Out-arc pancyclicity of vertices in tournaments

Yao, Guo and Zhang [T. Yao, Y. Guo, K. Zhang, Pancyclic out-arcs of a vertex in a tournament, 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. In this paper, we prove that every strong tournament with minimum out-degree at least two contains two such vertices. Yeo [A. Yeo, The number of pancyclic arcs in a k-strong tournament, J. Graph Theory 50 (2005) 212-219.] conjectured that every 2-strong tournament has three distinct vertices {x,y,z}, such that every arc out of x,y and z is pancyclic. In this paper, we also prove that Yeo’s conjecture is true.     

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.     

Vertex‐pancyclicity of hypertournaments

A hypertournament or a k‐tournament, on n vertices, 2≤k≤n, is a pair T=(V, E), where the vertex set V is a set of size n and the edge set E is the collection of all possible subsets of size k of V, called the edges, each taken in one of its k! possible permutations. A k‐tournament is pancyclic if there exists (directed) cycles of all possible lengths; it is vertex‐pancyclic if moreover the cycles can be found through any vertex. A k‐tournament is strong if there is a path from u to v for each pair of distinct vertices u and v. A question posed by Gutin and Yeo about the characterization of pancyclic and vertex‐pancyclic hypertournaments is examined in this article. We extend Moon's Theorem for tournaments to hypertournaments. We prove that if k≥8 and n≥k + 3, then a k‐tournament on n vertices is vertex‐pancyclic if and only if it is strong. Similar results hold for other values of k. We also show that when n≥7, k≥4, and n≥k + 2, a strong k‐tournament on n vertices is pancyclic if and only if it is strong. The bound n≥k+ 2 is tight. We also find bounds for the generalized problem when we extend vertex‐pancyclicity to require d edge‐disjoint cycles of each possible length and extend strong connectivity to require d edge‐disjoint paths between each pair of vertices. Our results include and extend those of Petrovic and Thomassen. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 338–348, 2010     

The number of vertices whose out-arcs are pancyclic in a 2-strong tournament

An arc going out from a vertex x in a digraph is called an out-arc of x. Yao et al. [Discrete Appl. Math. 99 (2000) 245-249] proved that every strong tournament contains a vertex x such that all out-arcs of x are pancyclic. Recently, Yeo [J. Graph Theory 50 (2005) 212-219] proved that each 3-strong tournament contains two such vertices. In this paper, we confirm that Yeo's result is also true for 2-strong tournaments. Our proof implies a polynomial algorithm to find two such vertices.     

Score certificates for tournaments

The score of a vertex in a tournament is its out-degree. A score certificate for a labeled tournament T is a labeled subdigraph D of T which together with the score sequence of T allows errorless reconstruction of T. In this paper we prove a general lower bound on the sizes of score certificates. Our main theorem can be stated as follows: Except for the regular tournaments on 3 and 5 vertices, every tournament T on n vertices requires at least n−1 arcs in a score certificate for T. This is best possible since every transitive tournament on n vertices has a score certificate with exactly n−1 arcs. © 1997 John Wiley & Sons, Inc.     

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.     

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     

Tournois sans intervalle acyclique

An interval X of a tournament T is a vertex subset of T such that any vertex not in X either dominates or is dominated by all of the vertices in X. We caracterize the tournaments such that the only non empty acyclic intervals are the singletons and which are critical for that property, that is whenever a vertex is removed at least one acyclic interval with more than 2 vertices is created. These tournaments are exactly those which are the composition of any tournament with circulant tournaments. That work on acyclic intervals was motivated by the study of tournaments for which no median order forced itself naturally. To cite this article: J.-F. Culus, B. Jouve, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

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.     

Sur la répartition des diamants dans un tournoi

A tournament T=(V,A) is a directed graph such that for every x,y∈V, where x≠y, (x,y)∈A if and only if (y,x)?A. For example, the 3-cycle is the tournament ({1,2,3}, {(1,2),(2,3),(3,1)}). Up to an isomorphism, there are two tournaments with 4 vertices and containing an unique 3-cycle which we call diamonds. We prove that for any tournament T defined on n?9 vertices, either T contains at least 2n?6 diamonds or the number of diamonds contained in T is equal to 0, n?3 or 2n?8. Following the characterization of the tournaments without diamonds due to Gnanvo and Ille (Z. Math. Logik Grundlag. Math. 38 (1992) 283–291) and to Lopez and Rauzy (Z. Math. Logik Grundlag. Math. 38 (1992) 27–37), we study the morphology of the tournaments defined on n?5 vertices and which contain exactly n?3 or 2n?8 diamonds. To cite this article: H. Bouchaala, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Lengths of cycles in halin graphs

A Halin graph is a plane graph H = T U C, where T is a plane tree with no vertex of degree two and at least one vertex of degree three or more, and C is a cycle connecting the endvertices of T in the cyclic order determined by the embedding of T. We prove that such a graph on n vertices contains cycles of all lengths l, 3 ≤ l n, except, possibly, for one even value m of l. We prove also that if the tree T contains no vertex of degree three then G is pancyclic.     

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.     

Arc-disjoint hamiltonian paths in non-round decomposable local tournaments

Thomassen proved that a strong tournament $T$ has a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices if and only if $T$ is not an almost transitive tournament of odd order, where an almost transitive tournament is obtained from a transitive tournament with acyclic ordering $u_1,u_2,\dots ,u_n$ (i.e., $u_i\to u_j$ for all $1\le i<j\le n$) by reversing the arc $u_1{u}_n$. A digraph $D$ is a local tournament if for every vertex $x$ of $D$, both the out-neighbors and the in-neighbors of $x$ induce tournaments. Bang-Jensen, Guo, Gutin and Volkmann split local tournaments into three subclasses: the round decomposable; the non-round decomposable which are not tournaments; the non-round decomposable which are tournaments. In 2015, we proved that every 2-strong round decomposable local tournament has a Hamiltonian path and a Hamiltonian cycle which are arc-disjoint if and only if it is not the second power of an even cycle. In this paper, we discuss the arc-disjoint Hamiltonian paths in non-round decomposable local tournaments, and prove that every 2-strong non-round decomposable local tournament contains a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices. This result combining with the one on round decomposable local tournaments extends the above-mentioned result of Thomassen to 2-strong local tournaments.     

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     

Monochromatic sinks in nearly transitive arc-colored tournaments

Let T be the set of all arc-colored tournaments, with any number of colors, that contain no rainbow 3-cycles, i.e., no 3-cycles whose three arcs are colored with three distinct colors. We prove that if T∈T and if each strong component of T is a single vertex or isomorphic to an upset tournament, then T contains a monochromatic sink. We also prove that if T∈T and T contains a vertex x such that T−x is transitive, then T contains a monochromatic sink. The latter result is best possible in the sense that, for each n≥5, there exists an n-tournament T such that (T−x)−y is transitive for some two distinct vertices x and y in T, and T can be arc-colored with five colors such that T∈T, but T contains no monochromatic sink.     

Vertex pancyclic in‐tournaments

An in‐tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. The topic of this paper is to investigate vertex k‐pancyclicity of in‐tournaments of order n, where for some 3 ≤ k ≤ n, every vertex belongs to a cycle of length p for every k ≤ p ≤ n. We give sharp lower bounds for the minimum degree such that a strong in‐tournament is vertex k‐pancyclic for k ≤ 5 and k ≥ n − 3. In the latter case, we even show that the in‐tournaments in consideration are fully (n − 3)‐extendable which means that every vertex belongs to a cycle of length n − 3 and that the vertex set of every cycle of length at least n − 3 is contained in a cycle of length one greater. In accordance with these results, we state the conjecture that every strong in‐tournament of order n with minimum degree greater than is vertex k‐pancyclic for 5 < k < n − 3, and we present a family of examples showing that this bound would be best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 84–104, 2001     

The partition of a strong tournament

A digraph T is strong if for every pair of vertices u and v there exists a directed path from u to v and a directed path from v to u. Denote the in-degree and out-degree of a vertex v of T by d^-(v) and d⁺(v), respectively. We define δ^-(T)=min_vV(T){d^-(v)} and δ⁺(T)=min_vV(T){d⁺(v)}. Let T₀ be a 7-tournament which contains no transitive 4-subtournament. In this paper, we obtain some conditions on a strong tournament which cannot be partitioned into two cycles. We show that a strong tournament T with n6 vertices such that TT₀ and max{δ⁺(T),δ^-(T)}3 can be partitioned into two cycles. Finally, we give a sufficient condition for a tournament to be partitioned into k cycles.     

On the number of Hamiltonian cycles in tournaments

The main results assert that the minimum number of Hamiltonian bypasses in a strong tournament of order n and the minimum number of Hamiltonian cycles in a 2-connected tournament of order n increase exponentially with n. Furthermore, the number of Hamiltonian cycles in a tournament increases at least exponentially with the minimum outdegree of the tournament. Finally, for each k?1 there are infinitely many tournaments with precisely k Hamiltonian cycles.     

The number of pancyclic arcs in a k‐strong tournament

A tournament is a digraph, where there is precisely one arc between every pair of distinct vertices. An arc is pancyclic in a digraph D, if it belongs to a cycle of length l, for all 3 ≤ l ≤ |V (D) |. Let p(D) denote the number of pancyclic arcs in a digraph D and let h(D) denote the maximum number of pancyclic arcs belonging to the same Hamilton cycle of D. Note that p(D) ≥ h(D). Moon showed that h(T) ≥ 3 for all strong non‐trivial tournaments, T, and Havet showed that h(T) ≥ 5 for all 2‐strong tournaments T. We will show that if T is a k‐strong tournament, with k ≥ 2, then p(T) ≥ 1/2, nk and h(T) ≥ (k + 5)/2. This solves a conjecture by Havet, stating that there exists a constant α_k, such that p(T) ≥ α_k n, for all k‐strong tournaments, T, with k ≥ 2. Furthermore, the second results gives support for the conjecture h(T) ≥ 2k + 1, which was also stated by Havet. The previously best‐known bounds when k ≥ 2 were p(T) ≥ 2k + 3 and h(T) ≥ 5. © 2005 Wiley Periodicals, Inc. J Graph Theory