On d-panconnected tournaments with large semidegrees

Discrete Mathematics, 2016

In this paper we relate the global irregularity and the order of a c-partite tournament T to the existence of certain cycles and the problem of finding the maximum strongly connected subtournament of T. In particular, we give results related to the following problem of Volkmann: How close to regular must a c-partite tournament be, to secure a strongly connected subtournament of order c?

Comptes Rendus Mathematique, 2010

Given a tournament T=(V,A), a subset X of $V$ is an interval of T provided that for every a, b in X and x\in V-X, (a,x) in A if and only if (b,x) in A. For example, $\emptyset$, {x}(x in V) and V are intervals of T, called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A critical tournament is an indecomposable tournament T of cardinality $\geq 5$ such that for any vertex x of T, the tournament T-x is decomposable. The critical tournaments are of odd cardinality and for all $n \geq 2$ there are exactly three critical tournaments on 2n+1 vertices denoted by $T_{2n+1}$, $U_{2n+1}$ and $W_{2n+1}$. The tournaments $T_{5}$, $U_{5}$ and $W_{5}$ are the unique indecomposable tournaments on 5 vertices. We say that a tournament T embeds into a tournament T' when T is isomorphic to a subtournament of T'. A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and $T_{5}$ embed into an indecomposable tournament T, then $W_{5}$ and $U_{5}$ embed into T. To conclude, we prove the following: given an indecomposable tournament T, with $\mid\!V(T)\!\mid \geq 7$, T is critical if and only if the indecomposable subtournaments on 7 vertices of T are isomorphic to one and only one of the tournaments $T_{7}$, $U_{7}$ and $W_{7}$.

Applied Mathematics and Computation, 1981

Journal of Combinatorial Theory, Series B, 1976

A family B of simple (that is, cycle-free) paths is a path decomposition of a tournament T if and only if B partitions the arcs of T. The path number of T, denoted pn(T), is the minimum value of I+' 1 over all path decompositions 9 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 5 k S nS/4, k an integer. It is also shown that if n is odd and Tis a tournament on n vertices, then (n + 1)/2 S pn(T) $ (nP -1)/4. Moreover, if k is an integer satisfying (i) (n + 1)/2 5 k 5 n -1 or (ii) n < k S (ns -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 5 k < (n* -1)/4.

European Journal of Combinatorics

In this work we present a version of the so called Chen and Chvátal's conjecture for directed graphs. A line of a directed graph D is defined by an ordered pair (u, v), with u and v two distinct vertices of D, as the set of all vertices w such that u, v, w belong to a shortest directed path in D containing a shortest directed path from u to v. A line is empty if there is no directed path from u to v. Another option is that a line is the set of all vertices. The version of the Chen and Chvátal's conjecture we study states that if none of previous options hold, then the number of distinct lines in D is at least its number of vertices. Our main result is that any tournament satisfies this conjecture as well as any orientation of a complete bipartite graph of diameter three.

Applied Mathematics and Computation

For k ≥ 2, a strongly connected digraph D is called λ k-connected if it contains a set of arcs W such that D − W contains at least k non-trivial strong components. The krestricted arc connectivity of a digraph D was defined by Volkmann as λ k (D) = min {| W | : W is a k-restricted arc-cut }. In this paper we bound λ k (T) for a family of bipartite tournaments T called projective bipartite tournaments. We also introduce a family of "good" bipartite oriented digraphs. For a good bipartite tournament T we prove that if the minimum degree of T is at least 1. 5 k − 1 then k (k − 1) ≤ λ k (T) ≤ k (N − 2 k − 2) , where N is the order of the tournament. As a consequence, we derive better bounds for circulant bipartite tournaments.

Journal of Information and Optimization Sciences, 1984

If a tournament T is said to satisfy O(p, q) condition, then d+(uHd-(v);;>p-q for each arc VII in T, wha re p = I VeT) I , q is an integer. Let T be a tournament satis fying O{p,q) condition. Alspach [1] proved that T is arc-pancyclic when q= 1. Zhu and Tian [2] proved that T is arc-pancyclic when q<2. Tn this paper, we prove that Tis arc-parcyclic when p-;' 3q+3. We also find a useful result (Lemma VII) which deals with the longest cycle with a pair of arcs across it. 1. INTRODUCT.ION Since the regular tournament had been pron:d to be arc-pancyclic by Alspach [1], many further results about the arc-pancyclic property in tournaments have been found. In this paper, \ve shall show that a kind of tournaments has this property. The tournaments we'll consider is confined with a degree condition which was suggested in [2]. This degree condition is defined as follows : If tournament T of order p is said to satisfy O(p,q) conditon, where q is an integer, then every arc uv of T satifies that

Congressus numerantium, 2002

Given an acyclic digraph D, we seek a smallest sized tournament T that has D as a minimum feedback arc set. The reversing number of a digraph is defined to be r(D) = |V (T )|−|V (D)| . The case where D is a tournament Tn was studied by Isaak in 1995 using an integer linear programming formulation. In particular, this approach was used to produce lower bounds for r(Tn), and it was conjectured that the given bounds were tight. We examine the class of tournaments where n = 2 k + 2 k−2 and show the known lower bounds for r(T n ) are best possible.

Canadian mathematical bulletin, 1970

A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph H there exists (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n (H). This conjecture has a directed equivalent version stating that for every tournament H there exists (H) > 0 such that every H−free n−vertex tournament T contains a transitive subtournament of size at least n (H). Recently the conjecture was proved for all six-vertex tournaments, except K6. In this paper we construct two infinite families of tournaments for which the conjecture is still open for infinitely many tournaments in these two families − the family of so-called super nebulas and the family of so-called super triangular galaxies. We prove that for every super nebula H1 and every ∆galaxy H2 there exist (H1, H2) such that every {H1, H2}−free tournament T contains a transitive subtournament of size at least |T | (H 1 ,H 2). We also prove that for every central triangular galaxy H there exist (K6, H) such that every {K6, H}−free tournament T contains a transitive subtournament of size at least |T | (K 6 ,H). And we give an extension of our results.