and h_k(n) := min{h(T), T k‐strong tournament of order n}. We conjecture that (for k ≥ 2) there exists a constant α_k> 0 such that p_k(n) ≥ α_kn and h_k(n) ≥ 2k+1. In this paper, we establish the later conjecture when k = 2. We then characterized the tournaments with h(T) = 4 and those with p(T) = 4. We also prove that for k ≥ 2, p_k(n) ≥ 2k+3. At last, we characterize the tournaments having exactly five pancyclic arcs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 87–110, 2004     

Subdivisions of Transitive Tournaments

We prove that, for r ≥ 2 andn ≥ n(r), every directed graph with n vertices and more edges than the r -partite Turán graph T(r, n) contains a subdivision of the transitive tournament on r + 1 vertices. Furthermore, the extremal graphs are the orientations ofT (r, n) induced by orderings of the vertex classes.     

Path numbers of tournaments

A family of simple (that is, cycle-free) paths is a path decomposition of a tournament T if and only if partitions the acrs of T. The path number of T, denoted pn(T), is the minimum value of | | over all path decompositions of T. In this paper it is shown that if n is even, then there is a tournament on n vertices with path number k if and only if n/2 k n²/4, k an integer. It is also shown that if n is odd and T is a tournament on n vertices, then (n + 1)/2 pn(T) (n² − 1)/4. Moreover, if k is an integer satisfying (i) (n + 1)/2 k n − 1 or (ii) n < k (n² − 1)/4 and k is even, then a tournament on n vertices having path number k is constructed. It is conjectured that there are no tournaments of odd order n with odd path number k for n k < (n² − 1)/4.     

Each 3-strong Tournament Contains 3 Vertices Whose Out-arcs Are Pancyclic

An arc in a tournament T with n ≥ 3 vertices is called pancyclic, if it is in a cycle of length k for all 3 ≤ k ≤ n. Yeo (Journal of Graph Theory, 50 (2005), 212–219) proved that every 3-strong tournament contains two distinct vertices whose all out-arcs are pancyclic, and conjectured that each 2-strong tournament contains 3 such vertices. In this paper, we confirm Yeo’s conjecture for 3-strong tournaments. The author is an associate member of “Graduiertenkolleg: Hierarchie und Symmetrie in mathematischen Modellen (DFG)” at RWTH Aachen University, Germany.     

An approximate version of Sumner?s universal tournament conjecture

Sumner?s universal tournament conjecture states that any tournament on 2n−2 vertices contains a copy of any directed tree on n vertices. We prove an asymptotic version of this conjecture, namely that any tournament on (2+o(1))n vertices contains a copy of any directed tree on n vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed Δ, any tournament on (1+o(1))n vertices contains a copy of any directed tree on n vertices with maximum degree at most Δ.     

The Number of Out-Pancyclic Vertices in a Strong Tournament

An arc in a tournament T with n ≥ 3 vertices is called pancyclic, if it belongs to a cycle of length l for all 3 ≤ l ≤ n. We call a vertex u of T an out-pancyclic vertex of T, if each out-arc of u is pancyclic in T. Yao et al. (Discrete Appl. Math. 99, 245–249, 2000) proved that every strong tournament contains an out-pancyclic vertex. For strong tournaments with minimum out-degree 1, Yao et al. found an infinite class of strong tournaments, each of which contains exactly one out-pancyclic vertex. In this paper, we prove that every strong tournament with minimum out-degree at least 2 contains three out-pancyclic vertices. Our result is best possible since there is an infinite family of strong tournaments with minimum degree at least 2 and no more than 3 out-pancyclic vertices.     

Domination Graphs with Nontrivial Components

 A tournament is an oriented complete graph. Vertices x and y dominate a tournament T if for all vertices z≠x,y, either (x,z) or (y,z) are arcs in T (possibly both). The domination graph of a tournament T is the graph on the vertex set of T containing edge {x,y} if and only if x and y dominate T. In this paper we determine which graphs containing no isolated vertices are domination graphs of tournaments. Received: May 20, 1998 Final version received: May 26, 1999     

Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its first outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2n − 2 contains every arborescence of order n > 1. This is a particular case of Sumner's conjecture: every tournament of order 2n − 2 contains every oriented tree of order n > 1. Using our method, we prove that every tournament of order (7n − 5)/2 contains every oriented tree of order n. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 244–256, 2000     

Outpaths of arcs in multipartite tournaments

1. IntroductionThroughout the paPer, we use the terminology and notation of [1] and [2]. Let D =(V(D), A(D)) be a digraPh. If xy is an arc of a digraPh D, then we say that x dominatesy, denoted by x - y. More generally, if A and B are two disjoint vertex sets of D such thatevery vertex of A dominates every vertex of B, then we say that A dominates B, denotedby A - B. The outset N (x) of a vertex x is the set of vertices dominated by x in D,and the inset N--(x) is the set of vertices d…     

Monochromatic Paths and at Most 2-Coloured Arc Sets in Edge-Coloured Tournaments

We call the tournament T an m-coloured tournament if the arcs of T are coloured with m-colours. If v is a vertex of an m-coloured tournament T, we denote by ξ(v) the set of colours assigned to the arcs with v as an endpoint. In this paper is proved that if T is an m-coloured tournament with |ξ(v)|≤2 for each vertex v of T, and T satisfies at least one of the two following properties (1) m≠3 or (2) m=3 and T contains no C₃ (the directed cycle of length 3 whose arcs are coloured with three distinct colours). Then there is a vertex v of T such that for every other vertex x of T, there is a monochromatic directed path from x to v. Received: April, 2003     

Feedback Vertex Sets in Tournaments

We study combinatorial and algorithmic questions around minimal feedback vertex sets (FVS) in tournament graphs. On the combinatorial side, we derive upper and lower bounds on the maximum number of minimal FVSs in an n‐vertex tournament. We prove that every tournament on n vertices has at most 1.6740ⁿ minimal FVSs, and that there is an infinite family of tournaments, all having at least 1.5448ⁿ minimal FVSs. This improves and extends the bounds of Moon (1971). On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal FVSs of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum‐sized FVS in a tournament.     

Score sets ink-partite tournaments

The set S of distinct scores (outdegrees) of the vertices of ak-partite tournamentT(X ₁, X₂, ···, X_k) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set ofn non-negative integers is a score set of somek-partite tournament for everyn ≥k ≥ 2.     

On the number of cycles in local tournaments

A digraph without loops, multiple arcs and directed cycles of length two is called a local tournament if the set of in-neighbors as well as the set of out-neighbors of every vertex induces a tournament. A vertex of a strongly connected digraph is called a non-separating vertex if its removal preserves the strong connectivity of the digraph in question.In 1990, Bang-Jensen showed that a strongly connected local tournament does not have any non-separating vertices if and only if it is a directed cycle. Guo and Volkmann extended this result in 1994. They determined the strongly connected local tournament with exactly one non-separating vertex. In the first part of this paper we characterize the class of strongly connected local tournaments with exactly two non-separating vertices.In the second part of the paper we consider the following problem: Given a strongly connected local tournament D of order n with at least n+2 arcs and an integer 3≤r≤n. How many directed cycles of length r exist in D? For tournaments this problem was treated by Moon in 1966 and Las Vergnas in 1975. A reformulation of the results of the first part shows that we have characterized the class of strongly connected local tournaments with exactly two directed cycles of length n−1. Among other things we show that D has at least n−r+1 directed cycles of length r for 4≤r≤n−1 unless it has a special structure. Moreover, we characterize the class of local tournaments with exactly n−r+1 directed cycles of length r for 4≤r≤n−1 which generalizes a result of Las Vergnas.     

The structure of 4‐strong tournaments containing exactly three out‐arc pancyclic vertices

Yao et al. (Discrete Appl Math 99 (2000), 245–249) proved that every strong tournament contains a vertex u such that every out‐arc of u is pancyclic and conjectured that every k‐strong tournament contains k such vertices. At present, it is known that this conjecture is true for k = 1, 2, 3 and not true for k?4. In this article, we obtain a sufficient and necessary condition for a 4‐strong tournament to contain exactly three out‐arc pancyclic vertices, which shows that a 4‐strong tournament contains at least four out‐arc pancyclic vertices except for a given class of tournaments. Furthermore, our proof yields a polynomial algorithm to decide if a 4‐strong tournament has exactly three out‐arc pancyclic vertices.