Mika Olsen

A conjecture of Neumann-Lara on infinite families of r-dichromatic circulant tournaments

In this paper, we exhibit infinite families of vertex critical r-dichromatic circulant tournaments for all r>=3. The existence of these infinite families was conjectured by Neumann-Lara [V. Neumann-Lara, Note on vertex critical 4-dichromatic circulant tournaments, Discrete Math. 170 (1997) 289-291], who later proved it for all r>=3 and r 7. Using different methods, we provide new constructions of such infinite families for all r>=3, which covers the case r=7 and thus settles the conjecture.

Characterization of asymmetric CKI- and KP-digraphs with covering number at most 3

A set N ? V ( D ) is said to be a kernel if N is an independent set and for every vertex x ? ( V ( D ) ? N ) there is a vertex y ? N such that x y ? A ( D ) . Let D be a digraph such that every proper induced subdigraph of D has a kernel. D is said to be kernel perfect digraph (KP-digraph) if the digraph D has a kernel and critical kernel imperfect digraph (CKI-digraph) if the digraph D does not have a kernel. In this paper we characterize the asymmetric CKI-digraphs with covering number at most 3 . Moreover, we prove that the only asymmetric CKI-digraphs with covering number at most 3 are: C ? 3 , C ? 5 and C ? 7 ( 1 , 2 ) . Several interesting consequences are obtained.

Infinite families of tight regular tournaments

In this paper, we construct inflnite families of tight regular tournaments. In particular, we prove that two classes of regular tournaments, tame molds and ample tournaments are tight. We exhibit an inflnite family of 3-dichromatic tight tournaments. With this family we positively answer to one case of a conjecture posed by V. Neumann-Lara. Finally, we show that any tournament with a tight mold is also tight.

On the acyclic disconnection of multipartite tournaments

The acyclic disconnection of a digraph D is the maximum number of components that can be obtained by deleting from D the set of arcs of an acyclic subdigraph. We give bounds for the acyclic disconnection of strongly connected bipartite tournaments and of regular bipartite tournaments. For the latter case, we exhibit an infinite family of tournaments with acyclic disconnection equal to 4.

On the acyclic disconnection and the girth

The acyclic disconnection, ω ? ( D ) , of a digraph D is the maximum number of connected components of the underlying graph of D - A ( D ? ) , where D ? is an acyclic subdigraph of D . We prove that ω ? ( D ) ? g - 1 for every strongly connected digraph with girth g ? 4 , and we show that ω ? ( D ) = g - 1 if and only if D ? C g for g ? 5 . We also characterize the digraphs that satisfy ω ? ( D ) = g - 1 , for g = 4 in certain classes of digraphs. Finally, we define a family of bipartite tournaments based on projective planes and we prove that their acyclic disconnection is equal to 3 . Then, these bipartite tournaments are counterexamples of the conjecture ω ? ( T ) = 3 if and only if T ? C ? 4 posed for bipartite tournaments by Figueroa et?al. (2012).

On a Conjecture of Víctor Neumann-Lara

Abstract We disprove the following conjecture due to Victor Neumann-Lara: for every couple of integers ( r , s ) such that r ≥ s ≥ 2 there is an infinite set of circulant tournaments T such that the dichromatic number and the acyclic disconnection of T are equal to r and s respectively. We show that for every integer s ≥ 2 there exists a sharp lower bound b ( s ) for the dichromatic number r such that for every r b ( s ) there is no circulant tournament T satisfying the conjecture with these parameters. We give an upper bound B ( s ) for the dichromatic number r such that for every r ≥ B ( s ) there exists an infinite set of circulant tournaments for which the conjecture is valid.

Bounds on the k -restricted arc connectivity of some bipartite tournaments

Abstract 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 k-restricted 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.

CKI-Digraphs, Generalized Sums and Partitions of Digraphs

A kernel of a digraph is a set of vertices which is both independent and absorbent. Let

Strong subtournaments and cycles of multipartite tournaments

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